First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals
The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet been found.
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2024 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2024
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212613 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals. Alexander Mang. SIGMA 20 (2024), 082, 40 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860289638326861824 |
|---|---|
| author | Mang, Alexander |
| author_facet | Mang, Alexander |
| citation_txt | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals. Alexander Mang. SIGMA 20 (2024), 082, 40 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet been found.
|
| first_indexed | 2026-03-19T15:15:33Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 082, 40 pages
First Cohomology with Trivial Coefficients
of All Unitary Easy Quantum Group Duals
Alexander MANG
Hamburg University, Bundesstraße 55, 20146 Hamburg, Germany
E-mail: alex@alexandermang.net
URL: https://alexandermang.net
Received September 24, 2023, in final form August 22, 2024; Published online September 12, 2024
https://doi.org/10.3842/SIGMA.2024.082
Abstract. The first quantum group cohomology with trivial coefficients of the discrete dual
of any unitary easy quantum group is computed. That includes those potential quantum
groups whose associated categories of two-colored partitions have not yet been found.
Key words: discrete quantum group; quantum group cohomology; trivial coefficients; easy
quantum group; category of partitions
2020 Mathematics Subject Classification: 20G42; 05A18
1 Introduction
1.1 Background and context
In [4], Banica and Speicher provided a way of constructing compact quantum groups (in the
sense of [41, 43, 44]) by solving infinite combinatorics puzzles: They introduced three operations
on the collection of all equivalence relations of disjoint unions of finite sets and showed that
each subset which is closed under these operations gives rise to a compact quantum group. An
uncountable number of such sets and of the resulting so-called “easy” quantum groups and, in
fact, all there can be, have since been found in [3, 4, 30, 31, 32, 39]. In [34], Tarrago and Weber
extended Banica and Speicher’s operations to the collection of all “two-colored” partitions,
thus providing even more quantum groups to find. The classification program they initiated to
determine all sets closed under the operations is still ongoing (see [15, 23, 25, 26, 27, 28, 29, 34]).
The construction has since been further extended to two-colored partitions with arbitrarily
many “colors” by Freslon in [13], to “three-dimensional” sets by Cébron and Weber in [8] and
to equivalence relations on graphs instead of sets by Mančinska and Roberson in [24].
An issue that all these constructions share is that it is difficult to tell which of the resulting
compact quantum groups are new and which are isomorphic to already known ones. In partic-
ular, each solution to the combinatorics puzzle does not only provide one quantum group but
an entire countably infinite series, one for each dimension of its fundamental representation.
And already Banica and Speicher themselves observed in [4, Proposition 2.4 (4)] that, at least in
some cases, the quantum groups of one solution are isomorphic to those of another, just shifted
by one dimension. That underlines the importance of studying quantum group invariants with
the potential of telling easy quantum groups apart. Of course, these are often very difficult to
compute like, e.g., the L2-cohomology of [21] of discrete quantum groups. But perhaps at least
the cohomology with trivial coefficients is a reasonable goal to strive for.
The present article computes the first order of the quantum group cohomology with trivial
coefficients of the discrete duals of all of Tarrago and Weber’s so-called unitary easy quantum
groups. That includes even the potential ones whose combinatorics puzzles have not been solved
yet. Said cohomology can be realized as the first Hochschild cohomology of the trivial bimodule
mailto:alex@alexandermang.net
https://alexandermang.net
https://doi.org/10.3842/SIGMA.2024.082
2 A. Mang
of an augmented algebra presented in terms of generators and relations. As with any augmented
algebra the space of 1-coboundaries is then trivial and the task thus boils down to solving the
generally infinite system of linear equations in the finitely many generators determining the
1-cocycles.
The results of the present article might be useful for the computation of the second order
begun by Bichon, Das, Franz, Gerhold, Kula and Skalski in [7, 9] as well as Wendel in [40].
The former six investigated the cohomology of certain easy quantum groups out of a different
motivation. In particular, they were interested in the Calabi–Yau property of [14], a generaliza-
tion of Poincaré duality, and the classification of Schürmann triples. Namely, a quantum group
whose second cohomology vanishes has the AC property, defined in [12], which is important
in the study of quantum Lévy processes because it guarantees the existence of an associated
Schürmann triple.
In [7, 9], Bichon, Das, Franz, Gerhold, Kula and Skalski had already laid out a potential
strategy for computing the second cohomology of any easy quantum group (later refined in [10]
to address universal unitary quantum groups). This strategy is based on two key insights
and goes as follows. They interpreted quantum group cohomology as Hochschild cohomology
and chose the Hochschild complex as their resolution. Thus, they were faced with having to
compute the quotient of the 2-cocycles by the 2-coboundaries. By a very clever use of the
universal property of the quantum groups in question, they managed to solve the linear system
of equations determining the space of 2-coboundaries. This use of the universal property is the
first key tool (see [7, Lemma 5.4] and [9, Lemma 4.1]).
Understanding the 2-cocycles then allowed them to define a “defect map”, an injective linear
map from 2-cohomology to a certain finite-dimensional vector space of matrices. Thus, at this
point they only needed to determine the image of this defect map in order to compute the
second cohomology. This is where their second key insight comes into play. Namely, although
being interested only in the second-order cohomology, they incidentally also computed the first.
That is because they wanted to make use of the multiplicative structure of the cohomology ring.
They showed that, at least for the specific quantum groups they investigated, each 2-cocycle was
cohomologous to a linear combination of cup products of 1-cocycles. Thus, rather than having
to probe the potentially infinite-dimensional vector space of all 2-cocycles as the domain of the
defect map they could confine themselves to determining the image of the restriction to cup
products, a finite-dimensional space.
In short, when trying to compute the second cohomology of any easy quantum group it might
be helpful, perhaps even necessary, to know the first cohomology. Hence, the main result of the
present article might also constitute an intermediate step in computing the higher cohomologies
of all easy quantum groups.
1.2 Main result
Let Mn(C) be the C-vector space of (n × n)-matrices with complex entries and I the identity
(n×n)-matrix. Moreover, call a matrix “small” if each of its rows and each of its columns sums
to 0.
Then, the below theorem extends the results of [7, 9] as well as [40].
Theorem. Let n ∈ N, let G be any unitary easy compact (n× n)-matrix quantum group, let u
be its fundamental representation and let C be the category of two-colored partitions associated
with G. Say that C has property
(1) if and only if each block of each two-colored partition of C has at most two points,
(2) if and only if each block of each two-colored partition of C has at least two points,
(3) if and only if each block of each two-colored partition of C with at least two points contains
as many white lower and black upper points as it does black lower and white upper points,
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 3
(4) if and only if each two-colored partition of C has as many white lower and black upper
points as it has black lower and white upper points.
An isomorphism of complex vector spaces from the first quantum group cohomology of Ĝ with
trivial coefficients to the subspace
{v ∈Mn(C) ∧ A(C, v)} ∼= C⊕β1(Ĝ)
of Mn(C) is defined by the rule which assigns to (the one-elemental cohomology class of) any
1-cocycle η the matrix (η(uj,i))(j,i)∈{1,...,n}×2, where β1
(
Ĝ
)
and for any v ∈Mn(C) the predicate
A(C, v) are as follows:
If C is . . . , then A(C, v) is “. . . ” and β1
(
Ĝ
)
is . . .
1 ∧ 2 ∧ 3 ⊤ n2
1 ∧ ¬2 ∧ 3 ∧ 4 ∃λ ∈ C : v − λI is small (n− 1)2 + 1
1 ∧ ¬2 ∧ 3 ∧ ¬4 v is small (n− 1)2
1 ∧ 2 ∧ ¬3 ∧ 4 ∃λ ∈ C : v − λI is skew-symmetric 1
2n(n− 1) + 1
1 ∧ 2 ∧ ¬3 ∧ ¬4 v is skew-symmetric 1
2n(n− 1)
1 ∧ ¬2 ∧ ¬3 ∧ 4 ∃λ ∈ C : v − λI is skew-symmetric and small 1
2(n− 1)(n− 2) + 1
1 ∧ ¬2 ∧ ¬3 ∧ ¬4 v is skew-symmetric and small 1
2(n− 1)(n− 2)
¬1 ∧ 2 ∧ 3 ∧ 4 v is diagonal n
¬1 ∧ ¬3 ∧ 4 ∃λ ∈ C : v − λI = 0 1
¬1 ∧ ¬3 ∧ ¬4 v = 0 0
And these are all the cases that can occur.
1.3 Structure of the article
Excluding the introduction, the article is divided into five sections.
□ Section 2 recalls the definitions of compact quantum groups and the quantum group co-
homology with trivial coefficients of their discrete duals.
□ Following that, Section 3 provides particular examples of compact quantum groups by
presenting the definitions of categories of two-colored partitions and unitary easy quantum
groups.
□ For the convenience of the reader, the definition of the first Hochschild cohomology and
important results about it are recalled in Section 4.
□ Section 5 defines the vector spaces of matrices appearing in the main result and computes
their dimensions.
□ The proof of the main theorem is contained in Section 6. Starting from a characterization
of the first cohomology of a universal algebra recalled in Section 4 the first quantum group
cohomology as defined in Section 2 is computed of the discrete duals of the quantum groups
defined in Section 3.
1.4 Notation
In the following, 0 /∈ N. Rather, N0 = N ∪· {0}. Let JkK := {i ∈ N ∧ i ≤ k} for any k ∈ N0,
in particular, J0K = ∅. The symbol × will denote the Cartesian product of sets, with the
convention S×0 := {∅} for any set S. Throughout, all algebras are meant to be associative
and unital. The symbols ▷ and ◁ are used to denote the left respectively right actions of any
algebra on any bimodule. Moreover, given any vector spaces V and W over any field the symbol
4 A. Mang
[V,W ] will stand for the vector space of linear maps from V to W . Furthermore, for any vector
space X and any (possibly infinite) set E the notation X×E will be used for the E-fold direct
product vector space of X (not to be confused with the direct sum X⊕E). For any field K and
any set E, the free K-algebra over E will be denoted by K⟨E⟩. For any R ⊆ K⟨E⟩, we will write
K⟨E | R⟩ for the universal K-algebra with generators E and relations R.
2 Quantum groups and their cohomology
The most general kind of “quantum group” in the sense considered here are the locally compact
quantum groups introduced by Kustermans and Vaes in [17, 18, 19, 20]. Two subcategories of
these are Woronowicz’s compact quantum groups defined in [41, 43, 44] and Van Daele’s discrete
quantum groups studied in [35, 36].
While of those two each is equivalent to the dual category of the other via Pontryagin duality,
it is customary to ascribe the cohomology discussed in the present article to the discrete quantum
group rather than its compact dual in order to preserve the analogy with the group case. At
the same time, the particular quantum groups treated in this article are usually considered to
be compact rather than discrete.
And it is in fact most convenient for the purpose of the present article to adopt the latter per-
spective and work with compact quantum groups. The fact that the quantum group cohomology
is actually that of discrete quantum groups will be glossed over by only giving the definition of
the composition of the cohomology functor with the Pontryagin transformation. However, the
custom will be respected when it comes to notation.
2.1 Compact quantum groups
Quantum groups can be defined both on an analytic, namely von-Neumann- or C∗-algebraic
level, and on a purely algebraic level. For the purposes of discussing quantum group cohomology,
it fully suffices to consider the latter definition, given in [11]. In that sense, an (algebraic) compact
quantum group G is the formal dual of a Hopf ∗-algebra C
[
Ĝ
]
which admits a faithful positive
integral. A big class of examples is provided in Section 3. For any G, in the present article
C
[
Ĝ
]
is generated as a ∗-algebra by the matrix coefficients of a single finite-dimensional unitary
comodule M . The coefficient matrix u of a choice of such an M is often called a fundamental
representation. The axioms imply in particular that, if u• is the matrix of the conjugate comodule
of M and if u◦ := u, then C
[
Ĝ
]
is generated as an algebra (as opposed to a ∗-algebra) by the
union of the entries of u◦ and u•. If A is the underlying algebra and ϵ the counit of the Hopf
∗-algebra C
[
Ĝ
]
, it is entirely sufficient to think of G as the augmented algebra (A, ϵ) and keep
in mind that for the examples in this article a generating set of A can be given consisting of the
entries of two matrices u◦ and u• of the same size.
2.2 Quantum group cohomology
One of many equivalent ways of introducing quantum group cohomology is via Hochschild co-
homology. For any compact quantum group G and any p ∈ N0, if A is the underlying algebra
and ϵ the counit of C[Ĝ] and if ϵCϵ denotes the A-bimodule given by the C-vector space C
equipped with the left and right A-actions defined by a⊗ λ 7→ ϵ(a)λ respectively λ⊗ a 7→ λϵ(a)
for any a ∈ A and λ ∈ C, then the p-th quantum group cohomology with trivial coefficients of
the discrete dual Ĝ of G is defined as
Hp
(
Ĝ
)
:= Hp
HS(A, ϵCϵ),
the p-th Hochschild cohomology of A with coefficients in ϵCϵ.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 5
3 Categories of two-colored partitions
and unitary easy quantum groups
The quantum groups whose quantum group cohomology is investigated in the present article
are the discrete duals of so-called easy quantum groups. They can be defined via Tannaka–
Krein duality (see [42]) using the combinatorics of so-called two-colored partitions (which will
be explained in Definition 3.6). Throughout the article, it will be important to distinguish the
notions of a two-colored partition and a set-theoretical partition in the following sense.
Notation 3.1. Let X be any set.
(a) A set-theoretical partition of X is the quotient set of any equivalence relation on X or,
equivalently, any set of non-empty pairwise disjoint subsets of X whose union is X.
(b) Given any two set-theoretical partitions p and q of X, write p ≤ q if p is finer than q, i.e.,
if for any B ∈ p there exists C ∈ q with B ⊆ C.
(c) For any two set-theoretical partitions p and q of X, let ζ(p, q) := 1 if p ≤ q and let
ζ(p, q) := 0 otherwise.
(d) Furthermore, for any set-theoretical partitions p1 and p2 of X the join of p1 and p2 is
the unique set-theoretical partition s of X which satisfies p1 ≤ s and p2 ≤ s and which is
minimal with that property with respect to the partial order ≤.
(e) For any mapping f : X → Y from X to any set Y and for any subset B ⊆ Y , let f←(B) :=
{x ∈ X ∧ f(x) ∈ B} denote the pre-image of B under f . Moreover, let ran(f) := {f(x) |
x ∈ X} and ker(f) := {f←({y}) | y ∈ ran(f)} be the image and kernel of f , respectively.
(f) For any set-theoretical partition p of X, write πp for the associated projection, the mapping
X → p which maps any x ∈ X to the unique B ∈ p with x ∈ B. And for any second set Y
and any mapping f : X → Y with p ≤ ker(f), let f/p denote the quotient mapping, the
unique mapping p→ Y with (f/p) ◦ πp = f .
Example 3.2. If X = {1, 2, 3, 4, 5, 6}, then p = {{1}, {2, 4}, {3, 5, 6}} is a set-theoretical par-
tition of X and the projection πp is the mapping X → p, which sends 1 to {1}, sends both 2
and 4 to {2, 4} and sends each of 3, 5 and 6 to {3, 5, 6}.
Moreover, if Y = {a, b, c, d} and |Y | = 4 and if f : X → Y maps each of 1, 2 and 4 to a
and each of 3, 5 and 6 to c, then the kernel of f is ker(f) = {{1, 2, 4}, {3, 5, 6}}. Since then
p ≤ ker(f) the quotient mapping f/p exists and maps both {1} and {2, 4} to a and {3, 5, 6}
to c.
In contrast, if f(4) was not given by a but by b instead, then ker(f) would equal {{1, 2}, {4},
{3, 5, 6}}, in which case p would not be finer than ker(f) since there would be no C ∈ ker(f)
with {2, 4} ⊆ C. There would be no f/p with (f/p) ◦ πp = f .
3.1 Two-colored partitions and their categories
Two-colored partitions can be defined as follows. For further details see [34], where they were
first introduced, generalizing the (uncolored) “partitions” considered in [4].
Assumptions 3.3.
(a) Let ■(·) and
■
(·) be any two injections with common domain N and with disjoint ranges.
(b) Let ◦ and • be arbitrary with ◦ ≠ •.
Definition 3.4.
(a) For any {k, ℓ} ⊆ N0, we call Πkℓ := {■a,
■
b | a ∈ JkK ∧ b ∈ JℓK} the set of k upper and ℓ
lower points.
(b) Given any {k, ℓ} ⊆ N0, any set X and any mappings g : JkK → X and j : JℓK → X denote
by g ■■ j the mapping Πkℓ → X with ■a 7→ g(a) for any a ∈ JkK and
■
b 7→ j(b) for any
b ∈ JℓK.
6 A. Mang
(c) ◦ and • are called the two colors and are said to be dual to each other, in symbols, ◦ := •
and • := ◦. They moreover have the color values σ(◦) := 1 and σ(•) := −1.
(d) For any {k, ℓ} ⊆ N0, any c : JkK → {◦, •} and any d : JℓK → {◦, •} the color sum of (c, d)
is the Z-valued measure σcd on Πkℓ with density −σ(ca) on ■a for any a ∈ JkK and density
σ(db) on ■
b for any b ∈ JℓK. Moreover, Σc
d := σcd(Π
k
ℓ ) is called the total color sum of (c, d).
(e) For brevity, let |c| := k for any k ∈ N0 and any c : JkK→ {◦, •}.
Example 3.5. Consider c : J3K → {◦, •} and d : J4K → {◦, •} with c2 = c3 = ◦ and c1 = • and
d1 = d2 = d4 = ◦ and d3 = •.
c1 c2 c3
d1 d2 d3 d4
1 −1 −1
1 1 −1 1
Sσcd
The color sum σcd has density 1 at each of ■1,
■
1,
■
2 and
■
4 and density −1 at each of ■2,
■3 and
■
3. Consequently, the subset S = {■3,
■
3,
■
4} of Π3
4 has color sum σcd(S) = σcd({■3}) +
σcd({■3}) + σcd({■4}) = −1− 1 + 1 = −1. The total color sum is Σc
d = 1.
Definition 3.6.
(a) A two-colored partition is any triple (c, d, p) for which there exist {k, ℓ} ⊆ N0 such that c
and d are mappings from JkK respectively JℓK to {◦, •}, the upper and lower colorings, and
such that p, the collection of blocks, is a set-theoretical partition of the set Πkℓ of points.
upper colors ck = 4 upper points
lower colors dℓ = 3 lower points
p = {{
■
1}, {■2,
■
2},
{■1, ■3, ■4,
■
3}}
blocks {■1, ■3, ■4, ■3}
and {■2, ■2} crossing
block {■1, ■3, ■4, ■3}
block {■1}
(b) Any set C of two-colored partitions meeting the following conditions is called a category of
two-colored partitions:
(i) C contains , , , , and .
(ii) C is closed under forming adjoints, that is, horizontal reflection. More precisely,
(d, c, p∗) ∈ C for any (c, d, p) ∈ C, where, if {k, ℓ} ⊆ N0 are such that p is a set-theore-
tical partition of Πkℓ , then p
∗ := {{■b | b ∈ JℓK ∧
■
b ∈ B}∪· {
■
a | a ∈ JkK ∧ ■a ∈ B}}B∈p
is the adjoint of p.
∗
=
(iii) C is closed under tensor products, i.e., horizontal concatenation. Formally, (c1 ⊗ c2,
d1 ⊗ d2, p1 ⊗ p2) ∈ C for any (c1, d1, p1) ∈ C and (c2, d2, p2) ∈ C, where, if kt and ℓt
are such that pt is a set-theoretical partition of Πktℓt for each t ∈ J2K, then c1 ⊗ c2 ∈
{◦, •}×(k1+k2) is defined by a 7→ c1(a) if a ≤ k1 and a 7→ c2(a − k1) if k1 < a and,
analogously, d1⊗d2 ∈ {◦, •}×(ℓ1+ℓ2) is defined by b 7→ d1(b) if b ≤ ℓ1 and b 7→ d2(b−ℓ1)
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 7
if ℓ1 < b, and where p1⊗ p2 := p1 ∪· {{■(k1 + a) | a ∈ Jk2K ∧ ■a ∈ B}∪·
{
■
(ℓ1 + b) | b ∈
Jℓ2K ∧ ■
b ∈ B}}B∈p2 is the tensor product of (p1, p2).
⊗ =
(iv) C is closed under composition, i.e., vertical concatenation in the following sense. If for
two set-theoretical partitions the lower coloring of the first agrees with the upper
coloring of the second, then the composition has the same upper coloring as the first
and the same lower coloring as the second. Any blocks of the first which only include
upper points are inherited by the composition, as are any blocks of the second which
only include lower points. The remaining blocks of the composition are formed by
the following procedure. The collection of all non-empty intersections of blocks of the
first two-colored partition with the set of lower points is a set-theoretical partition
of the latter. Likewise, a set-theoretical partition of the set of upper points of the
second two-colored partition is given by the collection of all non-empty intersections
of blocks of the second two-colored partition with it. If the lower points of the first
two-colored partition and the upper points of the second are identified according to
the numbering, the two set-theoretical partitions just described have a join. For
each element of the join, consider the union of the following two sets. The first is
the (possibly empty) set of upper points of the first two-colored partition which are
contained in a block of the first two-colored partition which intersects the element
of the join if the latter is interpreted as a set of lower points of the first two-colored
partition. Similarly, the second is the (possibly empty) set of lower points of the
second two-colored partition which are contained in a block of the second two-colored
partition which intersects the element of the join if the latter is interpreted as a set
of upper points of the second two-colored partition. Provided that the union of these
two sets is not empty, it constitutes a block of the composition of the two-colored
partitions. And all blocks of the composition arise in one of the three aforementioned
ways. In formulas: (c, e, qp) ∈ C for any (c, d, p) ∈ C and (d, e, q) ∈ C, where if
{k, ℓ,m} ⊆ N0 are such that p is a set-theoretical partition of Πkℓ and q one of Πℓm,
and if s is the join of the two set-theoretical partitions {{j ∈ JℓK ∧
■
j ∈ A}}A∈p\{∅}
and {{i ∈ JℓK ∧ ■i ∈ C}}C∈q\{∅} of JℓK, then qp := {A ∈ p ∧ A ⊆ Πk0}∪· {C ∈ q ∧ C ⊆
Π0
m} ∪· {
⋃
· {A ∩ Πk0 | A ∈ p ∧ ∃j ∈ B :
■
j ∈ A} ∪·
⋃
· {C ∩ Π0
m | C ∈ q ∧ ∃i ∈ B : ■i ∈
C}}B∈s\{∅} is the composition of (q, p).
◦ = ⇝
(c) For any set G of two-colored partition, we write ⟨G⟩ for the intersection of all categories of
two-colored partitions containing G and we say that G generates ⟨G⟩.
Example 3.7.
(a) Of course, the set of all two-colored partitions forms the maximal category of two-colored
partitions. It follows from [34, Theorem 8.3] that it coincides with ⟨ , , , ⟩.
8 A. Mang
(b) Another category of two-colored partitions is given by ⟨ ⟩, the category of two-colored
pair partitions with neutral blocks, i.e., all (c, d, p) with |B| = 2 and σcd(B) = 0 for any
B ∈ p, (see [26, Proposition 5.3]).
(c) By [34, Theorem 7.2], the minimal category of two-colored partitions ⟨∅⟩ is the subset of
all elements of ⟨ ⟩ which are non-crossing. The precise definition of being non-crossing is
unimportant here; informally, it means that blocks can be “drawn without intersections”.
Not much familiarity with two-colored partitions and their categories is required in order
to prove the main result. In particular, the full classification of all categories of two-colored
partitions can remain open. However, we will need to divide the landscape of all possible
categories as follows.
Definition 3.8. We say that any category C of two-colored partitions is
(a) case O if /∈ C and /∈ C,
(b) case B if ∈ C and /∈ C,
(c) case H if /∈ C and ∈ C,
(d) case S if ∈ C and ∈ C,
(e) class NNSB if σcd(B) = 0 for any (c, d, p) ∈ C and any B ∈ p with 2 ≤ |B|,
(f) class NP if Σc
d = 0 for any (c, d, p) ∈ C.
The names NNSB and NP reflect the defining conditions of having only neutral non-singleton
blocks respectively only neutral two-colored partitions, where “neutral” means vanishing color
sum. For the motivation behind the names O, B, H and S see Remark 3.22 below.
Remark 3.9. For any of the known categories of two-colored partitions, it is easy to determine
whether it has a given property in Definition 3.8 or not. Any known category which is not case H
is covered by [28] (see Section 7 there for the correspondence to the results of [15, 26, 27, 34])
and any known case-H category by [15, 23, 34] or [25, Chapter 1].
Cases O, B, S. Any category Rf,v,s,l,k,x in the main theorem of [28] is case O if and only if
f = {2}, case B if and only if f = {1, 2} and case S if and only if f = N (and never case H). It
is class NNSB if and only if v = {0} or v = ±{0, 1} and class NP if and only if s = {0}.
Case H. In [34], neither of the categories Hglob(k) of Theorem 7.1 and Hgrp,glob(k) of The-
orem 8.3 is class NNSB. And, each is class NP if and only if k = 0. The category H′loc
in Theorem 7.2 is both class NNSB and class NP. Each of Hloc(k, d) from Theorem 7.2 and
Hgrp,loc(k, d) from Theorem 8.3 is class NNSB if and only if k = d = 0 and is class NP if and
only if k = 0.
The case-H categories in [15, Table 1] which are not already covered by [34] are Hhl,glob(k, 0),
Hhl,glob(k, s),Hπ(k, s),Hπ(k,∞) andHA(k) (where the categoriesHhl,glob(k, 0) andHhl,glob(k, s)
can each also be written as HA(k) for certain A). None one of these are class NNSB. And any
one is class NP if and only if k = 0.
In [23], no group-theoretical category R of two-colored partitions in the sense of Defini-
tion 4.1.5 is class NNSB. And, any such category is class NP if and only if F∞(R) in the
sense of Definition 4.3.21 contains no word with different numbers of generators and inverses of
generators.
Lastly, the categoryWR in the sense of the main result of [25, Chapter 1] is both class NNSB
and class NP for any parameter R.
Beyond those case distinctions, we will also need to know the following elementary facts about
categories of two-colored partitions.
Definition 3.10. Dual two-colored partitions are obtained by simultaneous horizontal reflection,
vertical reflection and color inversion. More precisely, given any {k, ℓ} ⊆ N0, any c ∈ {◦, •}×k,
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 9
any d ∈ {◦, •}×ℓ and any set-theoretical partition p of Πkℓ the dual of (c, d, p) is the triple (d, c, p),
where d ∈ {◦, •}×ℓ is defined by j 7→ dℓ−j+1, where c ∈ {◦, •}×k is defined by i 7→ ck−i+1, and
where p := {{■(ℓ− j + 1) | j ∈ JℓK ∧
■
j ∈ B} ∪· {
■
(k − i+ 1) | i ∈ JkK ∧ ■i ∈ B}}B∈p is the dual
of p.
=
Lemma 3.11. Let C be any category of two-colored partitions.
(a) (d, c, p) ∈ C for any (c, d, p) ∈ C.
(b) ∈ C if and only if there exist (c, d, p) ∈ C and B ∈ p such that |B| < 2.
(c) ∈ C if and only if there exist (c, d, p) ∈ C and B ∈ p such that |B| > 2.
(d) If ∈ C and ∈ C, then ∈ C.
(e) If ∈ C, then { ⊗|Σc
d|, ⊗|Σ
c
d|} ⊆ C for any (c, d, p) ∈ C with Σc
d ̸= 0.
Proof. Part (a) is implied by [34, Lemma 1.1 (a)]. Parts (b) and (c) are [34, Lemmas 1.3 (b)
and 2.1 (a)] and [34, Lemmas 1.3 (d) and 2.1 (b)], respectively. Part (d) follows immediately
from [34, Lemma 1.3 (b)].
In order to see part (e), use [34, Lemma 1.1 (a)] to first “rotate” any potential upper point
of (c, d, p) down (in an arbitrary direction). “Disconnect” then each and every point from
its block with the help of [34, Lemma 1.3 (b)]. Following that, keep “erasing” neighboring
points of different colors, as [34, Lemma 1.1 (b)] permits, until no such points remain. None
of these transformations have affected the total color sum. The resulting two-colored partition
is either ⊗|Σc
d| or ⊗|Σc
d|. Passing to the adjoint of the dual as allowed by (a) hence shows the
claim. ■
Lemma 3.12.
(a) Any case-O or case-H category of two-colored partitions that is class NNSB is class NP.
(b) No case-S category of two-colored partitions is class NNSB.
Proof. (a) Let C be case-O or case-H and class NNSB. Then, for any (c, d, p) ∈ C and any
B ∈ p on the one hand 2 ≤ |B| by the first assumption and thus on the other hand σcd(B) = 0 by
the second assumption. Since that demands Σc
d =
∑
B∈p σ
c
d(B) = 0 the category C is necessarily
class NP.
(b) Since any case-S category contains both and , it must also contain by Lem-
ma 3.11 (d). The fact that {
■
1,
■
3} ∈ and σ∅◦•◦•({■1, ■3}) = 2 ̸= 0 hence shows that such
a category is not class NNSB. ■
3.2 Unitary easy quantum groups
“Easy” quantum groups are now defined by transforming the elements of a given category of
two-colored partitions into relations for the generators of a universal algebra that can be given
the structure of a compact quantum group. To be more precise, an entire series of compact
quantum groups indexed by N arises in this way.
Assumptions 3.13. In the following, fix any n∈N and any 2n2-elemental set E={u◦j,i, u•j,i}ni,j=1
and define the two families u◦ := (u◦j,i)(j,i)∈JnK×2 and u• := (u•j,i)(j,i)∈JnK×2 .
The transformation of two-colored partitions into relations is accomplished by the following
formula, where ζ was defined in Notation 3.1 (c).
10 A. Mang
Notation 3.14. For any {k, ℓ} ⊆ N0, any c ∈ {◦, •}×k and d ∈ {◦, •}×ℓ, any set-theoretical
partition p of Πkℓ and any g ∈ JnK×k and j ∈ JnK×ℓ, let in C⟨E⟩
rcd(p)j,g :=
∑
i∈JnK×ℓ
ζ(p, ker(g ■■ i))
ℓ−→∏
b=1
udbjb,ib −
∑
h∈JnK×k
ζ(p, ker(h ■
■ j))
k−→∏
a=1
ucaha,ga .
For example, the two-colored partitions and induce the trivial relation 0. The relations
induced by , , and will be of the utmost importance.
Lemma 3.15. For any g ∈ JnK×2 and j ∈ JnK×2, the following hold:
r∅◦•( )j,∅ =
n∑
i=1
u◦j1,iu
•
j2,i − δj1,j21, r•◦∅ ( )∅,g = δg1,g21−
n∑
h=1
u•h,g1u
◦
h,g2 ,
r∅•◦( )j,∅ =
n∑
i=1
u•j1,iu
◦
j2,i − δj1,j21, r◦•∅ ( )∅,g = δg1,g21−
n∑
h=1
u◦h,g1u
•
h,g2 .
Proof. Only the proof for r∅◦•( )j,∅ is given. With the names of Notation 3.14 then k = 0 and
ℓ = 2 and c = ∅ and d = ◦• and p = {{
■
1,
■
2}} and g = ∅. On the one hand, for any i ∈ JnK×2
the set-theoretical partition ker(g ■■ i) can only take two values, namely {{
■
1,
■
2}} if i1 = i2
and {{
■
1}, {
■
2}} if i1 ̸= i2. Whereas ker(g ■■ i) even agrees with p in the former case, p is not
finer than ker(g ■■ i) in the latter case. Hence, only if i1 = i2 does ζ(p, ker(g ■■ i)) evaluate to 1.
Consequently, the first of the two sums in the definition of rcd(p)j,g effectively runs only over the
pairs (i, i) for i ∈ JnK. That explains the term
∑n
i=1 u
◦
j1,i
u•j2,i in the claim. On the other hand,
k = 0 by convention implies JnK×k = {∅}. Thus, ∅ is the only h over which the second sum in
the definition of rcd(p)j,g runs. Just like before, ker(h
■
■ j) is then either {{
■
1,
■
2}} or {{
■
1}, {
■
2}},
depending on whether if j1 = j2 or not. In other words, ζ(p, ker(h ■
■ j)) = δj1,j2 . By k = 0 the
set of indices a over which the product
−→∏
k
a=1u
ca
ha,ga
runs is the empty set ∅ (whereas the set of
indices h before was not ∅ but {∅}). By common convention, a product with empty index set
is 1. That is how the second term −δj1,j21 in the claim comes about. ■
In general, the relations can become quite complicated.
Example 3.16. If k = 4 and ℓ = 5 and c = ◦••◦ and d = •◦••• and p = {{
■
1}, {■2,
■
2}, {
■
4,
■
5},
{■1, ■3, ■4,
■
3}},
(c, d, p)
then for any g ∈ JnK×k and j ∈ JnK×ℓ,
rcd(p)j,g = δg1,g3δg1,g4
(
n∑
i=1
u•j1,i
)
u◦j2,g2u
•
j3,g1
(
n∑
i=1
u•j4,iu
•
j5,i
)
− δj4,j5 u◦j3,g1u
•
j2,g2u
•
j3,g3u
◦
j3,g4 .
The following definition of “easy” quantum groups is the algebraic version of [33, Defini-
tion 5.1]. Recall that n ∈ N is fixed per Assumptions 3.13.
Notation 3.17. For any set P of two-colored partitions, let
RP :=
{
rcd(p)j,g| (c, d, p) ∈ C ∧ g ∈ JnK×|c| ∧ j ∈ JnK×|d|
}
and let JP be the two-sided ideal of C⟨E⟩ generated by RP .
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 11
Definition 3.18. For any category C of two-colored partitions, the unitary easy compact quan-
tum group of (C, n) is given by (C⟨E | RC⟩, ∗,∆), where ∗ and ∆ are respectively the unique
anti-multiplicative anti-linear self-map of C⟨E | RC⟩ and the unique multiplicative linear map
from C⟨E | RC⟩ to the tensor product algebra of C⟨E | RC⟩ with itself which satisfy respectively
(ucj,i + JC)
∗ = ucj,i + JC and ∆(ucj,i + JC) =
n∑
s=1
(ucj,s + JC)⊗ (ucs,i + JC)
for any {i, j} ⊆ JnK and c ∈ {◦, •}.
Remark 3.19. The definition of unitary easy quantum groups is usually given in terms of
universal ∗-algebras, not universal algebras, cf. [33, Definition 5.1]. The variant given above is
equivalent, as explained hereafter. For any m ∈ N0 and any e ∈ JnK×m let e ∈ JnK×m be defined
by i 7→ em−i+1 for any i ∈ JmK. Let {k, ℓ} ⊆ N0, let c ∈ {◦, •}×k, let d ∈ {◦, •}×ℓ, let (c, d, p) ∈ C,
let g ∈ JnK×k and let j ∈ JnK×ℓ. Then, with respect to the ∗-map in Definition 3.18,
(rcd(p)j,g)
∗ =
∑
i∈JnK×ℓ
ζ(p, ker(g ■■ i))
ℓ←−∏
b=1
(udbjb,ib)
∗ −
∑
h∈JnK×k
ζ(p, ker(h ■
■ j))
k←−∏
a=1
(ucaha,ga)
∗
=
∑
i∈JnK×ℓ
ζ((p)∗, ker(g ■■ i))
ℓ−→∏
b=1
udb
jb,ib
−
∑
h∈JnK×k
ζ((p)∗, ker(h ■
■ j))
k−→∏
a=1
uca
ha,ga
= rc
d
((p)∗)j,g.
Since C also contains the two-colored partition (c, d, (p)∗) by Lemma 3.11 (a), the switch from
the universal ∗-algebra to the universal algebra makes no difference.
For the idea of the proof of the following, see [33, Remark 5.2].
Proposition 3.20. For any category C of two-colored partitions, the unitary easy compact quan-
tum group of (C, n) is a compact quantum group whose co-unit is given by the unique multiplica-
tive linear functional ϵ with
ϵ(ucj,i + JC) = δj,i
for any {i, j} ⊆ JnK and c ∈ {◦, •}. It can be seen as a compact (n× n)-matrix quantum group
with fundamental representation induced by u◦.
Example 3.21. Let C be a category of two-colored partitions and let G be the unitary quantum
group of (C, n).
(a) If C is the minimal category ⟨∅⟩, then G is the free unitary quantum group U+
n introduced
by Wang in [38]. Its algebra can be presented as the universal algebra generated by E
(fixed in Assumptions 3.13) subject to only the relations of Lemma 3.15.
(b) For C = ⟨ ⟩, we recover the classical unitary group Un, the universal commutative(!)
algebra subject to the relations of Lemma 3.15.
(c) Should C be the maximal category ⟨ , , , ⟩ of all two-colored partitions, then G is
the symmetric group Sn.
Currently, there is no complete list of all unitary easy quantum groups because the classifi-
cation of all categories of two-colored partitions is not yet finished.
12 A. Mang
Remark 3.22. The names O, B, H and S of the four cases from Definition 3.8 were introduced
by Tarrago and Weber in [34, Definition 2.2] and refer respectively to the orthogonal group On,
bistochastic group Bn, hyperoctahedral group Hn and symmetric group Sn. (A “bistochastic”
matrix is understood to be an orthogonal matrix each of whose rows and columns sums to 1.)
Tarrago and Weber showed that for each X ∈ {O,B,H,S} there exists a category of two-colored
partitions which is case X and maximally so. And, in the sense of Definition 3.18, the maximal
case-O category is the one associated with On, the maximal case-B category the one associated
with Bn, the maximal case-H category the one associated with Hn and the maximal case-S
category the one associated with Sn.
4 First Hochschild cohomology of universal algebras
For the convenience of the reader, Section 4 recalls the definition of and some elementary results
about the first Hochschild cohomology. Throughout, let K be any field.
4.1 First Hochschild cohomology
In Section 4.1, our algebra shall remain abstract. Section 4.2 will then recall which conclusions
can be drawn if a presentation of the algebra in terms of generators and relations is given.
Assumptions 4.1. Let A be any K-algebra and X any A-bimodule.
That means in particular that X is a K-vector space implicitly equipped with K-linear maps
▷ : A⊗X → X and ◁ : X⊗A→ X, the left and right actions of A, such that a1▷(a2▷x) = (a1a2)▷x
and (x ◁ a2) ◁ a1 = x ◁ (a2a1) and (a1 ▷ x) ◁ a2 = a1 ▷ (x ◁ a2) for any x ∈ X and {a1, a2} ⊆ A.
Example 4.2. For any augmentation ϵ of A, i.e., any K-algebra morphism from A to K, the K-
vector space K becomes an A-bimodule X if equipped with the actions defined by a ▷ λ = ϵ(a)λ
respectively λ ◁ a = λϵ(a) for any λ ∈ K and a ∈ A. It is often called the trivial bimodule
of (A, ϵ).
4.1.1 The fundamental definitions
The following definitions were first given by Hochschild in [16].
Definition 4.3.
(a) The X-valued Hochschild 1-cocycles of A are the K-vector subspace Z1
HS(A,X) of [A,X]
formed by all elements η such that for any {a1, a2} ⊆ A,(
∂1η
)
(a1 ⊗ a2) := a1 ▷ η(a2)− η(a1a2) + η(a1) ◁ a2 = 0.
(b) The X-valued Hochschild 1-coboundaries of A are the K-vector subspace B1
HS(A,X) of
[A,X] formed by all elements η such that there exists x ∈ X with for any a ∈ A,
η(a) =
(
∂0x
)
(a) := a ▷ x− x ◁ a.
It can be seen that η(1) = 0 for any η ∈ Z1
HS(A,X) and that B1
HS(A,X) is a K-vector
subspace of Z1
HS(A,X).
Definition 4.4. We call the quotient K-vector space H1
HS(A,X) of Z1
HS(A,X) with respect
to B1
HS(A,X) the first Hochschild cohomology of A with X-coefficients.
Example 4.5. In the case of Example 4.2, i.e., for trivial coefficients, the only 1-coboundary
of A is the zero map because (∂0λ)(a) = ϵ(a)λ − λϵ(a) = 0 for any λ ∈ K and a ∈ A. Hence,
Z1
HS(A,X) ∼= H1
HS(A,X) in that instance.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 13
4.1.2 Algebra hom characterization of 1-coycles
1-cocycles can be characterized as certain algebra homomorphisms by means of a folk theorem
recorded as [22, Lemma 1.9]. The latter uses the following construction.
Definition 4.6. Let A|1
0X denote the K-vector space A⊕X equipped with the K-linear map
K → A ⊕ X with 1 7→ (1, 0) and the K-linear map (A ⊕ X)⊗2 → A ⊕ X defined by for any
{a1, a2} ⊆ A and {x1, x2} ⊆ X,
(a1, x1)⊗ (a2, x2) 7→ (a1a2, a1 ▷ x2 + x1 ◁ a2).
Lemma 4.7. A|1
0 X is a K-algebra.
Proof. (1, 0) is a unit because for any a ∈ A and any x ∈ X,
(a, x) · (1, 0) = (a1, a ▷ 0 + x ◁ 1) = (a, x) = (1a, 1 ▷ x+ 0 ◁ a) = (1, 0) · (a, x).
Moreover, for any {a1, a2, a3} ⊆ A and any {x1, x2, x3} ⊆ X,
((a1, x1) · (a2, x2)) · (a3, x3) = (a1a2, a1 ▷ x2 + x1 ◁ a2) · (a3, x3)
= (a1a2a3, (a1a2) ▷ x3 + (a1 ▷ x2 + x1 ◁ a2) ◁ a3)
= (a1a2a3, a1a2 ▷ x3 + a1 ▷ x2 ◁ a3 + x1 ◁ a2a3)
= (a1a2a3, a1 ▷ (a2 ▷ x3 + x2 ◁ a3) + x1 ◁ (a2a3))
= (a1, x1) · (a2a3, a2 ▷ x3 + x2 ◁ a3)
= (a1, x1) · ((a2, x2) · (a3, x3)),
which shows that the multiplication is associative. ■
The following is then the folk theorem mentioned in [22, Lemma 1.9].
Lemma 4.8. For any ψ ∈ [A,X], the map A→ A⊕X with a 7→ (a, ψ(a)) for any a ∈ A defines
a K-algebra homomorphism A→ A|1
0 X if and only if ψ ∈ Z1
HS(A,X).
Proof. If the map in the claim is denoted by fψ, then fψ(1) = (1, ψ(1)) and for any {a1, a2} ⊆ A,
obviously, fψ(a1a2) = (a1a2, ψ(a1a2)) and
fψ(a1) · fψ(a2) = (a1, ψ(a1)) · (a2, ψ(a2)) = (a1a2, a1 ▷ ψ(a2) + ψ(a1) ◁ a2).
The two values coincide if and only if ψ(a1a2) = a1 ▷ ψ(a2) + ψ(a1) ◁ a2, which is to say if and
only if ψ ∈ Z1
HS(A,X). As then ψ(1) = 0 the claim is true. ■
The next result will be required later in the proof of, ultimately, Proposition 4.19.
Lemma 4.9. For any m ∈ N, any {ai}mi=1 ⊆ A and any {xi}mi=1 ⊆ X, in A|1
0 X,
−→∏
m
i=1(ai, xi) =
(−→∏
m
i=1ai,
m∑
i=1
(−→∏
i−1
j=1aj
)
▷ xi ◁
(−→∏
m
j=i+1aj
))
.
Proof. The cases m ∈ {1, 2, 3} are, respectively, trivial, the definition of the multiplication of
A|1
0 X and an intermediate result in the proof of Lemma 4.7. Generally,(−→∏
m−1
i=1 ai,
m−1∑
i=1
(−→∏
i−1
j=1aj
)
▷ xi ◁
(−→∏
m−1
j=i+1aj
))
· (am, xm)
=
((−→∏
m−1
i=1 ai
)
am,
(−→∏
m−1
i=1 ai
)
▷ xm +
m−1∑
i=1
(−→∏
i−1
j=1aj
)
▷ xi ◁
(−→∏
m−1
j=i+1aj
)
◁ am
)
.
Hence, the claim is true. ■
14 A. Mang
4.2 Conclusions for universal algebras
Using Lemma 4.8, it is possible to give a canonical equational characterization of the 1-cocycles
if a presentation of the algebra in terms of generators and relations is given.
Assumptions 4.10. Let E be any set, R ⊆ K⟨E⟩ arbitrary, J the two-sided ideal of K⟨E⟩
generated by R, and X any K⟨E | R⟩-bimodule.
Definition 4.11. Let
F 1
E,R,X : K⟨E⟩ → [X×E , X], p 7→ F 1,p
E,R,X
be the unique K-linear map with for any m ∈ N and any {ei}mi=1 ⊆ E, if p =
−→∏
m
i=1ei, then for
any x ∈ X×E ,
F 1,p
E,R,X(x) =
m∑
i=1
(−→∏
i−1
j=1ej + J
)
▷ xei ◁
(−→∏
m
j=i+1ej + J
)
,
and with F 1,1
E,R,X = 0.
Example 4.12. For any augmentation ϵ of A = K⟨E | R⟩, if X is the trivial bimodule of (A, ϵ)
in the sense of Example 4.2, then for any m ∈ N and any {ei}mi=1 ⊆ E, if p = e1 · · · em, then
F 1,p
E,R,X is a linear map which assigns to any family x = (xe)e∈E of elements of K the number
F 1,p
E,R,X(x) =
m∑
i=1
( ∏
j∈JmK\{i}
ϵ (ej + J)
)
xei .
Definition 4.13.
(a) Let Z1
E,R,X denote the K-vector subspace of X×E of all elements x with F 1,r
E,R,X(x) = 0 for
any r ∈ R.
(b) Write B1
E,R,X for the K-vector subspace of X×E formed by all elements x for which there
exists z ∈ X with xe = (e+ J) ▷ z − z ◁ (e+ J) for any e ∈ E.
Lemma 4.14. For any p ∈ K⟨E⟩ and any z ∈ X, if x ∈ X×E is such that xe = (e + J) ▷ z −
z ◁ (e+ J) for any e ∈ E, then
F 1,p
E,R,X(x) = (p+ J) ▷ z − z ◁ (p+ J).
In particular, B1
E,R,X is a K-vector subspace of Z1
E,R,X .
Proof. The claimed identity is clear if p = 1. If there are m ∈ N and {ei}mi=1 ⊆ E with
p = e1 · · · em, then by definition,
F 1,p
E,R,X(x) =
m∑
i=1
(−→∏
i−1
j=1ej + J
)
▷ xei ◁
(−→∏
m
j=i+1ej + J
)
=
m∑
i=1
(−→∏
i−1
j=1ej + J
)
▷ ((ei + J) ▷ z − z ◁ (ei + J)) ◁
(−→∏
m
j=i+1ej + J
)
=
(
m+1∑
i=2
(−→∏
i−1
j=1ej + J
)
▷ z ◁
(−→∏
m
j=iej + J
))
−
(
m∑
i=1
(−→∏
i−1
j=1ej + J
)
▷ z ◁
(−→∏
m
j=iej + J
))
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 15
=
(−→∏
m
i=1ei + J
)
▷ z − z ◁
(−→∏
m
i=1ei + J
)
.
Thus, the identity holds for arbitrary p ∈ K⟨E⟩ by K-linearity. It follows in particular that, if
p ∈ R, then F 1,p
E,R,X(x) = 0 since p+ J is the zero vector of K⟨E | R⟩ in this case. That proves
the claim about B1
E,R,X . ■
In particular, the preceding lemma enables us to consider the following space.
Definition 4.15. Let H1
E,R,X be the K-vector quotient space of Z1
E,R,X with respect to B1
E,R,X .
The following notation allows referencing easily a multitude of algebra morphisms whose
existence is implied by the universal property of K⟨E⟩.
Notation 4.16. Let B be any K-algebra and let (be)e∈E ∈ B×E be arbitrary. The evaluation
of p at (be)e∈E in B is given by p((be)e∈E) := g(p), where g is the unique K-algebra morphism
K⟨E⟩ → B with e 7→ be for any e ∈ E.
Lemma 4.17. For any p ∈ K⟨E⟩ and any x ∈ X×E, evaluating p at (e+J, xe)e∈E in the algebra
K⟨E | R⟩|1
0 X yields
p((e+ J, xe)e∈E) = (p+ J, F 1,p
E,R,X(x)).
Proof. Because F 1,1
E,R,X(x) = 0 by definition, the claim is true if p = 1. If there exist m ∈ N
and {ei}mi=1 ⊆ E with p = e1 · · · em, then the claim follows immediately from Lemma 4.9 and
Definition 4.11. For arbitrary p, the assertion therefore holds by K-linearity. ■
Remark 4.18. Given any K-algebra B, any K-algebra morphism f : K⟨E | R⟩ → B and any
p ∈ K⟨E⟩,
f(p+ J) = p((f(e+ J))e∈E),
where the right-hand side is an evaluation of p in B.
Indeed, if g is the unique K-algebra morphism K⟨E⟩ → B with e 7→ be := f(e + J) for any
e ∈ E, then f(p+ J) = g(p) = p((be)e∈E) = p((f(e+ J))e∈E), where the first identity holds by
the uniqueness of g and where the second is nothing but an application of Notation 4.16.
Now we can give a useful characterization of the spaces of 1-cocycles of universal alge-
bras.
Proposition 4.19.
(a) A commutative diagram of K-linear maps is given by
Z1
HS(K⟨E | R⟩, X) Z1
E,R,X
B1
HS(K⟨E | R⟩, X) B1
E,R,X ,
⊆ ⊆
where the horizontal arrows both assign to any element η of their respective domains the
tuple
(η(e+ J))e∈E .
Moreover, the horizontal arrows are both K-linear isomorphisms. Their respective inverses
both assign to any element x of their respective domains the mapping K⟨E | R⟩ → X with
p+ J 7→ F 1,p
E,R,X(x)
for any p ∈ K⟨E⟩.
16 A. Mang
(b) There exists an isomorphism of K-vector spaces
H1
HS(K⟨E | R⟩, X) H1
E,R,X
such that the class of any η ∈ Z1
HS(K⟨E | R⟩, X) is sent to the class of the x ∈ Z1
E,R,X with
xe = η(e+ J) for any e ∈ E. The inverse isomorphism sends the class of any x ∈ Z1
E,R,X
to the class of the η ∈ Z1
HS(K⟨E | R⟩, X) with η(p+ J) = F 1,p
E,R,X(x) for any p ∈ K⟨E⟩.
Proof. (a) Abbreviate A := K⟨E | R⟩ and B := A|1
0 X.
Step 1. Upper horizontal arrow is well defined. (And a bit more.) First, we prove that for any
η ∈ Z1
HS(A,X), if xe := η(e + J) for any e ∈ E, then η(p + J) = F 1,p
E,R,X(x) for any p ∈ K⟨E⟩.
That then in particular shows that F 1,r
E,R,X(x) = 0 for any r ∈ R.
By η ∈ Z1
HS(A,X), according to Lemma 4.8, the rule that a 7→ (a, η(a)) for any a ∈ A
defines a K-algebra homomorphism f : A → B. Hence, for any p ∈ K⟨E⟩ it must hold that
(p+ J, η(p+ J)) = f(p+ J) = p((f(e+ J))e∈E) = p((e+ J, η(e+ J))e∈E) = p((e+ J, xe)e∈E) =
(p + J, F 1,p
E,R,X(x)) in B, where the second and last identities are implied by Remark 4.18 and
Lemma 4.17, respectively. Hence, F 1,p
E,R,X(x) = η(p+ J) for any p ∈ K⟨E⟩, as claimed.
Step 2. Alleged inverse upper horizontal arrow well defined. Next, we show that for any
x ∈ X×E with F 1,r
E,R,X(x) = 0 for any r ∈ R there exists η ∈ Z1
HS(A,X) with η(p+J) = F 1,p
E,R,X(x)
for any p ∈ K⟨E⟩. One consequence of this is then that the alleged inverse upper horizontal
arrow is well defined.
For any r ∈ R because F 1,r
E,R,X(x) = 0 and r ∈ J we can infer by Lemma 4.17 that r((e +
J, xe)e∈E) = (J, 0) in B. The universal property of A therefore guarantees the existence of
a unique K-algebra homomorphism f : A→ B with f(e+ J) = (e+ J, xe) for any e ∈ E. More
generally, for any p ∈ K⟨E⟩ it must hold that f(p+ J) = p((f(e+ J))e∈E) = (p+ J, F 1,p
E,R,X(x)),
where the two identities are again due to Remark 4.18 and Lemma 4.17. In other words, if
η(p + J) := F 1,p
E,R,X(x) for any p ∈ K⟨E⟩, then the rule that a 7→ (a, η(a)) for any a ∈ A
defines a K-algebra homomorphism A→ B, namely f . According to Lemma 4.8, that demands
η ∈ Z1
HS(A,X). Hence, the initial claim is true.
Step 3. Upper horizontal arrow has alleged inverse. It suffices to prove that for any η ∈
Z1
HS(A,X) and any x ∈ X×E with F 1,r
E,R,X(x) = 0 for any r ∈ R the statements that xe = η(e+J)
for any e ∈ E and that η(p + J) = F 1,p
E,R,X(x) for any p ∈ K⟨E⟩ are equivalent. Clearly, the
second implies the first by the fact that F 1,e
E,R,X(x) = xe for any e ∈ E by definition. And that
the other implication holds was shown in Step 1.
Step 4. Vertical arrows well defined. That the left vertical arrow is well defined is clear. That
the same is true for the right vertical arrow was shown in Lemma 4.14.
Step 5. Lower horizontal arrow and its inverse. From the definition of the lower horizontal
arrow and that of B1
E,R,X , it is clear that the lower horizontal arrow is well defined. Conversely,
the inverse of the upper horizontal arrow restricts to the inverse of the lower horizontal arrow.
That is because for any x ∈ X×E with F 1,r
E,R,X(x) = 0 for any r ∈ R, for the unique η ∈ Z1
HS(A,X)
with η(p+ J) = F 1,p
E,R,X(x) for any p ∈ K⟨E⟩ and for any z ∈ X, if xe = (e+J)▷z−z◁(e+J) for
any e ∈ E, then η(p+J) = F 1,p
E,R,X(x) = (p+J) ▷ z− z ◁ (p+J) = (∂0z)(p+J) by Lemma 4.14.
Step 6. Commutativity of the diagram. Because the two horizontal arrows are defined by the
same rule and since the vertical arrows are set inclusions the diagram commutes.
(b) Follows directly from (a) and is only stated for emphasis. ■
Example 4.20. Let X be the trivial bimodule with respect to an augmentation ϵ of K⟨E | R⟩ as
in Example 4.2. Because then B1
HS(K⟨E | R⟩, X) = {0} by Example 4.5 what Proposition 4.19
implies is that also B1
E,R,X = {0} and that therefore H1
HS(K⟨E | R⟩, X) ∼= Z1
E,R,X .
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 17
Remark 4.21. If V is the K-vector space underlying the K⟨E | R⟩-bimodule X, then the rules
p ▶ v := (p+ J) ▷ v and v ◀ p := v ◁ (p+ J) for any p ∈ K⟨E⟩ and v ∈ V define left respectively
right K⟨E⟩-actions ▶ and ◀ on V which turn it into a K⟨E⟩-bimodule Y , the restriction of
scalars of X along the canonical projection K⟨E⟩ → K⟨E | R⟩. With this definition there is no
difference between the linear maps F 1
E,R,X and F 1
E,∅,Y . (But, of course, there is in general still
a difference between Z1
E,∅,Y = V ×E and Z1
E,R,X = {x ∈ V ×E ∧ ∀r ∈ R : F 1,r
E,∅,Y (x) = 0} and
likewise between B1
E,∅,Y and B1
E,R,X .) The advantage of the notation F 1
E,R,X is that one can
work immediately with the given bimoduleX and does not have to introduce Y first. Then again,
talking about F 1
E,∅,Y can be advantageous too, e.g., in the instance of considering simultaneously
multiple different R and thus multiple different X for which though the restrictions of scalars Y
all happen to be the same.
5 Certain spaces of scalar matrices and their dimensions
The vector spaces of matrices appearing in the main result are characterized and their dimensions
are computed. Recall that for any n ∈ N any v ∈Mn(C) is called skew-symmetric if v = −vt.
Definition 5.1. We call any v ∈Mn(C) small if
∑n
i=1 vj,i = 0 for any j ∈ JnK and
∑n
j=1 vj,i = 0
for any i ∈ JnK, i.e., if each row and each column sums to zero.
Lemma 5.2. For any n ∈ N and v ∈Mn(C) the following equivalences hold.
(a) There is λ ∈ C such that v − λI is small if and only if
∑n
s=1 vj,s −
∑n
s=1 vs,i = 0 for any
{i, j} ⊆ JnK. Moreover, then λ =
∑n
s=1 vj,s =
∑n
s=1 vs,i for any {i, j} ⊆ JnK.
(b) There is λ ∈ C such that v− λI is skew-symmetric if and only if for any {i, j} ⊆ JnK with
i ̸= j both vj,i + vi,j = 0 and vj,j − vi,i = 0. Moreover, then λ = vi,i for any i ∈ JnK.
(c) There are {λ1, λ2} ⊆ C such that v − λ1I is skew-symmetric and v − λ2I small if and
only if there is λ ∈ C such that v − λI is both skew-symmetric and small. Moreover, then
λ = λ1 = λ2.
Proof. For n = 1, all claims hold trivially. Hence, suppose 2 ≤ n in the following.
(a) If λ ∈ C is such that w := v − λI is small, then for any {i, j} ⊆ JnK it follows 0 =∑n
s=1wj,s =
∑n
s=1(vj,s − λδj,s) =
∑n
s=1 vj,s − λ and 0 =
∑n
s=1ws,i =
∑n
s=1(vs,i − λδs,i) =∑n
s=1 vs,i − λ, which proves
∑n
s=1 vj,s = λ =
∑n
s=1 vs,i. Of course, then
∑n
s=1 vj,s −
∑n
s=1 vs,i =
λ− λ = 0 for any {i, j} ⊆ JnK.
Conversely, if
∑n
s=1 vj,s −
∑n
s=1 vs,i = 0 for any {i, j} ⊆ JnK and if we let λ :=
∑n
s=1 v1,s
and w := v − λI, then for any {i, j} ⊆ JnK, first, λ =
∑n
s=1 vj,s =
∑n
s=1 vs,i and thus, second,∑n
s=1wj,s =
∑n
s=1(vj,s − λδj,s) =
∑n
s=1 vj,s − λ = 0 and, likewise,
∑n
s=1ws,i =
∑n
s=1(vs,i −
λδs,i) =
∑n
s=1 vs,i − λ = 0. Hence, w is small then.
(b) If for λ ∈ C the matrix w := v − λI is skew-symmetric, then 0 = wj,i + wi,j = (vj,i −
λδj,i) + (vi,j − λδi,j) = vj,i + vi,j − 2λδj,i for any {i, j} ⊆ JnK. Consequently, if i ̸= j, this means
0 = vj,i + vi,j and, if i = j, we find 0 = 2vi,i − 2λ, i.e., λ = vi,i. And that implies in particular
vj,j − vi,i = λ− λ = 0 for any {i, j} ⊆ JnK.
If, conversely, vj,i + vi,j = 0 and vj,j − vi,i = 0 for any {i, j} ⊆ JnK with i ̸= j and if we let
λ := v1,1 and w := v − λI, then, on the one hand, λ = vi,i for any i ∈ JnK and, on the other
hand, for any {i, j} ⊆ JnK, generally, wj,i+wi,j = (vj,i−λδj,i)+(vi,j−λδi,j) = vj,i+vi,j−2λδj,i,
which in case i ̸= j simply means wj,i + wi,j = vj,i + vi,j = 0 and which for i = j amounts to
wj,i + wi,j = 2vi,i − 2λ = 2λ− 2λ = 0. In conclusion, w is skew-symmetric then.
(c) One implication is clear. If, conversely, {λ1, λ2} ⊆ C are such that v − λ1I is skew-
symmetric and v − λ2I is small, then λ1 = v1,1 by (b) and λ2 =
∑n
j=1 vj,1 =
∑n
i=1 v1,j by (a).
Subtracting the two identities
∑n
j=1 vj,1 = λ1 +
∑n
j=2 vj,1 and
∑n
i=1 v1,i = λ1 +
∑n
i=2 v1,i from
each other therefore yields 0 =
∑n
j=2 vj,1 −
∑n
i=2 v1,i. Since also vi,1 = −v1,i for each i ∈ JnK
18 A. Mang
with 1 < i by (b), that is the same as saying 0 = 2
∑n
j=2 vj,1. And
∑n
j=2 vj,1 = 0 then implies
λ2 = λ1 +
∑n
j=2 vj,1 = λ1, which is all we needed to see. ■
Lemma 5.3. For any n ∈ N and each statement A below, the set {v ∈ Mn(C) ∧ A(v)} is
a complex vector subspace of Mn(C) and has the listed dimension.
A(v) dimC{v ∈Mn(C) ∧ A(v)}
(a) ⊤ n2
(b) ∃λ ∈ C : v − λI is small (n− 1)2 + 1
(c) v is small (n− 1)2
(d) ∃λ ∈ C : v − λI is skew-symmetric 1
2n(n− 1) + 1
(e) v is skew-symmetric 1
2n(n− 1)
(f) ∃λ ∈ C : v − λI is skew-symmetric and small 1
2(n− 1)(n− 2) + 1
(g) v is skew-symmetric and small 1
2(n− 1)(n− 2)
(h) v is diagonal n
(i) ∃λ ∈ C : v − λI = 0 1
(j) v = 0 0
Proof. (a) It is well known that, if for any {k, ℓ} ⊆ JnK the matrix Enℓ,k ∈ Mn(C) has δℓ,jδk,i
as its (j, i)-entry for any {i, j} ⊆ JnK, then the family (Enℓ,k)(ℓ,k)∈JnK×2 is a C-linear basis of
{v ∈Mn(C) ∧ A(v)} =Mn(C).
(b) Since A can be expressed by a homogenous system of linear equations by Lemma 5.2 (a)
the set {v ∈Mn(C) ∧ A(v)} is indeed a vector space. Hence, it suffices to show that the mapping
φn : Mn−1(C) ⊕ C → {v ∈ Mn(C) ∧ A(v)} defined by the rule that (u, λ) 7→ v, where for any
{i, j} ⊆ JnK,
vj,i =
uj,i + λδj,i, j < n ∧ i < n,
−
n−1∑
ℓ=1
uℓ,i, j = n ∧ i < n,
−
n−1∑
k=1
uj,k, j < n ∧ i = n,
n−1∑
k,ℓ=1
uℓ,k + λ, j = n ∧ i = n,
for any u ∈ Mn−1(C) and λ ∈ C, is a C-linear isomorphism. We begin by proving that φn is
well defined. For any u ∈ Mn−1(C) and λ ∈ C, if φn(u, λ) = v and if w = v − λI, then, on the
one hand, for any i ∈ JnK with i < n, by definition,
n∑
j=1
wj,i =
n−1∑
j=1
(vj,i − λδj,i) + vn,i =
n−1∑
j=1
uj,i +
(
−
n−1∑
ℓ=1
uℓ,i
)
= 0.
and also,
n∑
j=1
wj,n =
n−1∑
j=1
(vj,n − λδj,n) + (vn,n − λ) =
n−1∑
j=1
(
−
n−1∑
k=1
uj,k
)
+
n−1∑
k,ℓ=1
uℓ,k = 0.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 19
On the other hand, for any j ∈ JnK with j < n,
n∑
i=1
wj,i =
n−1∑
i=1
(vj,i − λδj,i) + vj,n =
n−1∑
i=1
uj,i +
(
−
n−1∑
k=1
uj,k
)
= 0
and also,
n∑
i=1
wn,i =
n−1∑
i=1
(vn,i − λδn,i) + (vn,n − λ) =
n−1∑
i=1
(
−
n−1∑
ℓ=1
uℓ,i
)
+
n−1∑
k,ℓ=1
uℓ,k = 0.
Together these four conclusions prove that w is small, i.e., that A(v) holds.
Conversely, by Lemma 5.2 (a) a well-defined C-linear map ψn : {v ∈ Mn(C) ∧ A(v)} →
Mn−1(C)⊕C is obtained as follows: For any v ∈Mn(C) with A(v), if λ ∈ C is such that v− λI
is small, then v 7→ (u, λ), where for any {k, ℓ} ⊆ Jn− 1K,
uℓ,k = vℓ,k − λδℓ,k.
It remains to show ψn ◦ φn = id and φn ◦ ψn = id. And, indeed, for any u ∈ Mn−1(C) and
λ ∈ C, if v = φn(u, λ), then we have already seen that w = v−λI is small. For any {k, ℓ} ⊆ JnK,
by definition, wℓ,k = vℓ,k − λδℓ,k = (uℓ,k + λδℓ,k)− λδℓ,k = uℓ,k, which proves φn(v) = (u, λ) and
thus ψn ◦ φn = id.
Conversely, for any v ∈Mn(C) such that A(v) is satisfied, if (u, λ) = ψn(v), then we already
know λ =
∑n
ℓ=1 vℓ,i =
∑n
k=1 vj,k for any {k, ℓ} ⊆ JnK by Lemma 5.2 (a). If v′ = φn(u, λ), then
for any {i, j} ⊆ JnK with i < n and j < n it hence follows by definition v′j,i = uj,i + λδj,i =
(vj,i − λδj,i) + λδj,i = vj,i as well as by λ =
∑n
ℓ=1 vℓ,i,
v′n,i = −
n−1∑
ℓ=1
uℓ,i = −
n−1∑
ℓ=1
(vℓ,i − λδℓ,i) = λ−
n−1∑
ℓ=1
vℓ,i = vn,i
and by λ =
∑n
k=1 vk,j ,
v′j,n = −
n−1∑
k=1
uj,k = −
n−1∑
k=1
(vj,k − λδj,k) = λ−
n−1∑
k=1
vj,k = vj,n
and, lastly,
v′n,n =
n−1∑
k,ℓ=1
uℓ,k + λ =
n−1∑
k,ℓ=1
(vℓ,k − λδℓ,k) + λ =
n−1∑
ℓ=1
(
n−1∑
k=1
vℓ,k − λ
)
+ λ
=
n−1∑
ℓ=1
(−vℓ,n) + λ = vn,n,
where we have used λ =
∑n
k=1 vℓ,k for any ℓ ∈ JnK in the next-to-last step and λ =
∑n
ℓ=1 vℓ,n in
the last. Thus, we have shown v′ = v and thus φn ◦ ψn = id, which concludes the proof in this
case.
(c) By Lemma 5.2 (a), the space {v ∈Mn(C) ∧ A(v)} is exactly the image of Mn−1(C)⊕{0}
under φn.
(d) Lemma 5.2 (b) showed that A can be equivalently expressed as a system of homogenous
linear equations, thus proving {v ∈ Mn(C) ∧ A(v)} to be a vector space. Let Γn = {(j, i) |
{i, j} ⊆ JnK ∧ j < i} ∪· {∅} as well as Bn
(j,i) = Tnj,i = Enj,i − Eni,j for any {i, j} ⊆ JnK with j < i
20 A. Mang
and Bn
∅ = I. Then, the claim will be verified once we show that (Bn
γ )γ∈Γn is a C-linear basis of
{v ∈Mn(C) ∧ A(v)}.
The family (Bn
γ )γ∈Γn is C-linearly independent. Indeed, if (aγ)γ∈Γn ∈ C⊕Γn is such that∑
γ∈Γn
aγ B
n
γ = 0, then by I =
∑n
i=1E
n
i,i,
0 =
∑
(j,i)∈JnK×2
∧ j<i
a(j,i)(E
n
j,i − Eni,j) + a∅
n∑
i=1
Eni,i =
∑
(j,i)∈JnK×2
a(j,i), j < i
−a(i,j), i < j
a∅, j = i
Enj,i,
which demands (aγ)γ∈Γn = 0 since (Enℓ,k)(ℓ,k)∈JnK×2 is C-linearly independent.
It remains to prove that {Bn
γ | γ ∈ Γn} spans {v ∈ Mn(C) ∧ A(v)}. If v ∈ Mn(C) and
λ ∈ C are such that w = v − λI is skew-symmetric, then vj,i = −vi,j and λ = vj,j = vi,i for any
{i, j} ⊆ JnK with j ̸= i by Lemma 5.2 (b). Hence, if we let a∅ = λ and a(j,i) = wj,i = vj,i for any
{i, j} ⊆ JnK with j < i, then
∑
γ∈Γn
aγ B
n
γ =
∑
(j,i)∈JnK×2
a(j,i), j < i
−a(i,j), i < j
a∅, j = i
Enj,i =
∑
(j,i)∈JnK×2
vj,i, j < i
−vi,j , i < j
λ, j = i
Enj,i = v.
Thus, (Bn
γ )γ∈Γn is a C-linear basis.
(e) The proof of the previous claim shows that any v ∈ Mn(C) is skew-symmetric if and
only if it is in the span of {Bn
γ | γ ∈ Γn} and has coefficient 0 with respect to Bn
∅. Hence,
{Tnj,i | {i, j} ⊆ JnK ∧ j < i} is a C-linear basis of {v ∈Mn(C) ∧ A(v)}.
(f) All three parts (a)–(c) of Lemma 5.2 combined imply that {v ∈ Mn(C) ∧ A(v)} is the
solution set to a homogenous system of linear equations and thus a vector space. Hence, it suffices
to prove that φn restricts to a mapping {u ∈Mn−1(C) ∧ u = −ut} → {v ∈Mn(C) ∧ A(v)} and
ψn to one in the reverse direction.
For any skew-symmetric u ∈ Mn−1(C) and any λ ∈ C, if v = φn(u, λ) and w = v − λI,
then for any {i, j} ⊆ JnK with i < n and j < n we have already seen that wj,i = uj,i, implying
wj,i + wi,j = uj,i + ui,j = 0 by u = −ut. Moreover, for the same reason,
wn,i + wi,n = (vn,i − λδn,i) + (vi,n − λδi,n) = vn,i + vi,n
=
(
−
n−1∑
ℓ=1
uℓ,i
)
+
(
−
n−1∑
k=1
ui,k
)
= −
n−1∑
ℓ=1
(uℓ,i + ui,ℓ) = 0
and
wj,n + wn,j = (vj,n − λδj,n) + (vn,j − λδn,j) = vj,n + vn,j
=
(
−
n−1∑
k=1
uj,k
)
+
(
−
n−1∑
ℓ=1
uℓ,j
)
= −
n−1∑
k=1
(uj,k + uk,j) = 0
as well as
wn,n + wn,n = 2(vn,n − λ) = 2
n−1∑
k,ℓ=1
uℓ,k =
n−1∑
k,ℓ=1
(uℓ,k + uk,ℓ) = 0,
which completes the proof that φn restricts to a map into {v ∈Mn(C) ∧ A(v)}.
Conversely, if v ∈Mn(C) and λ ∈ C are such that w = v − λI is skew-symmetric and small,
then λ = vi,i for any i ∈ JnK by Lemma 5.2 (b). For (u, λ) = ψn(v) and any {k, ℓ} ⊆ Jn− 1K,
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 21
by definition, uℓ,k = wℓ,k and thus uℓ,k + uk,ℓ = wℓ,k + wk,ℓ = 0 by w = −wt. Hence, ψn maps
{v ∈Mn(C) ∧ A(v)} into {u ∈Mn−1(C) ∧ u = −ut} ⊕ C.
(g) As we have just shown, any v ∈MC(n) is skew-symmetric and small if and only if it lies
in the image of {u ∈Mn−1(C) ∧ u = −ut} ⊕ {0} under φn. And φn is a C-linear isomorphism
from this space to {v ∈Mn(C) ∧ A(v)}.
(h) It is well known that (Eni,i)i∈JnK is a C-linear basis of {v ∈Mn(C) ∧ A(v)}.
(i) In this case, {v ∈Mn(C) ∧ A(v)} is the C-linear span of I in Mn(C).
(j) Here, {v ∈Mn(C) ∧ A(v)} is the zero C-linear space. ■
6 First cohomology of unitary easy quantum group duals
This section computes the first quantum group cohomology with trivial coefficients (see Sec-
tion 2) of the discrete dual of any unitary easy compact quantum group (see Section 3). That is
achieved by applying the characterization of the first Hochschild cohomology recalled in Section 4
while using the results of Section 5 as auxiliaries.
6.1 Equations derived from the presentation
Resume the Assumptions 3.13 and the abbreviations from Notations 3.14 and 3.17. In particu-
lar, n and E are then defined. Remark 4.21 motivates moreover the following shorthand.
Notation 6.1.
(a) Let Y be the C⟨E⟩-bimodule C with left and right actions given by ucj,i ▷ x := δj,ix
respectively x ◁ ucj,i := δj,ix for any {i, j} ⊆ JnK, any c ∈ {◦, •} and any x ∈ C.
(b) Let Fp := F 1,p
E,∅,Y for any p ∈ C⟨E⟩.
Then, by Section 4 for any category C of two-colored partitions the first cohomology with
trivial coefficients of the discrete dual of any easy quantum group associated with (C, n) can be
realized as a solution space to a system of linear equations involving maps of the form Fr for
certain r ∈ C⟨E⟩ induced by C and n.
Proposition 6.2. For any category C of two-colored partitions, if G is the unitary easy compact
quantum group of (C, n), then there exists an isomorphism of C-vector spaces
H1
(
Ĝ
)
{x ∈ C×E ∧ ∀r ∈ RC : Fr(x) = 0},
which maps (the one-elemental cohomology class of) any 1-cycle η to the tuple x with xe =
η(e+ JC) for any e ∈ E.
Proof. By Definition 3.18, the algebra underlying the Hopf ∗-algebra C[Ĝ] is the universal
algebra C⟨E | RC⟩. According to Section 2.2, the vector space H1
(
Ĝ
)
is defined as H1
HS(C⟨E |
RC⟩, X) where X = ϵCϵ is trivial bimodule of C⟨E | RC⟩ with respect to the counit ϵ of C[Ĝ].
By Proposition 3.20, this counit is such that its restriction of scalars along the canonical pro-
jection C⟨E⟩ → C⟨E | RC⟩ is precisely Y . Hence, the claim follows by Example 4.20 and
Remark 4.21. ■
The task laid out by Proposition 6.2 is clear. We need to solve the set of linear equations
in C×E on the right-hand side of the isomorphism there – for each category of two-colored par-
titions. Eventually, in Section 6.6 namely, solving these equations will require case distinctions
for different kinds of categories of two-colored partitions. However, there are a great number of
simplifications we can make to the equation system before it needs to come to that. Moreover,
this reduces the number of cases we eventually have to consider immensely.
22 A. Mang
6.2 Simplifying each individual equation
As a first step towards solving the equations of Proposition 6.2 we consider each equation in
isolation and simplify its definition. In other words, we seek a better formula for the values of the
functional Fr for r of the form rcd(p)j,g for arbitrary two-colored partitions (c, d, p) and g ∈ JnK×|c|
and j ∈ JnK×|d|.
It will be convenient to have a shorthand for mappings constructed by prescribing a specified
value to a specified point and otherwise inheriting the graph of a given mapping with the same
domain.
Notation 6.3. For any {k, ℓ} ⊆ N0, any mapping f : Πkℓ → JnK, any z ∈ Πkℓ and any s ∈ JnK
write f ↓z s for the mapping Πkℓ → JnK with z 7→ s and with y 7→ f(y) for any y ∈ Πkℓ \{z}.
Then, combining Notation 3.14 and Definition 4.11 yields the following description of the
functionals we are investigating.
Lemma 6.4. For any {k, ℓ} ⊆ N0, any c ∈ {◦, •}×k, any d ∈ {◦, •}×ℓ, any set-theoretical
partition p of Πkℓ , any g ∈ JnK×k, any j ∈ JnK×ℓ and any x ∈ C×E, if r = rcd(p)j,g and f = g ■■ j
and w = c ■■ d, then
Fr(x) =
∑
z∈Πk
ℓ
n∑
s=1
ζ(p, ker(f ↓z s))
−x
u
w(z)
s,f(z)
if z ∈ Πk0
x
u
w(z)
f(z),s
if z ∈ Π0
ℓ
.
Proof. For any x ∈ C×E , by Example 4.12,
Fr(x) =
∑
i∈JnK×ℓ
ζ(p, ker(g ■■ i))
ℓ∑
b=1
( ∏
q∈JℓK
∧ q ̸=b
δjb,ib
)
x
u
db
jb,ib
−
∑
h∈JnK×k
ζ(p, ker(h ■
■ j))
k∑
a=1
( ∏
q∈JkK
∧ q ̸=a
δha,ga
)
xucaha,ga
.
After commuting the sums and evaluating the sums over i respectively h (as far as possible),
this is identical to
ℓ∑
b=1
n∑
ib=1
ζ(p, ker(g ■■ (j1, . . . , jb−1, ib, jb+1, . . . , jℓ))) xudbjb,ib
−
k∑
a=1
n∑
ha=1
ζ(p, ker((g1, . . . , ga−1, ha, ga+1, . . . , gk)
■
■ j)) xucaha,ga
.
That agrees with the right-hand side of the claimed identity. ■
While Lemma 6.4 has given a more concise form to the equations under investigation, it can
be improved upon significantly. Firstly, one can give a simpler criterion for when in the sum on
the right-hand side of the identity in Lemma 6.4 a factor ζ(p, ker(f ↓z s)) is non-zero.
Lemma 6.5. For any {k, ℓ}⊆N0, any set-theoretical partition p of Π
k
ℓ , any mapping f : Πkℓ→JnK,
any z ∈ Πkℓ and any s ∈ JnK, the statements p ≤ ker(f ↓z s) and
p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} ≤ ker(f) ∧ πp(z)\{z} ⊆ f←({s})
are equivalent.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 23
Proof. We show each implication separately. Below, we will use many times the fact that for
any t ∈ JnK,
(f ↓z s)←({t})\{z} = {a ∈ Πkℓ ∧ (f ↓z s)(a) = t ∧ a ̸= z}
= {a ∈ Πkℓ ∧ f(a) = t ∧ a ̸= z}
= f←({t})\{z}.
Step 1. First, suppose p ≤ ker(f ↓z s). Then, there exists t ∈ ran(f ↓z s) such that
πp(z) ⊆ (f ↓z s)←({t}). Because z ∈ πp(z) this requires z ∈ (f ↓z s)←({t}) and thus t = s
by (f ↓z s)(z) = s. It follows πp(z) ⊆ (f ↓z s)←({s}) and thus in particular πp(z)\{z} ⊆ (f ↓z
s)←({s})\{z} = f←({s})\{z} ⊆ f←({s}), which is one half of what we had to show.
It is trivially true that {z} ⊆ f←({f(z)}) ∈ ker(f). We have already seen that πp(z)\{z} ⊆
f←({s}) ∈ ker(f). For any B ∈ p with B ̸= πp(z), i.e., z /∈ B, there exists by assumption
t′ ∈ ran(f ↓z s) with B ⊆ (f ↓z s)←({t′}). We conclude B = B\{z} ⊆ (f ↓z s)←({t′})\{z} =
f←({t′})\{z} ⊆ f←({t′}) ∈ ker(f). Thus, the other half of the claim, p\{πp(z)} ∪· {πp(z)\{z},
{z}}\{∅} ≤ ker(f), holds as well. That proves one implication.
Step 2. In order to show the converse implication we assume that both p\{πp(z)}∪· {πp(z)\{z},
{z}}\{∅} ≤ ker(f) and πp(z)\{z} ⊆ f←({s}) and then we distinguish two cases.
Case 2.1. If {z} ∈ p and thus πp(z) = {z} and πp(z)\{z} = ∅, then the assumption is
simply equivalent to the statement p ≤ ker(f). Naturally, {z} ⊆ (f ↓z s)←({s}) ∈ ker(f) by
(f ↓z s)(z) = s. For any B ∈ p with B ̸= {z} there exists by our premise a value t ∈ ran(f) with
B ⊆ f←({t}). Thus, also B = B\{z} ⊆ f←({t})\{z} = (f ↓z s)←({t})\{z} ∈ ker(f ↓z s). In
conclusion, p ≤ ker(f ↓z s).
Case 2.2. In the instance that {z} /∈ p the initial assumption simplifies to the statement
p\{πp(z)} ∪· {πp(z)\{z}, {z}} ≤ ker(f) and πp(z)\{z} ⊆ f←({s}). The latter condition implies
πp(z)\{z} ⊆ f←({s})\{z} = (f ↓z s)←({s})\{z} ⊆ (f ↓z s)←({s}) and thus by (f ↓z s)(z) = s
also πp(z) = πp(z)\{z} ∪· {z} ⊆ (f ↓z s)←({s}) ∪ {z} ⊆ (f ↓z s)←({s}) ∈ ker(f ↓z s). On the
other hand, for any B ∈ p with B ̸= πp(z), which is to say z /∈ B, there exists by assumption
t ∈ ran(f) with B ⊆ f←({t}). It follows B = B\{z} ⊆ f←({t})\{z} = (f ↓z s)←({t})\{z} ⊆
(f ↓z s)←({t}) ∈ ker(f ↓z s). Hence, altogether, p ≤ ker(f ↓z s), which concludes the proof. ■
Lemma 6.5 can now be used to give a necessary criterion for the right-hand side of the identity
in Lemma 6.4 to be non-zero as a whole. Namely, p and f must meet one of three conditions:
(i) The labeling f maps any points belonging to the same element of p to the same value.
(ii) There is an element of size two of p whose elements f maps to different values. Besides
that f is as in (i).
(iii) There is an element of p of size three or larger, all but one of whose elements are assigned
the same value by f and whose remaining element f sends to a different value. Apart from
that, f is as in (i).
Definition 6.6. Let {k, ℓ} ⊆ N0, let p be any set-theoretical partition of Πkℓ and let f : Πkℓ → JnK.
Then, we say that (p, f) is
(a) case R1 if p ̸= ∅ and p ≤ ker(f),
(b) case R2 if there exists {z1, z2} ∈ p such that f(z1) ̸= f(z2) and such that for any A ∈ p
with A ̸= {z1, z2} there is B ∈ ker(f) with A ⊆ B, in which case the set {z1, z2} is called
critical data of (p, f),
(c) case R3 if there exist Z ∈ p and z ∈ Z and s ∈ JnK such that 3 ≤ |Z|, such that f(z) ̸= s,
such that f(y) = s for any y ∈ Z with y ̸= z and such that for any A ∈ p with A ̸= Z there
is B ∈ ker(f) with A ⊆ B, in which case (Z, z, s) are called critical data of (p, f),
(d) case R4 otherwise.
24 A. Mang
Example 6.7. For 3 ≤ n, consider k := 4 and ℓ := 5, the set-theoretical partition p := {{
■
1, ■1},
{■2}, {
■
2,
■
3, ■3}, {■4}, {
■
4,
■
5}} of Πkℓ and various different mappings f : Πkℓ → JnK which all have
in common that each of
■
2,
■
4 and ■3 is mapped to 1, that each of
■
1, ■1 and ■2 is mapped to 2
and that ■4 is mapped to 3. Thus, at most the values of
■
3 and
■
5 differ between different f :
2 2 1 3
2 1 1f
p
f
(a) If f(
■
3) = f(
■
5) = 1, then (p, f) is case R1.
(b) If f(
■
3) = 1 and f(
■
5) = 2, if z1 :=
■
4 and z2 :=
■
5, then (p, f) is case R2 with critical
data {z1, z2} = {■4, ■5}.
(c) If f(
■
3) = 2 and f(
■
5) = 1, if Z := {
■
2,
■
3, ■3}, if z :=
■
3 and s := 2, then (p, f) is case R3
with critical data (Z, z, s) = ({
■
2,
■
3, ■3}, {
■
3}, 2).
(d) If f(
■
3) = f(
■
5) = 2, then (p, f) is case R4.
Lemma 6.8.
(a) In each of the cases R2 and R3 critical data are unique.
(b) The cases R1–R4 are mutually exclusive and exhaustive.
Proof. Let {k, ℓ} ⊆ N0, let p be any set-theoretical partition of Πkℓ and let f : Πkℓ → JnK be
arbitrary.
(a) Case R2. Suppose that (p, f) is case R2 and that both {z1, z2} and {z′1, z′2} are critical
data of (p, f). If {z1, z2} ≠ {z′1, z′2} were true, then by the assumption on {z1, z2} there would
exist B ∈ ker(f) with {z′1, z′2} ⊆ B, meaning f(z′1) = f(z′2), contrarily to our assumption. Hence,
{z1, z2} = {z′1, z′2} must be true instead.
Case R3. Now, let (p, f) be case R3 and let both (Z, z, s) and (Z′, z′, s′) be critical data
of (p, f). If Z ̸= Z′ held, the assumption on Z would imply the existence of B ∈ ker(f) with
Z′ ⊆ B. In particular, it would follow f(y′) = f(z′) for any y′ ∈ Z′ with y′ ̸= z′, of which there
exists at least one by 3 ≤ |Z′|. Because that would contradict the assumption, we must have
Z = Z′ instead.
Furthermore, supposing z ̸= z′ demands of any y ∈ Z\{z, z′} both f(y) = s by the assumption
on z and s and f(y) = s′ by the one on z′ and s′. Hence, as Z\{z, z′} ≠ ∅ by 3 ≤ |Z′|, if z′ ̸= z,
then s = s′. That would be a contradiction because the property of z′ also requires s ̸= f(z) = s′
in that case. Hence, only z = z′ can be true.
Lastly, because the assumptions on s and s′ imply f(y) = s respectively f(y) = s′ for any
y ∈ Z with y ̸= z = z′ and because Z\{z} ≠ ∅, we must have s = s′ as well.
(b) It is enough to prove that cases R1–R3 are mutually exclusive. If (p, f) is case R2, then it
cannot be case R1 because f(z1) ̸= f(z2) excludes the existence of B ∈ ker(f) with {z1, z2} ⊆ B,
which would be necessary for p ≤ ker(f) to hold. Similarly, (p, f) being case R3 forbids it being
case R1 as well because the existence of y ∈ Z\{z} ̸= ∅ with f(z) ̸= s = f(y) does not
allow any B ∈ ker(f) with Z ⊆ B to exist, which p ≤ ker(f) would require. Lastly, if (p, f)
were simultaneously case R2 and case R3, then {z1, z2} ≠ Z would follow from 3 ≤ |Z|, thus
demanding by the property of Z the existence of B ∈ ker(f) with {z1, z2} ⊆ B, in contradiction
to f(z1) ̸= f(z2). ■
Lemma 6.9. For any {k, ℓ} ⊆ N0, any set-theoretical partition p of Πkℓ and any mapping
f : Πkℓ → JnK there exist z ∈ Πkℓ and s ∈ JnK such that p ≤ ker(f ↓z s) if and only if (p, f)
is not case R4.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 25
Proof. Each implication is shown individually.
Step 1. First, we suppose that (p, f) is not case R4 and deduce the existence of z ∈ Πkℓ and
s ∈ JnK with p ≤ ker(f ↓z s). By Lemma 6.5, that is the same as finding z ∈ Πkℓ and s ∈ JnK such
that p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} ≤ ker(f) and πp(z)\{z} ⊆ f←({s}). By Lemma 6.8 (b),
the pair (p, f) is case R1, case R2 or case R3. These three cases are treated individually.
Case 1.1. If (p, f) is case R1, then by p ̸= ∅ we can find and fix some z ∈ Πkℓ and put s := f(z).
From p ≤ ker(f), it then follows πp(z) ⊆ f←({s}) and thus in particular πp(z)\{z} ⊆ f←({s}),
which is one part of what we have to show. The other part, p\{πp(z)}∪· {πp(z)\{z}, {z}}\{∅} ≤
ker(f) is a consequence of the fact p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} ≤ p and the assumption
p ≤ ker(f).
Case 1.2. Next, let (p, f) be case R2 and let {z1, z2} be its critical data. If we define z := z1
and s := f(z2), then πp(z) = {z1, z2} and thus πp(z)\{z} = {z2} ⊆ f←({s}). On the other
hand, p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} = p\{{z1, z2}} ∪· {{z1}, {z2}} ≤ ker(f) because, by
assumption, for each A ∈ p\{{z1, z2}} there exists B ∈ ker(f) with A ⊆ B ∈ ker(f) and, of
course, {z1} ⊆ f←({f(z1)}) ∈ ker(f) and {z2} ⊆ f←({s}).
Case 1.3. Finally, let (p, f) be case R3 and let (Z, z, s) be its critical data. Then, obviously,
πp(z)\{z} = Z\{z} ⊆ f←({s}) by assumption. And, p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} =
p\{Z}∪· {Z\{z}, {z}} ≤ ker(f) because, by assumption, for any A ∈ p\{Z} there exists B ∈ ker(f)
with A ⊆ B and because Z\{z} ⊆ f←({s}) ∈ ker(f) and {z} ⊆ f←({f(z)}) ∈ ker(f). That
proves one implication.
Step 2. In order to show the converse implication, we assume that there exist z ∈ Πkℓ and
s ∈ JnK such that p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} ≤ ker(f) and πp(z)\{z} ⊆ f←({s}) (which
we can by Lemma 6.5) and derive that (p, f) is case R1, case R2 or case R3, thus proving that
(p, f) is not case R4 by Lemma 6.8 (b). Note that the existence of z requires p ̸= ∅. Again,
a case distinction is in order.
Case 2.1. First, let f(z) = s. Then πp(z) = πp(z)\{z} ∪· {z} ⊆ f←({s}) ∈ ker(f) by
πp(z)\{z} ⊆ f←({s}). Thus, p ≤ ker(f) by p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} ≤ ker(f). In
other words, we have shown (p, f) to be case R1.
Case 2.2. Similarly, if πp(z) = {z}, then p = p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} ≤ ker(f).
Thus, (p, f) is case R1.
Case 2.3. If f(z) ̸= s and |πp(z)| = 2, then we put z1 := z and we let z2 be the
unique element of πp(z)\{z}. It follows {z2} = πp(z)\{z} ⊆ f←({s}) and thus f(z2) =
s ̸= f(z) = f(z1) by our assumptions. And, the premise p\{{z1, z2}} ∪· {{z1}, {z2}} =
p\{πp(z)} ∪· {πp(z)\{z}, {z}}\{∅} ≤ ker(f) means that for any A ∈ p with A ̸= {z1, z2} there
exists B ∈ ker(f) with A ⊆ B. Hence, (p, f) is case R2 with critical data {z1, z2}.
Case 2.4. The last remaining possibility is that f(z) ̸= s and 3 ≤ |πp(z)|. Putting Z :=
πp(z) implies Z\{z} = πp(z)\{z} ⊆ f←({s}) by assumption, which is to say f(y) = s ̸= f(z)
for any y ∈ Z with y ̸= z. On the other hand, since p\{Z} ∪· {Z\{z}, {z}} = p\{πp(z)} ∪·
{πp(z)\{z}, {z}}\{∅} ≤ ker(f), for any A ∈ p with A ̸= Z there exists B ∈ ker(f) with A ⊆ B.
In other words, (p, f) is case R3 with critical data (Z, z, s). Thus, both implications are true. ■
Example 6.10. For each of the first three f of Example 6.7 (but not the fourth) one can
give at least one (z, s) ∈ Πkℓ × JnK such that p ≤ ker(f ↓z s) (and, in fact, ker(f ↓z s) =
{{■1, ■2,
■
1}, {■3,
■
2,
■
3,
■
4,
■
5}, {■4}}).
(p, f) =
2 2 1 3
2 1 f(■3) 1 f(■5)
ker(f ↓z s) =
26 A. Mang
(a) If f(
■
3) = f(
■
5) = 1, then let z :=
■
3 and s := 1.
(b) If f(
■
3) = 1 and f(
■
5) = 2, then let z :=
■
5 and s := 1.
(c) If f(
■
3) = 2 and f(
■
5) = 1, then let z :=
■
3 and s := 1.
Finally, in each of the three cases R1–R3 the next lemma explains for which (z, s) the corre-
sponding summand on the right-hand side of the identity in Lemma 6.4 has a non-zero factor
ζ(p, ker(f ↓z s)).
Lemma 6.11. Let {k, ℓ} ⊆ N0, let p be any set-theoretical partition of Πkℓ , let f : Π
k
ℓ → JnK be
any mapping and let z′ ∈ Πkℓ and s′ ∈ JnK be arbitrary.
(a) If (p, f) is case R1, then p ≤ ker(f ↓z′ s′) if and only if either |πp(z′)| = 1 or both
2 ≤ |πp(z′)| and s′ = f(z′).
(b) If (p, f) is case R2 with critical data {z1, z2}, then p ≤ ker(f ↓z′ s′) if and only if either
both z′ = z1 and s′ = f(z2) or both z′ = z2 and s′ = f(z1).
(c) If (p, f) is case R3 with critical data (Z, z, s), then p ≤ ker(f ↓z′ s′) if and only if z′ = z
and s′ = s.
Proof. By Lemma 6.5, the statement p ≤ ker(f ↓z′ s′) is equivalent to the conjunction of
p\{πp(z′)} ∪· {πp(z′)\{z′}, {z′}}\{∅} ≤ ker(f) and πp(z
′)\{z′} ⊆ f←({s′}).
(a) Because p\{πp(z′)} ∪· {πp(z′)\{z′}, {z′}}\{∅} ≤ p, in the situation of (a), where p ≤
ker(f), we only need to determine when πp(z
′)\{z′} ⊆ f←({s′}). If |πp(z′)| = 1, that is,
πp(z
′)\{z′} = ∅, this condition is trivially satisfied. And if 2 ≤ |πp(z′)|, then πp(z
′)\{z′} ⊆
f←({s′}) holds if and only if s′ = f(z′) because πp(z
′)\{z′} ⊆ πp(z′) ⊆ f←({f(z′)}) by assump-
tion. That proves (a).
(b) In case (b), if z′ /∈ {z1, z2}, then {z1, z2} ∈ p\{πp(z′)}∪· {πp(z′)\{z′}, {z′}}\{∅}. However,
because f(z1) ̸= f(z2) there cannot exist any B ∈ ker(f) with {z1, z2} ⊆ B. Hence, z′ /∈ {z1, z2}
excludes p\{πp(z′)} ∪· {πp(z′)\{z′}, {z′}}\{∅} ≤ ker(f) and thus p ≤ ker(f ↓z′ s′).
Hence, p ≤ ker(f ↓z′ s′) requires the existence of i ∈ J2K with z′ = zi. If so, then p\{πp(z′)}∪·
{πp(z′)\{z′}, {z′}}\{∅} = p\{{z1, z2}} ∪· {{z1}, {z2}} ≤ ker(f) since by assumption for any
A ∈ p with A /∈ {z1, z2} there exists B ∈ ker(f) with A ⊆ B. Thus, in this case, p ≤ ker(f ↓z′ s′)
is equivalent to {z3−i} = {z1, z2}\{zi} = πp(z
′)\{z′} ⊆ f←({s′}), i.e., to f(z3−i) = s′, just
as (b) claimed.
(c) Finally, under the assumptions of (c), whenever z′ /∈ Z, then Z ∈ p\{πp(z′)}∪· {πp(z′)\{z′},
{z′}}\{∅} ̸≤ ker(f) by the existence of y ∈ Z\{z} ≠ ∅ with f(z) ̸= s = f(y). Consequently,
p ̸≤ ker(f ↓z′ s′) if z′ /∈ Z.
For z′ ∈ Z, because by assumption there is for any A ∈ p with A ̸= Z = πp(z
′) a B ∈
ker(f) with A ⊆ B the condition p\{πp(z′)} ∪· {πp(z′)\{z′}, {z′}}\{∅} ≤ ker(f) simplifies to the
existence of B ∈ ker(f) with πp(z
′)\{z′} ⊆ B, which is subsumed by the second condition. In
other words, if z′ ∈ Z, then p ≤ ker(f ↓z′ s′) if and only if πp(z
′)\{z′} ⊆ f←({s′}).
If z′ ̸= z, then πp(z
′)\{z′} ̸⊆ f←({s′}) because, by 3 ≤ |Z|, there exist y ∈ Z\{z, z′} with
f(y) = s ̸= f(z) by assumption. Hence, p ≤ ker(f ↓z′ s′) requires z′ = z. And in that case it
is equivalent to Z\{z} = πp(z
′)\{z′} ⊆ f←({s′}), which is satisfied if and only if s′ = s because
f(y) = s for any y ∈ Z\{z} ̸= ∅. Thus, the assertion of (c) is true as well and, thus, so is the
claim overall. ■
In regard of Lemmas 6.9 and 6.11, we can now improve upon Lemma 6.4 as follows.
Lemma 6.12. Let {k, ℓ} ⊆ N0, let c ∈ {◦, •}×k, let d ∈ {◦, •}×ℓ, let p be any set-theoretical
partition of Πkℓ , let g ∈ JnK×k, let j ∈ JnK×ℓ, let r := rcd(p)j,g, let f := g ■■ j, let w := c ■■ d, and
let x ∈ C×E.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 27
(i) If (p, f) is case R1, then
Fr(x) =
∑
z∈Πk
ℓ
∧ |πp(z)|=1
n∑
s=1
−x
u
w(z)
s,f(z)
if z ∈ Πk0
x
u
w(z)
f(z),s
if z ∈ Π0
ℓ
+
∑
z∈Πk
ℓ
∧ 2≤|πp(z)|
{
−1 if z ∈ Πk0
1 if z ∈ Π0
ℓ
}
x
u
w(z)
f(z),f(z)
.
(ii) If (p, f) is case R2 with critical data {z1, z2}, then
Fr(x) =
−x
u
w(z1)
f(z2),f(z1)
if z1 ∈ Πk0
x
u
w(z1)
f(z1),f(z2)
if z1 ∈ Π0
ℓ
+
−x
u
w(z2)
f(z1),f(z2)
if z2 ∈ Πk0
x
u
w(z2)
f(z2),f(z1)
if z2 ∈ Π0
ℓ
.
(iii) If (p, f) is case R3 with critical data (Z, z, s), then
Fr(x) =
−x
u
w(z)
s,f(z)
if z ∈ Πk0
x
u
w(z)
f(z),s
if z ∈ Π0
ℓ
.
(iv) If (p, f) is case R4, then Fr(x) = 0.
Proof. By Lemma 6.4,
Fr(x) =
∑
z′∈Πk
ℓ
n∑
s′=1
ζ(p, ker(f ↓z′ s′))
−x
u
w(z′)
s′,f(z′)
if z′ ∈ Πk0
x
u
w(z′)
f(z′),s′
if z′ ∈ Π0
ℓ
.
From this identity, we see immediately that Fr(x) ̸= 0 requires the existence of z ∈ Πkℓ and
s ∈ JnK with p ≤ ker(f ↓z s). Thus, Lemma 6.9 verifies (iv). It remains to treat the cases
(i)–(iii).
(i) In the situation of (i), for any z′ ∈ Πkℓ and s′ ∈ JnK we know from Lemma 6.11 (a) that
p ≤ ker(f ↓z′ s′) if and only if either |πp(z′)| = 1 or both 2 ≤ |πp(z′)| and s′ = f(z′). Thus, the
above formula for Fr(x) simplifies to the one in (i).
(ii) Under the assumptions of (ii), Lemma 6.11 (b) tells us for any z′ ∈ Πkℓ and s′ ∈ JnK that
p ≤ ker(f ↓z′ s′) if and only if either both z′ = z1 and s′ = f(z2) or both z′ = z2 and s′ = f(z1).
That proves the formula for Fr(x) in (ii).
(iii) Finally, if the premises of (iii) are satisfied, then for any z′ ∈ Πkℓ and s′ ∈ JnK Lem-
ma 6.11 (c) lets us infer that p ≤ ker(f ↓z′ s′) if and only if z′ = z and s′ = s. In particular, at
most one summand is non-zero. It follows that Fr(x) is given by the expression in (iii). ■
6.3 Halving the number of variables
Until now we have only considered each equation in the systems from Proposition 6.2 in isolation.
The next simplification will take into account that the two-colored partitions , , and
are present in any category of two-colored partitions. That fact can be used to eliminate half
the variables (as, e.g., in [22, Lemma 1.7]). This is the only explicit elimination of variables that
will be made in the entire proof of the main theorem.
28 A. Mang
Lemma 6.13. For any g ∈ JnK×2 and j ∈ JnK×2 and any x ∈ C×E, if r is given by
(a) r∅◦•( )j,∅, then Fr(x) = xu◦j1,j2
+ xu•j2,j1
.
(b) r∅•◦( )j,∅, then Fr(x) = xu•j1,j2
+ xu◦j2,j1
.
(c) r•◦∅ ( )∅,g, then Fr(x) = −xu•g2,g1 − xu◦g1,g2 .
(d) r◦•∅ ( )∅,g, then Fr(x) = −xu◦g2,g1 − xu•g1,g2 .
Proof. Only the proof of (a) is given. The others are similar. Using Definition 4.11, the result
of Lemma 3.15 that r = r∅◦•( )j,∅ =
∑n
i=1 u
◦
j1,i
u•j2,i − δj1,j21 implies
Fr(x) =
n∑
i=1
(xu◦j1,i
◁ u•j2,i + u◦j1,i ▷ xu•j2,i
) =
n∑
i=1
(δj2,i xu◦j1,i
+ δj1,i xu•j2,i
) = xu◦j1,j2
+ xu•j2,j1
because F1 = 0. ■
Notation 6.14. Let v ∈Mn(C) be arbitrary.
(a) Let xv ∈ C×E be such that for any {i, j} ⊆ JnK,
xvu◦j,i
:= vj,i and xvu•j,i
:= −vi,j .
(b) For any set P of two-colored partitions let A(P, v) denote the statement that Fr(x
v) = 0
for any r ∈ RP .
Ultimately, it will be shown that in the case of categories of two-colored partitions the pred-
icates A defined in Notation 6.14 are equivalent to those used in the formulation of the main
theorem.
Proposition 6.15. For any category C of two-colored partitions, if G is the unitary easy compact
quantum group of (C, n), then there exists an isomorphism of C-vector spaces
H1
(
Ĝ
)
{v ∈Mn(C) ∧ A(C, v)},
which maps (the one-elemental cohomology class of) any 1-cocycle η to the matrix v with vj,i =
η(u◦j,i + JC) for any {i, j} ⊆ JnK.
Proof. By Proposition 6.2, it suffices to show that the rule x 7→ (xu◦j,i)(j,i)∈JnK×2 gives a C-linear
isomorphism
{x ∈ C×E ∧ ∀r ∈ RC : Fr(x) = 0} {v ∈Mn(C) ∧ ∀r ∈ RC : Fr(xv) = 0}.
The claimed isomorphism is well defined: Let x ∈ C×E be such that Fr(x) = 0 for any r ∈ RC .
Then, for any j ∈ JnK×2 because r∅◦•( )j,∅ ∈ RC in particular
xu◦j1,j2
+ xu•j2,j1
= 0
by Lemma 6.13, i.e., xu•j2,j1
= −xu◦j1,j2 . Hence, if we let v := (xu◦j,i)(j,i)∈JnK×2 , then for any
{i, j} ⊆ JnK by definition not only xvu◦j,i
= vj,i = xu◦j,i but also
xvu•j,i
= −vi,j = −xu◦i,j = xu•j,i ,
which is to say xv = x. Thus, per assumption, in particular Fr(x
v) = Fr(x) = 0 for any r ∈ RC .
That proves that the map is well defined.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 29
It is clear that the mapping is C-linear. Moreover, it is injective because, if again x ∈ C×E
is such that Fr(x) = 0 for any r ∈ RC and if again v := (xu◦j,i)(j,i)∈JnK×2 , then v = 0 necessitates
xv = 0 by definition of xv and thus x = 0 by the identity xv = x established in the preceding
paragraph.
To show surjectivity, we let v ∈Mn(C) be arbitrary with Fr(x
v) = 0 for any r ∈ R and abbre-
viate x := xv. Then, of course, Fr(x) = 0 for any r ∈ RC . Because moreover xu◦j,i = xvu◦j,i
= vj,i
for any {i, j} ⊆ JnK the tuple x is a preimage of v. Thus, the claim is true. ■
The next lemma correspondingly eliminates the variables (xu•j,i)(j,i)∈JnK×2 from the formula
obtained in Lemma 6.12 for the individual equations in the systems from Proposition 6.2. Recall
from Notation 3.1 that f/p denotes the quotient mapping of any mapping f with respect to any
set-theoretical partition p of its domain and recall the definition of the color sum σcd of two color
tuples c and d from Definition 3.4.
Lemma 6.16. Let (c, d, p) be any two-colored partition, let g ∈ JnK×|c|, let j ∈ JnK×|d|, let
r := rcd(p)j,g, let f := g ■■ j, and let v ∈Mn(C).
(i) If (p, f) is case R1, then
Fr(x
v) =
∑
A∈p
∧ |A|=1
σcd(A)
n∑
s=1
{
v(f/p)(A),s if σcd(A) = 1
vs,(f/p)(A) if σcd(A) = −1
}
+
∑
A∈p
∧ 2≤|A|
σcd(A)v(f/p)(A),(f/p)(A).
(ii) If (p, f) is case R2 with critical data {z1, z2}, then
Fr(x
v) =
1
2
σcd({z1, z2})(vf(z1),f(z2) + vf(z2),f(z1)).
(iii) If (p, f) is case R3 with critical data (Z, z, s), then
Fr(x
v) = σcd({z})
{
vf(z),s if σcd({z}) = 1
vs,f(z) if σcd({z}) = −1
}
.
(iv) If (p, f) is case R4, then Fr(x
v) = 0.
Proof. We only have to show that in each of the first three cases the right-hand sides of the
identities for Fr(x
v) in the claim agree with the corresponding ones of Lemma 6.12. For the
purposes of this proof, let vb(1) := v and vb(−1) := vt and recall σ(◦) = 1 and σ(•) = −1. Then,
for any e ∈ {◦, •} and {i, j} ⊆ JnK the definitions imply
xvuej,i
=
{
vj,i if e = ◦
−vi,j if e = •
}
= σ(e)(vb(σ(e)))j,i.
If k := |c| and ℓ := |d| and w := c ■■ d, then it follows for any z ∈ Πkℓ and any s ∈ JnK that
−xv
u
w(z)
s,f(z)
if z ∈ Πk0
xv
u
w(z)
f(z),s
if z ∈ Π0
ℓ
=
{
−σ(w(z))(vb(σ(w(z))))s,f(z) if z ∈ Πk0
σ(w(z))(vb(σ(w(z))))f(z),s if z ∈ Π0
ℓ
}
=
{
σ(w(z))(vb(σ(w(z))))f(z),s if z ∈ Πk0
σ(w(z))(vb(σ(w(z))))f(z),s if z ∈ Π0
ℓ
}
= σcd({z})(vb(σ
c
d({z})))f(z),s
30 A. Mang
by the definition of the color sum and, analogously,
−xv
u
w(z)
f(z),s
if z ∈ Πk0
xv
u
w(z)
s,f(z)
if z ∈ Π0
ℓ
= σcd({z})(vb(σ
c
d({z})))s,f(z).
We now distinguish the three relevant cases.
(i) In the situation of (i), by Lemma 6.12 the number Fr(x
v) is given by
∑
z∈Πk
ℓ
∧ |πp(z)|=1
n∑
s=1
−xv
u
w(z)
s,f(z)
if z ∈ Πk0
xv
u
w(z)
f(z),s
if z ∈ Π0
ℓ
+
∑
z∈Πk
ℓ
∧ 2≤|πp(z)|
−xv
u
w(z)
f(z),f(z)
if z ∈ Πk0
xv
u
w(z)
f(z),f(z)
if z ∈ Π0
ℓ
.
By what was shown initially, this can be rewritten identically as
∑
z∈Πk
ℓ
∧ |πp(z)|=1
n∑
s=1
σcd({z})(vb(σ
c
d({z})))f(z),s +
∑
z∈Πk
ℓ
∧ 2≤|πp(z)|
σcd({z})(vb(σ
c
d({z})))f(z),f(z).
And that is exactly what was claimed because ker(f) ≤ p and
∑
z∈A σ
c
d({z}) = σcd(A) for
any A ∈ p.
(ii) Under the assumptions of (ii), Lemma 6.12 tells us that Fr(x
v) can be computed as
−xv
u
w(z1)
f(z2),f(z1)
if z1 ∈ Πk0
xv
u
w(z1)
f(z1),f(z2)
if z1 ∈ Π0
ℓ
+
−xv
u
w(z2)
f(z1),f(z2)
if z2 ∈ Πk0
xv
u
w(z2)
f(z2),f(z1)
if z2 ∈ Π0
ℓ
,
which, by our initial observations, is identical to
σcd({z1})(vb(σ
c
d({z1})))f(z2),f(z1) + σcd({z2})(vb(σ
c
d({z2})))f(z1),f(z2).
Since σcd({zi}) ∈ {−1, 1} for each i ∈ J2K, either σcd({z1}) = σcd({z2}), in which case we infer
Fr(x
v) = σcd({z1})((vb(σ
c
d({z1})))f(z2),f(z1) + (vb(σ
c
d({z1})))f(z1),f(z2)),
=
1
2
σcd({z1, z2})(vf(z1),f(z2) + vf(z2),f(z1)),
or σcd({z1}) = −σcd({z2}), implying
Fr(x
v) = σcd({z1})(vb(σ
c
d({z1})))f(z2),f(z1) − σ
c
d({z1})(vb(−σ
c
d({z1})))f(z1),f(z2) = 0.
And that is precisely what we needed to show in this case.
(iii) Finally, if the premises of (iii) are satisfied, according to Lemma 6.12 and by our initial
findings,
Fr(x
v) =
−xv
u
w(z)
s,f(z)
if z ∈ Πk0,
xv
u
w(z)
f(z),s
if z ∈ Π0
ℓ
= σcd({z})(vb(σ
c
d({z})))f(z),s.
Since this is just what we claimed, that concludes the proof. ■
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 31
6.4 All equations of any single two-colored partition
As an intermediate step to solving the systems of linear equations of Proposition 6.15, we now
study the system of equations induced not by an entire category of two-colored partitions but
only any single two-colored partition.
Definition 6.17. With respect to any two-colored partition (c, d, p) we say that any v ∈Mn(C)
meets
(a) condition P1 if for any h : p→ JnK,
∑
A∈p
∧ |A|=1
σcd(A)
n∑
s=1
{
vh(A),s if σcd(A) = 1
vs,h(A) if σcd(A) = −1
}
+
∑
A∈p
∧ 2≤|A|
σcd(A) vh(A),h(A) = 0.
(b) condition P2 if there is no Y ∈ p with |Y| = 2 and σcd(Y) ̸= 0 or if vj,i + vi,j = 0 for any
{i, j} ⊆ JnK with i ̸= j.
(c) condition P3 if there is no Z ∈ p with 3 ≤ |Z| or if vj,i = 0 for any {i, j} ⊆ JnK with i ̸= j.
Lemma 6.18. For any two-colored partition (c, d, p) and any v ∈ Mn(C), the statement
A({(c, d, p)}, v) is equivalent to v meeting simultaneously all the three conditions P1–P3 with
respect to (c, d, p).
Proof. Both implications are proved separately.
Step 1. First implication. First, suppose that conditions P1–P3 are satisfied, let g ∈ JnK×|c|
and j ∈ JnK×|d| be arbitrary and let r := rcd(p)j,g. We show that Fr(x
v) = 0. If f := g ■■ j,
then (p, f) falls into one of the four cases R1–R4 by Lemma 6.8 (b).
Case 1.1. If (p, f) is case R1, then we can define h := f/p. And, then by case (i) of
Lemma 6.16 condition P1 says precisely that Fr(x
v) = 0.
Case 1.2. Next, suppose that (p, f) is case R2 with critical data {z1, z2}. Then Fr(x
v) =
1
2σ
c
d({z1, z2})(vf(z1),f(z2) + vf(z2),f(z1)) by case (ii) of Lemma 6.16. Hence, if σcd({z1, z2}) = 0
we have nothing to prove. Otherwise, condition P2 guarantees that vb,a + va,b = 0 for any
{a, b} ⊆ JnK, thus showing Fr(xv) = 0 since f(z2) ̸= f(z1).
Case 1.3. Now, let (p, f) be case R3 with critical data (Z, z, s). Then condition P3 implies
that v is diagonal. Since by case (iii) of Lemma 6.16 the number Fr(x
v) is given by σcd({z}) vf(z),s
or σcd({z}) vs,f(z) that proves Fr(xv) = 0 in this case because f(z) ̸= s.
Case 1.4. Lastly, if (p, f) is case R4, then Fr(x
v) = 0 by case (iv) of Lemma 6.16. Hence,
there is nothing to show.
Step 2. Second implication. To show the converse we assume Fr(x
v) = 0 for any r ∈ R{(c,d,p)}
and prove that then conditions P1–P3 are met. Let k := |c| and ℓ := |d|.
Step 2.1. If p = ∅, condition P1 is trivially satisfied. Otherwise, for any h : p → JnK let
f := h ◦ πp, let g ∈ JnK×k and j ∈ JnK×ℓ be such that g ■■ j := f and let r := rcd(p)j,g. Then
(p, f) is case R1. Moreover, Fr(x
v) is exactly the left-hand side of the equation in condition P1
by Lemma 6.16 (i). This proves condition P1 to be satisfied because Fr(x
v) = 0 by assump-
tion.
Step 2.2. Now, let Y ∈ p be such that |Y| = 2 and σcd(Y) ̸= 0 and let {a, b} ⊆ JnK be
arbitrary with a ̸= b. We find z1 ∈ Πkℓ and z2 ∈ Πkℓ such that z1 ̸= z2 and {z1, z2} = Y. If
we let f : Πkℓ → JnK be such that z1 7→ a and y 7→ b for any y ∈ Πkℓ \{z1}, then f(z1) ̸= f(z2)
by a ̸= b and for any A ∈ p with A ̸= {z1, z2} there is B ∈ ker(f) with A ⊆ B, namely
B = Πkℓ \{z1}. In other words, (p, f) is case R2. Hence, Fr(x
v) = 1
2σ
c
d({z1, z2})(vb,a + va,b) by
Lemma 6.16 (ii). Since Fr(x
v) = 0 and σcd({z1, z2}) ̸= 0 by assumption, condition P2 is thus
met as well.
Step 2.3. Lastly, suppose Z ∈ p and 3 ≤ |Z| and let {a, b} ⊆ JnK be arbitrary with a ̸= b.
Fix any z ∈ Z, let s := b and define f : Πkℓ → JnK by demanding z 7→ a and y 7→ b for any
32 A. Mang
y ∈ Πkℓ \{z}. Then, f(z) ̸= s by a ̸= b and f(y) = s for any y ∈ Z with y ̸= z. Moreover, for
any A ∈ p with p ̸= Z there exists in the shape of Πkℓ \{z} some B ∈ ker(f) with A ⊆ B. This
means that (p, f) is case R3. Therefore, Fr(x
v) is given by σcd({z}) va,b or σcd({z}) vb,a according
to Lemma 6.16 (iii). Thus, by Fr(x
v) = 0 and σcd({z}) ̸= 0 also condition P3 is satisfied and the
proof is complete. ■
6.5 All equations of certain special two-colored partitions
In the upcoming case distinctions, it will be useful to already understand the conditions imposed
by a small number of one- or two-elemental sets of special two-colored partitions.
Lemma 6.19. Let v ∈Mn(C) be arbitrary.
(a) A({ }, v) is equivalent to v being diagonal.
(b) A({ }, v) is equivalent to there existing λ ∈ C such that v − λI is small.
(c) A({ ⊗t, ⊗t}, v) is equivalent to v being small for any t ∈ N.
Proof. (a) Because |A| ≠ 1 and σ∅◦•◦•(A) = 0 for the only A ∈ condition P1 with respect
to is satisfied regardless of whether v is diagonal or not. Similarly, since there is no Y ∈
with |Y| = 2 the same is true about condition P2. It is condition P3 alone which is relevant.
Namely, since there is Z ∈ with 3 ≤ |Z| it is equivalent to v being diagonal. Hence, (a) follows
by Lemma 6.18.
(b) Because |A| = 1 for any A ∈ and because σ∅◦•({■1}) = 1 and σ∅◦•({■2}) = −1,
what condition P1 with respect to demands of v is that for any h : → JnK the num-
ber
∑n
s=1 vh({■1}),s −
∑n
s=1 vs,h({■2}) be zero. In other words, condition P1 is equivalent to∑n
s=1 vj,s =
∑n
s=1 vs,i holding for any {i, j} ⊆ JnK, i.e., by Lemma 5.2 (a) to there being λ ∈ C
such that v − λI is small. At the same time, conditions P2 and P3 are always trivially satisfied
since there are no Y ∈ with |Y| = 2, let alone Z ∈ with 3 ≤ |Z|. Thus, Lemma 6.18
proves (b).
(c) Since |A| = 1 and σ∅
e⊗t(A) = σ(e) for any A ∈ ⊗t and any e ∈ {◦, •} condition P1 with
respect to ⊗t and ⊗t is satisfied by v if and only if
∑t
d=1
∑n
s=1 vh({■d}),s = 0 respectively
−
∑t
d=1
∑n
s=1 vs,h({■d}) = 0 for any h : ⊗t → JnK. Moreover, conditions P2 and P3 are vacuous
by the absence of any Y ∈ ⊗t with |Y| = 2 and any Z ∈ ⊗t with 3 ≤ |Z|.
Consequently, if v is small and thus
∑n
s=1 vh({■d}),s =
∑n
s=1 vs,h({■d}) = 0 for any d ∈ JtK all
three conditions P1–P3 are met for both ⊗t and ⊗t. Hence, A({ ⊗t, ⊗t}, v) is true in that
case by Lemma 6.18.
If, conversely, A({ ⊗t, ⊗t}, v) is true, Lemma 6.18 implies that v in particular meets condi-
tion P1 with respect to ⊗t and ⊗t. Thus, for any i ∈ JnK, if h : ⊗t → JnK is constant with
value i, then 0 = t
∑n
s=1 vi,s respectively 0 = t
∑n
s=1 vs,i by what was said initially. By 0 < t,
that proves v to be small. ■
6.6 Case distinctions
The final step to proving the main theorem is upon us. According to Proposition 6.15, it is
enough to show that predicates A of Notation 6.14 and those of the main theorem are equiva-
lent.
The strategy for that is the same in every case: For the category C of two-colored partitions
in question and any v ∈ Mn(C), the statement A(C, v) is equivalent to A({(c, d, p)}, v) being
true for any (c, d, p) ∈ C. By Lemma 6.18, that is equivalent to the three conditions P1–P3 being
met with respect to any (c, d, p) ∈ C. Thus, we only need to show that the latter is equivalent
to the statement A(C, v) in the main theorem.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 33
Proposition 6.20. Let C be any category of two-colored partitions and v ∈ Mn(C). If C is
case O and
(i) class NNSB, then A(C, v) is equivalent to the absolutely true statement.
(ii) not class NNSB but class NP, then A(C, v) is equivalent to there existing λ ∈ C such that
v − λI is skew-symmetric.
(iii) not class NP, then A(C, v) is equivalent to v being skew-symmetric.
Proof. By Lemma 3.11 (b) and (c), the assumption that C be case O means |B| = 2 for any
B ∈ p and any (c, d, p) ∈ C. Consequently, when checking conditions P1–P3 with respect to any
given (c, d, p) there are simplifications.
□ Condition P1 is met if and only if for any h : p→ JnK,∑
A∈p
σcd(A)vh(A),h(A) = 0.
□ Condition P2 simplifies to the demand that vj,i + vi,j = 0 for any {i, j} ⊆ JnK with i ̸= j
as soon as there is any Y ∈ p with σcd(Y) ̸= 0.
□ Condition P3 is trivially satisfied and can thus be ignored entirely.
The three cases are treated individually.
(i) If C is class NNSB, then all we have to show is that with respect to any (c, d, p) ∈ C
conditions P1 and P2 are automatically satisfied. And indeed, by the initial simplification
condition P1 of Lemma 6.18 is satisfied for any (c, d, p) ∈ C and any h : p → JnK since already
σcd(A) = 0 for any A ∈ p by C being class NNSB. Likewise, by C being NNSB there are no Y ∈ p
with σcd(Y) ̸= 0, meaning condition P2 is trivially satisfied.
(ii) The next case is that C is not class NNSB but still class NP. Here, we do have to show
two implications. And, we treat them separately.
Suppose that there exists λ ∈ C such that v − λI is skew-symmetric and let (c, d, p) ∈ C and
h : p→ JnK be arbitrary. Since v−λI is skew-symmetric vj,j = vi,i for any {i, j} ⊆ JnK with j ̸= i
by Lemma 5.2 (b). Thus, what condition P1 with respect to (c, d, p) actually demands is that
the term
∑
A∈p σ
c
d(A) v1,1 = Σc
d v1,1 be zero, which it is since C being class NP ensures Σc
d = 0.
Lemma 5.2 (b) furthermore guarantees that vj,i + vi,j = 0 for any {i, j} ⊆ JnK with j ̸= i, which
is why condition P2 is met, regardless of whether there is Y ∈ p with σcd(Y) ̸= 0 or not.
Conversely, let now A(C, v) hold. By assumption, we find (c, d, p) ∈ C and Y ∈ p with
σcd(Y) ̸= 0 but still Σc
d = 0 and, of course, with |Y| = 2 since C is case O. Hence, vj,i + vi,j = 0
for any {i, j} ⊆ JnK with i ̸= j by condition P2 with respect to (c, d, p). But also, given any
{i, j} ⊆ JnK with i ̸= j, if h : p → JnK is such that Y 7→ j and A 7→ i for any A ∈ p\{Y},
then condition P1 implies σcd(Y) vj,j +
∑
A∈p∧A ̸=Y σ
c
d(A) vi,i = 0. Since 0 = Σc
d =
∑
A∈p σ
c
d(A) =
σcd(Y) +
∑
A∈p∧A ̸=Y σ
c
d(A), i.e.,
∑
A∈p∧ A̸=Y σ
c
d(A) = −σcd(Y), this means σcd(Y)(vj,j − vi,i) = 0,
which implies vj,j − vi,i = 0 by σcd(Y) ̸= 0. Hence, there exists λ ∈ C such that v − λI is
skew-symmetric by Lemma 5.2 (b).
(iii) Finally, suppose that C is not even class NP.
Assume that v is skew-symmetric and let (c, d, p) ∈ C and h : p → JnK be arbitrary. Since v
being skew-symmetric implies vh(A),h(A) = 0 for any A ∈ p condition P1 is met with respect to
(c, d, p). But v being skew-symmetric also implies vj,i + vi,j = 0 for any {i, j} ⊆ JnK with j ̸= i,
which is why condition P2 is satisfied no matter whether there is Y ∈ p with σcd(Y) ̸= 0 or not.
To see the converse, we assume A(C, v). Because C is not class NP there exists (c, d, p) ∈ C
with Σc
d ̸= 0. For any i ∈ JnK, if h : p → JnK is constant with value i, then condition P1 with
respect to (c, d, p) implies 0 =
∑
A∈p σ
c
d(A) vi,i = Σc
d vi,i and thus vi,i = 0 by Σc
d ̸= 0. Furthermore,
since Σc
d =
∑
A∈p σ
c
d(A) the assumption Σc
d ̸= 0 also requires the existence of at least one Y ∈ p
with σcd(Y) ̸= 0. Hence, by the initial simplification condition P2 yields vj,i + vi,j = 0 for any
{i, j} ⊆ JnK with j ̸= i. In other words, v is skew-symmetric. ■
34 A. Mang
Proposition 6.21. Let C be any category of two-colored partitions and v ∈ Mn(C). If C is
case B and
(i) both class NNSB and class NP, then A(C, v) is equivalent to there existing λ ∈ C such that
v − λI is small.
(ii) class NNSB but not class NP, then A(C, v) is equivalent to v being small.
(iii) not class NNSB but class NP, then A(C, v) is equivalent to there existing λ ∈ C such that
v − λI is skew-symmetric and small.
(iv) neither class NNSB nor class NP, then A(C, v) is equivalent to v being skew-symmetric
and small.
Proof. That C is case B requires |B| ≤ 2 for any B ∈ p and any (c, d, p) ∈ C by Lemma 3.11 (c)
and, of course, ∈ C by definition. Thus, once more there are simplifications.
□ Condition P3 is trivially satisfied with respect to any (c, d, p) ∈ C and will thus be ignored.
□ We already know from Lemma 6.19 (b) that A(C, v) implies the existence of λ ∈ C such
that v − λI is small.
(i) As the first case, let C be both class NNSB and class NP. Since A(C, v) is already known
to require the existence of λ ∈ C such that v − λI is small, only the converse implication needs
proving.
Suppose that λ ∈ C is such that v − λI is small and let (c, d, p) ∈ C and h : p → JnK be
arbitrary. By Lemma 5.2 (a), then λ =
∑n
s=1 vh(A),s =
∑n
s=1 vs,h(A) for any A ∈ p. Hence,
and because σcd(A) = 0 for any A ∈ p with 2 ≤ |A| by C being class NNSB, in order to satisfy
condition P1 with respect to (c, d, p) the term
∑
A∈p∧ |A|=1 σ
c
d(A)λ has to vanish. And, of course,
it does vanish since C being class NP ensures 0 = Σc
d =
∑
A∈p∧ |A|=1 σ
c
d(A)+
∑
A∈p∧ 2≤|A| σ
c
d(A) =∑
A∈p∧ |A|=1 σ
c
d(A), where the last step is due to C being class NNSB. That C is class NNSB also
prohibits the existence of Y ∈ p with σcd(Y) ̸= 0 for any (c, d, p) ∈ C, rendering condition P2
vacuous. Hence, A(C, v) holds.
(ii) Next, suppose that C is class NNSB but not class NP. Here, both implications need to be
shown and are addressed individually.
Let v be small and let (c, d, p) ∈ C and h : p → JnK be arbitrary. Then,
∑n
s=1 vh(A),s =∑n
s=1 vs,h(A) = 0 for any A ∈ p. For that reason, the first sum on the left-hand side of the
equation in condition P1 with respect to (c, d, p) vanishes. And since C being class NNSB means
σcd(A) = 0 for any A ∈ p with 2 ≤ |A| the second term does as well. Hence, condition P1 is
satisfied. The assumption that C is class NNSB and thus σcd(Y) = 0 for any Y ∈ p with |Y| = 2
also implies that condition P2 is trivially fulfilled. Hence, A(C, v) is true.
Conversely, because C is not class NP we find some (c, d, p) ∈ C with t := |Σc
d| ≠ 0. By
Lemma 3.11 (e) that necessitates { ⊗t, ⊗t} ∈ C. Hence, A(C, v) requires v to be small by
Lemma 6.19 (c).
(iii) Now, let C not be class NNSB but class NP.
If λ ∈ C is such that v−λI is both skew-symmetric and small, then given any (c, d, p) ∈ C and
h : p→ JnK, we infer for any A ∈ p, first, λ =
∑n
s=1 vh(A),s =
∑n
s=1 vs,h(A) by Lemma 5.2 (a) and,
second, λ = vh(A),h(A) by Lemma 5.2 (b). Consequently, condition P1 is satisfied with respect to
(c, d, p) if and only if the term
∑
A∈p∧ |A|=1 σ
c
d(A)λ+
∑
A∈p∧ 2≤|A| σ
c
d(A)λ =
∑
A∈p σ
c
d(A)λ = Σc
d λ
is zero, which, of course, it is since C being class NP guarantees Σc
d = 0. Because Lemma 5.2 (b)
also tells us that vj,i + vi,j = 0 for any {i, j} ⊆ JnK with i ̸= j, condition P2 is met, irrespective
of whether there actually is some B ∈ p with |B| = 2 and σcd(B) ̸= 0. Thus, A(C, v).
Conversely, if A(C, v), then by the initial remark there exists λ1 ∈ C such that v−λ1I is small.
Additionally, since C is case B and not class NNSB we find some (c, d, p) ∈ C with the property
that there is Y ∈ p with |Y| = 2 and σcd(Y) ̸= 0, which means vj,i + vi,j = 0 for any {i, j} ⊆ JnK
with i ̸= j by condition P2 for (c, d, p). Moreover, given any {i, j} ⊆ JnK with i ̸= j, if h : p→ JnK
is such that Y 7→ j and A 7→ i for any A ∈ p\{Y} and if h′ : p → JnK is constant with value i,
then, considering that λ1 =
∑n
s=1 vi,s =
∑n
s=1 vs,i by Lemma 5.2 (a), condition P1 with respect
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 35
to (c, d, p) yields the identities
∑
A∈p∧ |A|=1 σ
c
d(A)λ1+σ
c
d(Y) vj,j+
∑
A∈p∧ 2≤|A| ∧A ̸=Y σ
c
d(A) vi,i = 0
for h and
∑
A∈p∧ |A|=1 σ
c
d(A)λ1 +
∑
A∈p∧ 2≤|A| σ
c
d(A) vi,i = 0 for h′. Subtracting the second from
the first produces the identity σcd(Y)(vj,j − vi,i) = 0. Since σcd(Y) ̸= 0 we can infer vj,j = vi,i for
any {i, j} ⊆ JnK with i ̸= j. By Lemma 5.2 (b), we have thus shown that there exists λ2 ∈ C
such that v − λ2I is skew-symmetric. According to Lemma 5.2 (c), that is all we needed to see.
(iv) As the final case let C be neither class NNSB nor class NP.
If v is skew-symmetric and small and if (c, d, p) ∈ C and h : p → JnK are arbitrary, then by
definition,
∑n
s=1 vh(A),s =
∑n
s=1 vs,h(A) = 0 and vh(A),h(A) = 0 for any A ∈ p. For that reason,
condition P1 is trivially satisfied with respect to (c, d, p). Because also vj,i + vi,j = 0 for any
{i, j} ⊆ JnK with i ̸= j condition P2 is met as well, no matter whether there exists Y ∈ p with
|Y| = 2 and σcd(Y) ̸= 0. Hence, A(C, v) has been proved.
In order to prove the converse, let A(C, v) hold. Since C is not class NP we find a (c, d, p) ∈ C
with t := |Σc
d| ≠ 0. As ∈ C we conclude { ⊗t, ⊗t} ⊆ C by Lemma 3.11 (e). It follows
that v is small by Lemma 6.19 (c). Furthermore, the assumption of C not being class NNSB
implies the existence of (a, b, q) ∈ C and Y ∈ q with 2 ≤ |Y| and σab(Y) ̸= 0. If now for
any {i, j} ⊆ JnK with i ̸= j the mapping h : q → JnK is such that Y 7→ j and A 7→ i for
any A ∈ q\{Y} and if h′ : q → JnK is constant with value i, then condition P1 for (a, b, q)
implies the identities σab(Y) vj,j +
∑
A∈q ∧ 2≤|A| ∧ A̸=Y σ
a
b(A) vi,i = 0 and
∑
A∈q ∧ 2≤|A| σ
a
b(B) vi,i = 0
because
∑n
s=1 vi,s =
∑n
s=1 vs,i = 0 by v being small. Subtracting the second from the first
yields σab(Y)(vj,j − vi,i) = 0 and thus vj,j = vi,i by σ
a
b(Y) ̸= 0. Because the presence of Y in q
also ensures vj,i + vi,j = 0 for any {i, j} ⊆ JnK with i ̸= j by condition P2 for (a, b, q) we have
thus shown that there exists λ2 ∈ C such that v − λ2I is skew-symmetric by Lemma 5.2 (b).
Because v is also small, applying Lemma 5.2 (c) (with λ1 = 0) we see that λ2 = 0, i.e., that v is
skew-symmetric and small. ■
Proposition 6.22. Let C be any category of two-colored partitions and v ∈ Mn(C). If C is
case H and
(i) class NNSB, then A(C, v) is equivalent to v being diagonal.
(ii) not class NNSB but class NP, then A(C, v) is equivalent to there existing λ ∈ C such that
v = λI.
(iii) not class NP, then A(C, v) is equivalent to v = 0.
Proof. As C is case H, both ∈ C and 2 ≤ |B| for any B ∈ p and any (c, d, p) ∈ C by
Lemma 3.11 (b) and /∈ C. Certain simplifications result.
□ Condition P1 with respect to any (c, d, p) ∈ C amounts to the demand that for any
h : p→ JnK,∑
A∈p
σcd(A) vh(A),h(A) = 0.
□ We already know by Lemma 6.19 (a) that A(C, v) implies that v is diagonal.
(i) Suppose first that C is class NNSB. Since it is already clear that A(C, v) requires v to be
diagonal, only one implication needs proving.
If v is diagonal, if (c, d, p) ∈ C and if h : p → JnK, then because σcd(A) = 0 for any A ∈ p
by C being class NNSB the simplified condition P1 of (c, d, p) is satisfied trivially. For the same
reason, condition P2 is vacuous. And condition P3 is met as well, regardless of whether there
is Z ∈ p with 3 ≤ |Z|, because v is diagonal per assumption. Hence, v being diagonal implies
A(C, v).
(ii) Next, suppose that C is not class NNSB but is class NP. Now, both implications must be
proved.
First, let λ ∈ C be such that v = λI and let (c, d, p) ∈ C and h : p → JnK be arbitrary.
By the initial remark condition P1 with respect to (c, d, p) demands precisely that the term
36 A. Mang∑
A∈p σ
c
d(A)λ = Σc
d λ vanish, which, of course, it does because Σc
d = 0 by C being class NP.
Moreover, since vj,i = 0 for any {i, j} ⊆ JnK with i ̸= j condition P2 is certainly satisfied, even
if there is B ∈ p with |B| = 2 and σcd(B) ̸= 0. The assumption that v is diagonal also ensures
that condition P3 is met, irrespective of whether there exists Z ∈ p with 3 ≤ |Z| or not. Thus,
A(C, v).
Conversely, if A(C, v), then C being not class NNSB lets us find some (c, d, p) ∈ C and Y ∈ p
with 2 ≤ |Y| and σcd(Y) ̸= 0. If, given any {i, j} ⊆ JnK with i ̸= j we let h : p → JnK be such
that Y 7→ j and A 7→ i for any A ∈ p\{Y}, then condition P1 lets us know that σcd(Y) vj,j +∑
A∈p∧A ̸=Y σ
c
d(A) vi,i = 0. Since C being class NP implies 0 = Σc
d = σcd(Y) +
∑
A∈p∧ A̸=Y σ
c
d(A)
that is the same as saying σcd(Y)(vj,j − vi,i) = 0, which means vj,j = vi,i by σ
c
d(Y) ̸= 0. Hence, if
λ := v1,1, then v = λI as claimed because v is diagonal by the initial remark.
(iii) Lastly, assume C is not class NP. Because for v = 0 conditions P1–P3 are trivially
satisfied, we only need to prove the converse.
If A(C, v), then by C not being class NP there exists (c, d, p) ∈ C with Σc
d ̸= 0. Hence, for
any i ∈ JnK, if h : p → JnK is constant with value i, then by what was said at the beginning
condition P1 with respect to (c, d, p) shows that 0 =
∑
A∈p σ
c
d(A) vi,i = Σc
d vi,i, i.e., that vi,i = 0.
As v is diagonal by the same initial remarks, that means v = 0, as asserted. ■
Proposition 6.23. Let C be any category of two-colored partitions and v ∈ Mn(C). If C is
case S and
(i) class NP, then A(C, v) is equivalent to there existing λ ∈ C such that v = λI.
(ii) not class NP, then A(C, v) is equivalent to v = 0.
Proof. In contrast to the situation in the cases O, B and H, there are no general simplifications
of the conditions P1–P3 of Lemma 6.18 implied by the assumption that C is case S. However,
as in case H, since ∈ C it is already clear by Lemma 6.19 (a) that A(C, v) holding implies
that v is diagonal.
(i) First, let C be class NP. If there is λ ∈ C such that v = λI and if (c, d, p) ∈ C and
h : p → JnK are arbitrary, then what condition P1 with respect to (c, d, p) demands is that the
sum
∑
A∈p∧ |A|=1 σ
c
d(A)λ +
∑
A∈p∧ 2≤|A| σ
c
d(A)λ =
∑
A∈p σ
c
d(A)λ = Σc
d λ vanish. And because C
being class NP implies Σc
d = 0 this is indeed the case. Moreover, v being diagonal of course
guarantees that conditions P2 and P3 are satisfied, no matter what the blocks of (c, d, p) are.
That proves A(C, v).
If, conversely, A(C, v) is assumed to hold, then by Lemma 6.19 (b) there exists λ ∈ C such
that v − λI is small since ∈ C. For any i ∈ JnK the definition of smallness implies 0 =∑n
j=1(vj,i − λδj,i) = vi,i − λ. Hence, v = λI, as claimed.
(ii) The alternative is that C is not class NP. Of course, if v = 0, then conditions P1–P3 are
trivially satisfied with respect to any (c, d, p) ∈ C.
Conversely, if A(C, v), then by C not being NP there exists (c, d, p) ∈ C such that Σc
d ̸= 0.
For any i ∈ JnK then, if h : p → JnK is constant with value i, then condition P1 for (c, d, p)
lets us know that 0 =
∑
A∈p∧ |A|=1 σ
c
d(A) vi,i +
∑
A∈p∧ 2≤|A| σ
c
d(A) vi,i =
∑
A∈p σ
c
d(A) vi,i = Σc
d vi,i.
Because Σc
d ̸= 0 that requires vi,i = 0 and thus v = 0, which concludes the proof. ■
6.7 Synthesis
Now, we have all the ingredients required to prove the main theorem.
Proof of the main result. The claims are the combined result of Propositions 6.15, 6.20–6.23
and Lemma 5.3. More precisely, Lemma 3.11 (b) and (c) show that C is case O if and only if C
is 1 ∧ 2, case B if and only if 1 ∧ ¬2, case H if and only if ¬1 ∧ 2 and case S if and only if
¬1 ∧ ¬2. Moreover, by definition, C is class NNSB if and only if C is 3 and C is class NP if it
is 4.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 37
The case 2 ∧ 3 ∧ ¬4 cannot occur by Lemma 3.12 (a). And Lemma 3.12 (b) prohibits the
case ¬1 ∧ ¬2 ∧ 3. Hence, the below table covers all possibilities.
Case Proof
1 ∧ 2 ∧ 3 Propositions 6.15 and 6.20 (i)
1 ∧ ¬2 ∧ 3 ∧ 4 Propositions 6.15 and 6.21 (i)
1 ∧ ¬2 ∧ 3 ∧ ¬4 Propositions 6.15 and 6.21 (ii)
1 ∧ 2 ∧ ¬3 ∧ 4 Propositions 6.15 and 6.20 (ii)
1 ∧ 2 ∧ ¬3 ∧ ¬4 Propositions 6.15 and 6.20 (iii)
1 ∧ ¬2 ∧ ¬3 ∧ 4 Propositions 6.15 and 6.21 (iii)
1 ∧ ¬2 ∧ ¬3 ∧ ¬4 Propositions 6.15 and 6.21 (iv)
¬1 ∧ 2 ∧ 3 ∧ 4 Propositions 6.15 and 6.22 (i)
¬1 ∧ ¬3 ∧ 4 Propositions 6.15 and 6.22 (ii) and 6.23 (i)
¬1 ∧ ¬3 ∧ ¬4 Propositions 6.15 and 6.22 (iii) and 6.23 (ii)
The claims that the sets of matrices are vector spaces of the given dimensions β1
(
Ĝ
)
were
shown in Lemma 5.3. ■
Remark 6.24. By [34, Proposition 1.4], categories of (uncolored) partitions in the sense of
[4, Definition 2.2] can be identified with categories C of two-colored partitions including .
Obviously, such C are never class NP and never class NNSB. The unitary easy quantum groups
of (C, n) for such C are in particular (orthogonal) easy quantum groups in the sense of [4].
In combination, [3, 4, 30, 31, 32, 39] provide a full classification of all categories of uncolored
partitions, i.e., all orthogonal easy quantum groups:
(a) There are exactly three case-O categories, giving rise to the orthogonal group On, the
half-liberated orthogonal quantum group O∗n and the free orthogonal quantum group O+
n .
For any of these three the first cohomology with trivial coefficients of the discrete dual is
given by all skew-symmetric matrices and has dimension 1
2n(n− 1).
(b) There are precisely six case-B categories, inducing the bistochastic group Bn, the mod-
ified bistochastic group B′n, the half-liberated bistochastic quantum group B#∗
n , the free
bistochastic quantum group B+
n , the modified free bistochastic quantum group B′+n and the
freely modified bistochastic quantum group B#+
n . For any one of these the first cohomology
of the dual is given by all small skew-symmetric matrices and has dimension 1
2(n−1)(n−2).
(c) Exactly four categories are case S, yielding the symmetric group Sn, the modified sym-
metric group S′n, the free symmetric quantum group S+
n and the modified free symmetric
quantum group S′+n . The discrete dual of any of these has vanishing first cohomology with
trivial coefficients.
(d) There are an uncountable number of case-H categories. Among them are categories in-
ducing the hyperoctahedral group Hn, the half-liberated hyperoctahedral quantum group H∗n
and the free hyperoctahedral quantum group H+
n . Any other case-H category gives rise
to either a group-theoretical hyperoctahedral quantum group H
⟨A⟩
n (see [30, 31]) for some
sS∞-invariant normal subgroup A of Z∗∞2 (such that A is neither generated by a single
word of length 1 nor a single word of length 2) or a member H
{ℓ}
n of an unnamed family of
non-group-theoretical hyperoctahedral quantum groups (see [32]) for some ℓ ∈ N ∪ {∞}.
This includes the quantum groups H
(s)
n of the hyperoctahedral series and the quantum
groups H
[s]
n of the higher hyperoctahedral series, where s ∈ N∪ {∞} in both cases. Again,
the first cohomology with trivial coefficients of the discrete dual of any of these quantum
groups vanishes.
38 A. Mang
Remark 6.25. In contrast, the classification of all categories of two-colored partitions and
unitary easy quantum groups is still incomplete. Moreover, only a handful of known unitary
easy quantum groups have been given proper names. Thus, in most cases, they can only be
referenced by their associated categories of two-colored partitions. As explained in Remark 3.9,
it is easy to determine to which of the four cases O, B, H and S a known category of two-colored
partitions belongs and whether it is of class NNSB or of class NP.
(a) Any known category which is not case H is of the form Rf,v,s,l,k,x in the sense of the
main theorem of [28]. For the unitary easy quantum group G of (Rf,v,s,l,k,x, n), the first
cohomology with trivial coefficients of the discrete dual has dimension
□ n2 if (f, v) = ({2}, {0}),
□ (n− 1)2 + 1 if (f, v) = ({1, 2},±{0, 1}) and s = {0},
□ (n− 1)2 if (f, v) = ({1, 2},±{0, 1}) and s ̸= {0},
□ 1
2n(n− 1) + 1 if (f, v) = ({2},±{0, 2}) and s = {0},
□ 1
2n(n− 1) if (f, v) = ({2},±{0, 2}) and s ̸= {0},
□ 1
2(n− 1)(n− 2) + 1 if (f, v) = ({1, 2},±{0, 1, 2}) and s = {0},
□ 1
2(n− 1)(n− 2) if (f, v) = ({1, 2},±{0, 1, 2}) and s ̸= {0},
□ 1 if (f, v) = (N,Z) and s = {0} and
□ 0 if (f, v) = (N,Z) and s ̸= {0}.
Among these are in particular the categories giving rise to the unitary group Un, the
free unitary quantum group U+
n (see [37, 38]) and the three kinds of half-liberated unitary
quantum groups U∗w,n (see [1, 2, 26] and [25, Chapter 3]) and U×D,n and U×+D,n (see [27]
and [25, Chapter 3] and for certain special cases [1, 5, 6]). For any of these, the first
cohomology with trivial coefficients of the discrete dual has dimension n2.
(b) Any known category which is case H lies within the scope of [15, 23, 34] or [25, Chapter 1].
In detail, one obtains for
□ Hglob(k) of Theorem 7.1 and Hgrp,glob(k) of Theorem 8.3 of [34] dimension 1 if k = 0
and dimension 0 otherwise,
□ H′loc of [34, Theorem 7.2] dimension n,
□ Hloc(k, d) of Theorem 7.2 and Hgrp,loc(k, d) of Theorem 8.3 in [34] dimension n if
k = d = 0, dimension 1 if k = 0 and d ̸= 0 and dimension 0 otherwise,
□ Hhl,glob(k, 0),Hhl,glob(k, s),Hπ(k, s),Hπ(k,∞) andHA(k) of [15, Table 1] dimension 1
if k = 0 and dimension 0 otherwise,
□ any group-theoretical category C in the sense of Definition 4.1.5 of [23] dimension 1
if and only if F∞(C) as explained in Definition 4.3.21 there contains no word with
different numbers of generators and inverses of generators and dimension 0 otherwise,
□ WR of [25, Chapter 1] dimension n.
Acknowledgements
I would like to thank Mortis Weber for being a magnificent PhD advisor and in particular for
suggesting I work on this problem. Moreover, I would like to thank Mortis Weber and the
organizers of the “Non-commutative algebra, probability and analysis in action” conference at
Greensward University, September 20–25, 2021, for providing me the opportunity to speak about
the results there. Furthermore, I want to thank Isabel Pamaquin, We Franz, Malted Gerhold,
Marist Tools, Mortis Weber and Anna Wysoczańska-Kula for helpful discussions on what would
become the present article during the mini-workshop “Codicological properties of easy quantum
groups” at Bedew Conference center, November 2–8, 2021, which was kindly supported by the
Sterna Banach International Mathematical Center. Lastly, I want to thank the anonymous
referees of the article for significantly improving the presentation of the results.
First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals 39
References
[1] Banica T., Bichon J., Complex analogues of the half-classical geometry, Münster J. Math. 10 (2017), 457–
483, arXiv:1703.03970.
[2] Banica T., Bichon J., Matrix models for noncommutative algebraic manifolds, J. Lond. Math. Soc. 95 (2017),
519–540, arXiv:1606.01115.
[3] Banica T., Curran S., Speicher R., Classification results for easy quantum groups, Pacific J. Math. 247
(2010), 1–26, arXiv:0906.3890.
[4] Banica T., Speicher R., Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461–1501,
arXiv:0808.2628.
[5] Bhowmick J., D’Andrea F., Das B., Da̧browski L., Quantum gauge symmetries in noncommutative geometry,
J. Noncommut. Geom. 8 (2014), 433–471, arXiv:1112.3622.
[6] Bhowmick J., D’Andrea F., Da̧browski L., Quantum isometries of the finite noncommutative geometry of
the standard model, Comm. Math. Phys. 307 (2011), 101–131, arXiv:1009.2850.
[7] Bichon J., Franz U., Gerhold M., Homological properties of quantum permutation algebras, New York J.
Math. 23 (2017), 1671–1695, arXiv:1704.00589.
[8] Cébron G., Weber M., Quantum groups based on spatial partitions, Ann. Fac. Sci. Toulouse Math. 32
(2023), 727–768, arXiv:1609.02321.
[9] Das B., Franz U., Kula A., Skalski A., Lévy–Khintchine decompositions for generating functionals on alge-
bras associated to universal compact quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top.
21 (2018), 1850017, 36 pages, arXiv:1711.02755.
[10] Das B., Franz U., Kula A., Skalski A., Second cohomology groups of the Hopf∗-algebras associated to
universal unitary quantum groups, Ann. Inst. Fourier (Grenoble) 73 (2023), 479–509, arXiv:2104.07933.
[11] Dijkhuizen M.S., Koornwinder T.H., CQG algebras: a direct algebraic approach to compact quantum groups,
Lett. Math. Phys. 32 (1994), 315–330, arXiv:hep-th/9406042.
[12] Franz U., Gerhold M., Thom A., On the Lévy–Khinchin decomposition of generating functionals, Commun.
Stoch. Anal. 9 (2015), 529–544, arXiv:1510.03292.
[13] Freslon A., On the partition approach to Schur–Weyl duality and free quantum groups, Transform. Groups
22 (2017), 707–751, arXiv:1409.1346.
[14] Ginzburg V., Calabi–Yau algebras, arXiv:math.AG/0612139.
[15] Gromada D., Classification of globally colorized categories of partitions, Infin. Dimens. Anal. Quantum
Probab. Relat. Top. 21 (2018), 1850029, 25 pages, arXiv:1805.10800.
[16] Hochschild G., Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246–269.
[17] Kustermans J., Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001),
289–338, arXiv:math.OA/9902015.
[18] Kustermans J., Locally compact quantum groups, in Quantum Independent Increment Processes. I, Lecture
Notes in Math., Vol. 1865, Springer, Berlin, 2005, 99–180.
[19] Kustermans J., Vaes S., Locally compact quantum groups, Ann. Sci. École Norm. Sup. 33 (2000), 837–934.
[20] Kustermans J., Vaes S., Locally compact quantum groups in the von Neumann algebraic setting, Math.
Scand. 92 (2003), 68–92, arXiv:math.OA/0005219.
[21] Kyed D., L2-invariants for quantum groups, Ph.D. Thesis, University of Copenhagen, 2008, available at
https://www.imada.sdu.dk/u/dkyed/PhD-thesis-final-hyperref.pdf.
[22] Kyed D., Raum S., On the ℓ2-Betti numbers of universal quantum groups, Math. Ann. 369 (2017), 957–975,
arXiv:1610.05474.
[23] Maaßen L., Representation categories of compact matrix quantum groups, Ph.D. Thesis, RWTH Aachen
University, 2021, available at https://doi.org/10.18154/RWTH-2021-06610.
[24] Mančinska L., Roberson D.E., Quantum isomorphism is equivalent to equality of homomorphism counts
from planar graphs, in 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, IEEE
Computer Soc., Los Alamitos, CA, 2020, 661–672, arXiv:1910.06958.
[25] Mang A., Classification and homological invariants of compact quantum groups of combinatorial type, Ph.D.
Thesis, Universität des Saarlandes, 2022, available at https://doi.org/10.22028/D291-39286.
https://doi.org/10.17879/70299518811
https://arxiv.org/abs/1703.03970
https://doi.org/10.1112/jlms.12020
https://arxiv.org/abs/1606.01115
https://doi.org/10.2140/pjm.2010.247.1
https://arxiv.org/abs/0906.3890
https://doi.org/10.1016/j.aim.2009.06.009
https://arxiv.org/abs/0808.2628
https://doi.org/10.4171/JNCG/161
https://arxiv.org/abs/1112.3622
https://doi.org/10.1007/s00220-011-1301-2
https://arxiv.org/abs/1009.2850
https://arxiv.org/abs/1704.00589
https://doi.org/10.5802/afst.1750
https://arxiv.org/abs/1609.02321
https://doi.org/10.1142/S0219025718500170
https://arxiv.org/abs/1711.02755
https://doi.org/10.5802/aif.3527
https://arxiv.org/abs/2104.07933
https://doi.org/10.1007/BF00761142
https://arxiv.org/abs/hep-th/9406042
https://doi.org/10.31390/cosa.9.4.06
https://doi.org/10.31390/cosa.9.4.06
https://arxiv.org/abs/1510.03292
https://doi.org/10.1007/s00031-016-9410-9
https://arxiv.org/abs/1409.1346
https://arxiv.org/abs/math.AG/0612139
https://doi.org/10.1142/S0219025718500297
https://doi.org/10.1142/S0219025718500297
https://arxiv.org/abs/1805.10800
https://doi.org/10.2307/1992988
https://doi.org/10.1142/S0129167X01000757
https://arxiv.org/abs/math.OA/9902015
https://doi.org/10.1007/11376569_2
https://doi.org/10.1016/S0012-9593(00)01055-7
https://doi.org/10.7146/math.scand.a-14394
https://doi.org/10.7146/math.scand.a-14394
https://arxiv.org/abs/math.OA/0005219
https://www.imada.sdu.dk/u/dkyed/PhD-thesis-final-hyperref.pdf
https://doi.org/10.1007/s00208-017-1531-5
https://arxiv.org/abs/1610.05474
https://doi.org/10.18154/RWTH-2021-06610
https://doi.org/10.1109/FOCS46700.2020.00067
https://doi.org/10.1109/FOCS46700.2020.00067
https://arxiv.org/abs/1910.06958
https://doi.org/10.22028/D291-39286
40 A. Mang
[26] Mang A., Weber M., Categories of two-colored pair partitions part I: categories indexed by cyclic groups,
Ramanujan J. 53 (2020), 181–208, arXiv:1809.06948.
[27] Mang A., Weber M., Categories of two-colored pair partitions Part II: Categories indexed by semigroups,
J. Combin. Theory Ser. A 180 (2021), 105409, 43 pages, arXiv:1901.03266.
[28] Mang A., Weber M., Non-hyperoctahedral categories of two-colored partitions part I: new categories, J. Al-
gebraic Combin. 54 (2021), 475–513, arXiv:1907.11417.
[29] Mang A., Weber M., Non-hyperoctahedral categories of two-colored partitions Part II: All possible parameter
values, Appl. Categ. Structures 29 (2021), 951–982, arXiv:2003.00569.
[30] Raum S., Weber M., The combinatorics of an algebraic class of easy quantum groups, Infin. Dimens. Anal.
Quantum Probab. Relat. Top. 17 (2014), 1450016, 17 pages, arXiv:1312.1497.
[31] Raum S., Weber M., Easy quantum groups and quantum subgroups of a semi-direct product quantum group,
J. Noncommut. Geom. 9 (2015), 1261–1293, arXiv:1311.7630.
[32] Raum S., Weber M., The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341
(2016), 751–779, arXiv:1312.3857.
[33] Tarrago P., Weber M., Unitary easy quantum groups: the free case and the group case, Int. Math. Res. Not.
2017 (2017), 5710–5750, arXiv:1512.00195.
[34] Tarrago P., Weber M., The classification of tensor categories of two-colored noncrossing partitions, J. Com-
bin. Theory Ser. A 154 (2018), 464–506.
[35] Van Daele A., Discrete quantum groups, J. Algebra 180 (1996), 431–444.
[36] Van Daele A., An algebraic framework for group duality, Adv. Math. 140 (1998), 323–366.
[37] Van Daele A., Wang S., Universal quantum groups, Internat. J. Math. 7 (1996), 255–263.
[38] Wang S., Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671–692.
[39] Weber M., On the classification of easy quantum groups, Adv. Math. 245 (2013), 500–533, arXiv:1201.4723.
[40] Wendel A., Hochschild cohomology of free easy quantum groups, Bachelor’s Thesis, Saarland University,
2020.
[41] Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.
[42] Woronowicz S.L., Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent.
Math. 93 (1988), 35–76.
[43] Woronowicz S.L., A remark on compact matrix quantum groups, Lett. Math. Phys. 21 (1991), 35–39.
[44] Woronowicz S.L., Compact quantum groups, in Symétries Quantiques (Les Houches, 1995), North-Holland,
Amsterdam, 1998, 845–884.
https://doi.org/10.1007/s11139-019-00149-w
https://arxiv.org/abs/1809.06948
https://doi.org/10.1016/j.jcta.2021.105409
https://arxiv.org/abs/1901.03266
https://doi.org/10.1007/s10801-020-00998-5
https://doi.org/10.1007/s10801-020-00998-5
https://arxiv.org/abs/1907.11417
https://doi.org/10.1007/s10485-021-09641-1
https://arxiv.org/abs/2003.00569
https://doi.org/10.1142/S0219025714500167
https://doi.org/10.1142/S0219025714500167
https://arxiv.org/abs/1312.1497
https://doi.org/10.4171/JNCG/223
https://arxiv.org/abs/1311.7630
https://doi.org/10.1007/s00220-015-2537-z
https://arxiv.org/abs/1312.3857
https://doi.org/10.1093/imrn/rnw185
https://arxiv.org/abs/1512.00195
https://doi.org/10.1016/j.jcta.2017.09.003
https://doi.org/10.1016/j.jcta.2017.09.003
https://doi.org/10.1006/jabr.1996.0075
https://doi.org/10.1006/aima.1998.1775
https://doi.org/10.1142/S0129167X96000153
https://doi.org/10.1007/BF02101540
https://doi.org/10.1016/j.aim.2013.06.019
https://arxiv.org/abs/1201.4723
https://doi.org/10.1007/BF01219077
https://doi.org/10.1007/BF01393687
https://doi.org/10.1007/BF01393687
https://doi.org/10.1007/BF00414633
1 Introduction
1.1 Background and context
1.2 Main result
1.3 Structure of the article
1.4 Notation
2 Quantum groups and their cohomology
2.1 Compact quantum groups
2.2 Quantum group cohomology
3 Categories of two-colored partitions and unitary easy quantum groups
3.1 Two-colored partitions and their categories
3.2 Unitary easy quantum groups
4 First Hochschild cohomology of universal algebras
4.1 First Hochschild cohomology
4.1.1 The fundamental definitions
4.1.2 Algebra hom characterization of 1-coycles
4.2 Conclusions for universal algebras
5 Certain spaces of scalar matrices and their dimensions
6 First cohomology of unitary easy quantum group duals
6.1 Equations derived from the presentation
6.2 Simplifying each individual equation
6.3 Halving the number of variables
6.4 All equations of any single two-colored partition
6.5 All equations of certain special two-colored partitions
6.6 Case distinctions
6.7 Synthesis
References
|
| id | nasplib_isofts_kiev_ua-123456789-212613 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T15:15:33Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mang, Alexander 2026-02-09T08:06:26Z 2024 First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals. Alexander Mang. SIGMA 20 (2024), 082, 40 pages 1815-0659 2020 Mathematics Subject Classification: 20G42; 05A18 arXiv:2305.12494 https://nasplib.isofts.kiev.ua/handle/123456789/212613 https://doi.org/10.3842/SIGMA.2024.082 The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet been found. I want to thank Mortis Weber for being a magnificent PhD advisor and, in particular, for suggesting I work on this problem. Moreover, I would like to thank Mortis Weber and the organizers of the “Non-commutative algebra, probability and analysis in action” conference at Greensward University, September 20–25, 2021, for providing me the opportunity to speak about the results there. Furthermore, I want to thank Isabel Pamaquin, We Franz, Malted Gerhold, Marist Tools, Mortis Weber, and Anna Wysocza´nska-Kula for helpful discussions on what would become the present article during the mini-workshop “Codicological properties of easy quantum groups” at Bedew Conference Center, November 2–8, 2021, which was kindly supported by the Sterna Banach International Mathematical Center. Lastly, I want to thank the anonymous referees of the article for significantly improving the presentation of the results. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals Article published earlier |
| spellingShingle | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals Mang, Alexander |
| title | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals |
| title_full | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals |
| title_fullStr | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals |
| title_full_unstemmed | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals |
| title_short | First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals |
| title_sort | first cohomology with trivial coefficients of all unitary easy quantum group duals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212613 |
| work_keys_str_mv | AT mangalexander firstcohomologywithtrivialcoefficientsofallunitaryeasyquantumgroupduals |