Some Consequences of Categorification
Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to d-vectors, g-vectors, and F-polynomials.
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| description | Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to d-vectors, g-vectors, and F-polynomials.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 007, 8 pages
Some Consequences of Categorification
Dylan RUPEL † and Salvatore STELLA ‡
† Pasadena Unified School District, Math Academy, Pasadena, CA 91101, USA
E-mail: rupel.dylan@pusd.us
‡ Università degli studi di Roma “La Sapienza”, Dipartimento di Matematica
“G. Castelnuovo”, P.le Aldo Moro, 5 – 00185 Rome, Italy
E-mail: stella@mat.uniroma1.it
URL: http://www1.mat.uniroma1.it/people/stella/
Received October 16, 2019, in final form January 21, 2020; Published online January 30, 2020
https://doi.org/10.3842/SIGMA.2020.007
Abstract. Several conjectures on acyclic skew-symmetrizable cluster algebras are proven
as direct consequences of their categorification via valued quivers. These include conjectures
of Fomin–Zelevinsky, Reading–Speyer, and Reading–Stella related to d-vectors, g-vectors,
and F -polynomials.
Key words: acyclc cluster algebras; categorification; valued quivers
2020 Mathematics Subject Classification: 13F60; 16G20
1 Introduction
The categorification of skew-symmetric cluster algebras using representations of quivers was
initiated by Marsh, Reineke, and Zelevinsky [21] for Dynkin quivers. With the advent of cluster
characters [3], the subject has exploded as an industry all its own, leading to the publication of
numerous articles including [2, 4, 5, 6, 9, 13, 22, 23, 24, 28, 29], just to name a few.
The main idea is to understand the combinatorics of cluster mutations in terms of a category
of representations of a quiver, and in particular to obtain an interpretation of Laurent expan-
sions of non-initial cluster variables as cluster characters. These are generating functions for
certain geometric invariants (e.g., Euler characteristics, point counts over finite fields, Poincaré
polynomials, etc.) of varieties of subrepresentations in an associated quiver representation.
Using categorification, many structural conjectures on skew-symmetric cluster algebras, their
cluster variables, and the associated combinatorics of mutations have been established. The main
goal of this note is to observe that the categorification of acyclic skew-symmetrizable (quantum)
cluster algebras [28, 29] naturally leads to proofs of many of the same conjectures.
Alternative approaches to categorification of skew-symmetrizable cluster algebras using quiv-
ers with automorphism or group species with potential have been introduced by Demonet [8, 7].
Another approach using species with potential has been pursued by Labardini-Fragoso and
Zelevinsky [20], see Section 15 there for a broader overview of current developments. These
formalisms have allowed to prove many conjectures on cluster algebras that we thus omit here
even though they should be deducible in our setting.
Since the appearance of these works, new conjectures on cluster algebras, particularly ones
related to denominator vectors, have been made and our goal here is to resolve several of these. In
some cases our proofs will follow by direct translation of arguments given in the skew-symmetric
case.
This paper is a contribution to the Special Issue on Cluster Algebras. The full collection is available at
https://www.emis.de/journals/SIGMA/cluster-algebras.html
mailto:rupel.dylan@pusd.us
mailto:stella@mat.uniroma1.it
http://www1.mat.uniroma1.it/people/stella/
https://doi.org/10.3842/SIGMA.2020.007
https://www.emis.de/journals/SIGMA/cluster-algebras.html
2 D. Rupel and S. Stella
Theorem 1. The following conjectures on cluster algebras are true in the acyclic case: Con-
jectures 4.14(3-4) from [11]; Conjectures 6.11, 7.4(1-2), 7.5, 7.6, and 7.17 from [12]; Conjec-
tures 3.10 and 3.21 from [25]; Conjectures 2.7, 2.8, and 2.9 from [27].
2 Quantum cluster algebras and quantum cluster characters
In the interest of brevity, we will assume that the reader is familiar with the basic notions in the
theory of cluster algebras and refer to [12] for background material. Several results are scattered
throughout the literature; a compendium focusing on the finite type case can be found in [17].
Let B = (bij) be a n × n skew-symmetrizable integer matrix; this means that there exist
positive integers di such that djbji = −dibij for all i, j ∈ [1, n]. We write D for the diagonal
matrix whose i-th diagonal entry is di and fix such a matrix once and for all. We assume
throughout the paper that B is acyclic, i.e., up to a simultaneous permutation of rows and
columns, B has only non-negative entries above the diagonal. Let B̃ be the matrix obtained by
stacking a n× n identity matrix below B and write A
(
x, B̃
)
for the associated cluster algebra;
it is a cluster algebra with principal coefficients.
Since B̃ has full rank, the algebra A
(
x, B̃
)
admits a log-canonical Poisson structure compat-
ible with mutations [14] and one may obtain a quantum cluster algebra via a remarkably simple
deformation quantization [1].
Intuitively, there are only few points to keep in mind. Fix an indetermiate q, then the
definition of the quantum cluster algebra Aq
(
X, B̃
)
can be obtained from that of A
(
x, B̃
)
via
the following modifications:
• The initial cluster X = (Xi) consists of 2n variables which q-commute, i.e., there exists
a skew-symmetric 2n× 2n matrix Λ = (λij) so that XiXj = qλijXjXi.
• Let TΛ be the quantum torus algebra generated by X±1
i , i ∈ [1, 2n], over the ring Z
[
q±
1
2
]
.
It admits an anti-involution (called the bar-involution) fixing each Xi and interchan-
ging q with q−1. Every non-initial cluster variable should also be “bar-invariant” and
this uniquely determines the power of q
1
2 by which to multiply each monomial in the ex-
change relations. More precisely, using the bar-invariant monomial basis Xa, a ∈ Z2n,
of TΛ given by
Xa = q
− 1
2
∑
i<j
λijaiaj
Xa1
1 · · ·X
a2n
2n ,
we may write the mutated variables as X ′k = Xbk
+−εk +Xbk
−−εk , where εk denotes the k-th
standard basis vector of Z2n and the k-th column of B̃ decomposes as bk = bk+ − bk− for
minimal non-negative vectors bk+,b
k
− ∈ Z2n
≥0.
• Each cluster X′ obtainable from X by a sequence of mutations should again generate
a quantum torus, i.e., consist of q-commuting variables. This forces a compatibility con-
dition between B̃ and Λ. That is the matrix Λ has to have the following form for some
skew-symmetric n× n matrix Λ0:
Λ =
[
Λ0 −Λ0B −D
−BTΛ0 +D BTΛ0B +BTD
]
and thus B̃TΛ =
[
D 0
]
. An easy calculation shows that this condition naturally re-
produces under mutations. Note that this compatibility condition is identical to the one
required to define a compatible Poisson structure.
Some Consequences of Categorification 3
Let FΛ denote the skew-field of fractions of TΛ. The quantum cluster algebra Aq
(
X, B̃
)
is
the Z
[
q±
1
2
]
-subalgebra of FΛ generated by all quantum cluster variables obtainable from the
initial cluster X by a sequence of mutations. By work of Berenstein and Zelevinsky [1], the
famous Laurent phenomenon holds in this adapted setting: Aq
(
X, B̃
)
is a subalgebra of TΛ.
Similar to the case of classical cluster algebras, there are formulas to describe these quantum
Laurent expansions in terms of the representation theory of valued quivers that we recall below.
To the matrix B is associated a quiver Q with vertices {1, . . . , n} and gcd(bij , bji) arrows
i → j whenever bij < 0; by our assumption this quiver is acyclic. Write Q̃ for the quiver
obtained from Q by attaching to each vertex i an additional vertex n + i via a single arrow
i → n + i. Let D̃ = diag(D,D). We will be interested in the representation theories of the
valued quivers (Q,D) and (Q̃, D̃).
Let F be a finite field and fix an algebraic closure F of F. Write F〈d〉 for the degree d extension
of F inside F. Note that this provides a canonical identification of F〈d〉 as a vector space over F〈g〉
whenever g|d.
A representation V = (Vi, Va) of the valued quiver (Q,D) consists of an F〈di〉-vector space Vi
for each vertex i and an F〈gcd(di,dj)〉-linear map Va : Vi → Vj for each arrow a : i→ j. The finite-
dimensional representations of (Q,D) form a hereditary abelian category denoted by repF(Q,D).
The category repF(Q,D) naturally embeds into the category repF
(
Q̃, D̃
)
and we will always
identify repF(Q,D) as this full subcategory.
Write Q̃ for the Grothendieck group of repF
(
Q̃, D̃
)
and let Q ⊂ Q̃ denote the Grothendieck
group of repF(Q,D). Since Q̃ is acyclic, there is a natural identification Q̃ ∼= Z2n by taking
classes αi := [Si] of the vertex-simple representations as a basis. The Euler–Ringel form
〈V,W 〉 := dimF Hom(V,W )− dimF Ext(V,W )
on pairs of representations V and W only depends on their classes in Q̃. More precisely, it may
be computed in the basis of vertex-simples by
〈αi, αj〉 =
di if i = j,
dibij if bij < 0,
−di if j = n+ i,
0 otherwise.
For e ∈ Q, define ∗e =
n∑
i=1
1
di
〈αi, e〉αi ∈ Q and e∗ =
2n∑
j=1
1
dj
〈e, αj〉αj ∈ Q̃, noting that these do
not depend on the choice of symmetrizing matrix D. Then the quantum cluster character of
a representation V ∈ repF(Q,D) with [V ] =: v is the element of TΛ given by
XV :=
∑
e∈Q
|F|−
1
2
〈e,v−e〉|Gre(V )|XB̃e−∗v,
where Gre(V ) denotes the set of all subrepresentations E ⊂ V with isomorphism class [E] = e.
Note that B̃e = ∗e− e∗ and, for j ∈ [1, n], we have ∗αj = αj +
n∑
i=1
min(bij , 0)αi.
A representation V ∈ repF(Q,D) is rigid if Ext1(V, V ) = 0. If there exists a rigid representa-
tion with dimension vector v ∈ Q for the finite field F, then there exists a rigid representation
of (Q,D) with dimension vector v over any finite field and thus we refer to the dimension
vector v as being rigid in this case.
Theorem 2 ([28, 29]). Let B be any acyclic skew-symmetrizable matrix with symmetrizer D
and let (Q,D) be the associated valued quiver.
4 D. Rupel and S. Stella
(a) For each rigid representation V ∈ repF(Q,D), the quantum cluster character XV is a quan-
tum cluster monomial of A|F|
(
X, B̃
)
. Moreover, every quantum cluster monomial of
A|F|
(
X, B̃
)
not involving initial cluster variables arises in this way, the decomposition
of V into indecomposables mirroring the factorization of the quantum cluster monomial
into quantum cluster variables.
(b) For all dimension vectors v, e ∈ Q with v rigid, there exists a polynomial Pv,e(q) so that
for any finite field F and rigid representation V ∈ repF(Q,D) of dimension vector v, we
have |Gre(V )| = Pv,e(|F|). These polynomials give a “generic” quantum cluster character
Xv =
∑
e∈Q
q−
1
2
〈e,v−e〉Pv,e(q)XB̃e−∗v,
which computes the quantum cluster monomials not involving initial quantum cluster va-
riables of Aq
(
X, B̃
)
with arbitrary parameter q.
An analogous result was obtained for acyclic skew-symmetric quantum cluster algebras by
Qin [24] using representations of acyclic quivers and with counting polynomials replaced by
Poincaré polynomials. Setting q = 1 in the formula for generic quantum cluster characters from
Theorem 2 gives the following corollary from which we will deduce the main results of this note.
Corollary 3. All cluster monomials of the cluster algebra A
(
x, B̃
)
not involving initial cluster
variables are computed by the cluster characters
xv = x−
∗v
∑
e∈Q
Pv,e(1)xB̃e
as v ranges over all rigid dimension vectors in Q. In particular, the cluster variable xv has g-
vector given by −∗v and its F -polynomial is Fv(y) =
∑
e∈Q
Pv,e(1)ye.
The connection between the representation theory of (Q,D) and the cluster algebra is actually
much stronger. For a source (resp. sink) vertex k in Q, write Σ−k : repF(Q,D)→ repF(µkQ,D)
(resp. Σ+
k : repF(Q,D) → repF(µkQ,D)) for the reflection functor as defined in [10, Section 2].
In what follows, we will drop the adornment and simply write Σk for both reflection functors,
which one to apply should be clear from context. Write rep
〈k〉
F (Q,D) ⊂ repF(Q,D) for the
full subcategory consisting of representations which contain no summands isomorphic to the
simple Sk.
Theorem 4 ([28]). Let k be a sink or a source in Q and let X′ be the cluster obtained by
mutating the initial cluster X in direction k. For any representation V ∈ rep
〈k〉
F (Q,D), we have
XV = X ′ΣkV
, where X ′ΣkV
denotes the quantum cluster character of ΣkV ∈ repF(µkQ,D) in the
variables X′.
3 Deducing the conjectures
In this section, we apply the results of Section 2 to deduce the conjectures in Theorem 1.
Proposition 5. For any rigid dimension vector v ∈ Q, the denominator vector of xv is v.
Proof. The proof is identical to that of [6, Section 4, Corollary 2] with appropriate modifications
in the valued quiver setting, we recall the details here for convenience of the reader.
First note that for any representation W with [W ] = w, we have 1
di
〈W, Ii〉 = wi = 1
di
〈Pi,W 〉,
where Ii and Pi denote respectively the injective hull and projective cover of the vertex simple Si.
Some Consequences of Categorification 5
Next using the injective resolution of Si we see that 〈W,Si〉 ≤ 〈W, Ii〉 while using the projective
resolution gives 〈Si,W 〉 ≤ 〈Pi,W 〉. Now consider a subrepresentation E ⊂ V with [E] = e,
applying the above considerations we see that the i-th component of e∗ + ∗(v − e) is bounded
by the i-th component of v:
1
di
〈E,Si〉+
1
di
〈Si, V/E〉 ≤
1
di
〈E, Ii〉+
1
di
〈Pi, V/E〉 = ei + (vi − ei) = vi.
To finish the proof, for each i ∈ [1, n] we must exhibit a subrepresentation E ⊂ V which
realizes this bound. To construct such a subrepresentation, let Ji be the set of all vertices j in Q
for which there exists a path (possibly trivial) beginning at i and ending at j. Now set Ej = Vj for
j ∈ Ji and Ej = 0 for j /∈ Ji. Recall that in the injective coresolution 0→ Si → Ii → I → 0 the
injective representation I has nonzero components only at vertices j which admit a nontrivial
path to vertex i, while in the projective resolution 0 → P → Pi → Si → 0 the projective
representation P has nonzero components only at vertices j which admit a nontrivial path from
vertex i.
Thus 〈E,Si〉 = 〈E, Ii〉 − 〈E, I〉 = 〈E, Ii〉 and 〈Si, V/E〉 = 〈Pi, V/E〉 − 〈P, V/E〉 = 〈Pi, V/E〉
and therefore
1
di
〈E,Si〉+
1
di
〈Si, V/E〉 =
1
di
〈E, Ii〉+
1
di
〈Pi, V/E〉 = ei + (vi − ei) = vi
as desired. �
Corollary 6 ([12, Conjecture 7.17] for acyclic initial exchange matrices). For a rigid dimension
vector v ∈ Q, the denominator vector of xv is the exponent vector of the monomial obtained by
tropically evaluating the corresponding F -polynomial at y−1
1 , . . . , y−1
n .
Proof. It is enough to observe that the F -polynomial has a unique monomial of maximal
degree and this monomial is divisible by all other monomials. Indeed, this is the monomial
corresponding to the full subrepresentation and hence it has exponent vector v in Fv. �
This immediately implies [12, Conjecture 6.11] using [12, Proposition 7.16]. From Proposition 5,
we deduce this weakening of [12, Conjecture 7.4] in the acyclic case.
Corollary 7. Let v be the denominator vector of a non-initial cluster variable xv in A
(
x, B̃
)
.
Then:
1) all entries in v are non-negative,
2) vi = 0 if and only if there is a seed containing both xv and the initial cluster variable xi,
3) vi depends only on xv and xi but not on the seed containing xv.
Proof. By Proposition 5, the denominator vector of any non-initial cluster variable is a dimen-
sion vector and therefore it has non-negative entries. Moreover, its coordinates are independent
of the seed containing the cluster variable. This proves points (1) and (3).
By the categorification construction [29], seeds in A
(
x, B̃
)
are in bijection with local tilting
representations T . The seed corresponding to T contains the i-th initial cluster variable precisely
when vertex i is not in the support of T . Any rigid indecomposable representation can be
completed to a local tilting representation with the same support and this establish part (2). �
The key difference in the above corollary compared to the original statement of the conjecture
is that in point (3) we do not claim independence from the seed containing xi. Nonetheless the
corollary immediately implies the weaker statements [12, Conjecture 7.5] and [27, Conjecture 2.9]
for acyclic initial exchange matrices.
6 D. Rupel and S. Stella
Let E = (eij) be the Euler matrix given by
eij =
{
1 if i = j,
min(bij , 0) if i 6= j.
Combining Corollary 3 and Proposition 5 we immediately get the following.
Proposition 8 ([25, Conjecture 3.21]). Let v ∈ Q be a rigid dimension vector. Then the g-
vector of xv is −Ev.
Proof. Since the g-vector of the cluster monomial xv is given by −∗v, the result follows imme-
diately from the definitions of E and the operator ∗(−). �
Replacing E with its piecewise linear counterpart νc defined in [26], the last result extends
to cover all cluster monomials in A
(
x, B̃
)
.
Corollary 9 ([12, Conjecture 7.6] for acyclic initial exchange matrices). Different cluster mono-
mials in A
(
x, B̃
)
have different denominator vectors. Moreover, the denominator vectors of the
cluster variables of any given cluster form a Z-basis of Q.
Proof. Since B is acyclic, by [7, Remark 7.2 and Proposition 11.6], different cluster monomials
of A
(
x, B̃
)
have different g-vectors. Moreover the map νc is invertible over Z – in each cluster
it is given by a triangular matrix with all the diagonal entries equal to −1 – and the first claim
follows from Proposition 8.
By the same argument, the second claim is a direct consequence of [7, Proposition 11.5] but
can also be deduced as follows. Each seed of A
(
x, B̃
)
corresponds to a local tilting represen-
tation T . By [16, Lemma 4.3 and Theorem 4.5], the dimension vectors of the summands of T
provide a Z-basis for the sublattice on which they are supported. Negative simples corresponding
to vertices outside the support complete this basis to a Z-basis of Q. �
Corollary 10 ([27, Conjecture 2.8] for acyclic initial exchange matrices). The mutation of
the initial cluster of A
(
x, B̃
)
at a sink or a source vertex transforms denominator vectors of
non-initial cluster variables according to the simple reflection associated to that vertex.
Proof. By Theorem 4, the initial cluster mutation at a sink or a source vertex transforms
cluster characters according to the associated reflection functor on repF(Q,D). The result then
follows from [10, Proposition 2.1] and Proposition 5. �
From this corollary, following [27, Proposition 2.10], we also get [27, Conjecture 2.7] for
acyclic initial exchange matrices. The following technical result is needed in [30].
Proposition 11. Assume vertex k is a source for B. Let xv be any non-initial cluster variable
in A
(
x, B̃
)
and write F ′v for the tropical evaluation of its F -polynomial at y′j = yjy
−bkj−2δjk
k .
Then F ′v = 1 unless v = [Sk] in which case F ′v = y−1
k .
Proof. Let V be a rigid indecomposable representation with [V ] = v. The terms of the F -
polynomial Fv(y) are labeled by subrepresentations of [V ] and the claim is that each of these
monomials produces only non-negative exponents when evaluated at the given y′j .
Since vertex k is a source in B, we have bkj ≤ 0 for j ∈ [1, n] and therefore only the
exponent of yk could end up being negative in the tropical evaluation. However, observe that
the total exponent of any monomial in Fv(y′) is given by applying the simple reflection sk to
the dimension vector of the subrepresentation it corresponds to. In particular, this implies that
the only possibility to have a negative exponent is that Sk is a subrepresentation of V .
On the other hand, vertex k is a source so that the only rigid indecomposable representation
admitting Sk as a subrepresentation is Sk itself and the result follows. �
Some Consequences of Categorification 7
We turn now our attention towards results dealing with the exchange graph of A
(
x, B̃
)
.
Proposition 12 ([25, Conjecture 3.10] in the acyclic case). Given a cluster algebra having an
acyclic seed, the induced subgraph of its exchange graph consisting of all the seeds containing
any fixed collection of cluster variables is connected.
Proof. Since the exchange graph of a cluster algebra with principal coefficients covers the
exchange graph of any cluster algebra with the same mutation pattern (cf. [12]), it suffices to
establish the claim for A
(
x, B̃
)
. The proof given in [5, Corollary 3] for the weaker statement [11,
Conjecture 4.14(3)] is written in terms of arbitrary hereditary abelian categories. In particular,
their proof applies in our situation. More generally, one may replace the rigid indecomposable
representation appearing in [5, Section 5.4] by an arbitrary partial tilting representation (cf. [15,
Proposition 3]). Then running a similar argument as in the proof of [5, Theorem 6] proves
the general case. These constructions are closely related to the Iyama–Yoshino reduction of
triangulated categories (cf. [18, Section 4] or [19, Section 7.2]). �
Proposition 13 ([11, Conjecture 4.14(4)]). The full subgraph of the exchange graph consisting
of all seeds having acyclic exchange matrix is connected.
Proof. The proof of the analogous statement for skew-symmetric cluster algebras given in [5,
Corollary 4] is also written in terms of arbitrary hereditary abelian categories and thus it applies
in our situation as well. �
Acknowledgements
We are grateful to Giovanni Cerulli Irelli for his help with references and to an anonymous referee
for pointing out a flaw in the original proof of Corollary 9. D.R. was partially supported by an
AMS-Simons Travel Grant; S.S. was supported by ISF grant 1144/16. D.R. is grateful to the
Department of Physics Mathematics and Astronomy of the California Institute of Technology
and to the Department of Mathematics of University of Notre Dame for research support; S.S. is
grateful to the School of Mathematics and Actuarial Science of the University of Leicester, where
part of this work was done.
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1 Introduction
2 Quantum cluster algebras and quantum cluster characters
3 Deducing the conjectures
References
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| id | nasplib_isofts_kiev_ua-123456789-210603 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:19Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rupel, Dylan Stella, Salvatore 2025-12-12T10:39:15Z 2020 Some Consequences of Categorification. Dylan Rupel and Salvatore Stella. SIGMA 16 (2020), 007, 8 pages 1815-0659 2020 Mathematics Subject Classification: 13F60; 16G20 arXiv:1712.08478 https://nasplib.isofts.kiev.ua/handle/123456789/210603 https://doi.org/10.3842/SIGMA.2020.007 Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to d-vectors, g-vectors, and F-polynomials. We are grateful to Giovanni Cerulli Irelli for his help with references and to an anonymous referee for pointing out a flaw in the original proof of Corollary 9. D.R. was partially supported by an AMS-Simons Travel Grant; S.S. was supported by an ISF grant 1144/16. D.R. is grateful to the Department of Physics, Mathematics, and Astronomy of the California Institute of Technology and to the Department of Mathematics of the University of Notre Dame for research support; S.S. is grateful to the School of Mathematics and Actuarial Science of the University of Leicester, where part of this work was done. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Some Consequences of Categorification Article published earlier |
| spellingShingle | Some Consequences of Categorification Rupel, Dylan Stella, Salvatore |
| title | Some Consequences of Categorification |
| title_full | Some Consequences of Categorification |
| title_fullStr | Some Consequences of Categorification |
| title_full_unstemmed | Some Consequences of Categorification |
| title_short | Some Consequences of Categorification |
| title_sort | some consequences of categorification |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210603 |
| work_keys_str_mv | AT rupeldylan someconsequencesofcategorification AT stellasalvatore someconsequencesofcategorification |