Mixed vs Stable Anti-Yetter-Drinfeld Contramodules
We examine the cyclic homology of the monoidal category of modules over a finite-dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules....
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| description | We examine the cyclic homology of the monoidal category of modules over a finite-dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter-Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter-Drinfeld contramodules (recently introduced).
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 026, 10 pages
Mixed vs Stable Anti-Yetter–Drinfeld Contramodules
Ilya SHAPIRO
Department of Mathematics and Statistics, University of Windsor,
401 Sunset Avenue, Windsor, Ontario N9B 3P4, Canada
E-mail: ishapiro@uwindsor.ca
URL: http://http://web2.uwindsor.ca/math/ishapiro/
Received November 09, 2020, in final form March 04, 2021; Published online March 17, 2021
https://doi.org/10.3842/SIGMA.2021.026
Abstract. We examine the cyclic homology of the monoidal category of modules over a finite
dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference
between the recently introduced mixed anti-Yetter–Drinfeld contramodules and the usual
stable anti-Yetter–Drinfeld contramodules. Namely, we show that Sweedler’s Hopf algebra
provides an example where mixed complexes in the category of stable anti-Yetter–Drinfeld
contramodules (previously studied) are not equivalent, as differential graded categories to the
category of mixed anti-Yetter–Drinfeld contramodules (recently introduced).
Key words: Hopf algebras; homological algebra; Taft algebras
2020 Mathematics Subject Classification: 16E35; 16T05; 18G90; 19D55
1 Introduction
Cyclic (co)homology was introduced independently by Boris Tsygan and Alain Connes in the
1980s. It has since been generalized, applied to many fields, and now reaches into many different
settings. Our investigations in this paper focus on the equivariant flavour that began with
Connes–Moscovici [6] and was generalized into Hopf-cyclic cohomology by Hajac–Khalkhali–
Rangipour–Sommerhäuser [8, 9] and Jara–Stefan [11] (independently). Roughly speaking, the
original theory defines cohomology groups for an associative algebra that play the role of the
de Rham cohomology in the noncommutative setting. The equivariant version considers an alge-
bra with an action of a Hopf algebra. It turns out that just as in the de Rham cohomology, one
has coefficients in the Hopf setting; it is an interesting fact that, unlike the de Rham setting,
Hopf-cyclic cohomology requires coefficients, i.e., there are no canonical trivial coefficients. These
coefficients are known as stable anti-Yetter–Drinfeld modules, due to their similarity to the usual
Yetter–Drinfeld modules. It turns out that the more natural, from a conceptual point of view,
version of coefficients are stable anti-Yetter–Drinfeld contramodules [5]. It is the desire to under-
stand the coefficients themselves that motivated a series of papers by the author of the present
one. This paper is a natural next step.
This paper is a descendant of [20], where it is shown that the classic stable anti-Yetter–
Drinfeld contramodules are simply objects in the naive cyclic homology category of HM, the
monoidal category of modules over the Hopf algebra H. It is furthermore conjectured there,
that the new coefficients introduced (mixed anti-Yetter–Drinfeld contramodules) are obtained
via the true cyclic homology category; this makes exact the analogy between the de Rham
coefficients in the geometric setting and the Hopf-cyclic coefficients. Namely, while the latter
are obtained from the cyclic homology of HM, the former are shown in [4] to arise from the cyclic
homology of quasi-coherent sheaves on the space X. More precisely, in [20], a category of mixed
anti-Yetter–Drinfeld contramodules is defined by analogy with the derived algebraic geometry
case of [4]. This new generalization is conceptual, and furthermore allows the expression of the
mailto:ishapiro@uwindsor.ca
http://http://web2.uwindsor.ca/math/ishapiro/
https://doi.org/10.3842/SIGMA.2021.026
2 I. Shapiro
Hopf-cyclic cohomology of an algebra A with coefficients in M as an Ext (in this category)
between ch(A), the Chern character object associated to A, and M itself. Even if one takes M
to be a stable anti-Yetter–Drinfeld contramodule, the object ch(A) is truly a mixed anti-Yetter–
Drinfeld contramodule. It is conjectured that mixed anti-Yetter–Drinfeld contramodules are the
objects in the cyclic homology category of HM.
The comparison in [20] between anti-Yetter–Drinfeld contramodules and the cyclic homology
category of HM involves a monad on HM with a central element σ. When we talk about
the S1-action we mean the action of this central element on the category of modules over the
monad. It is this description that allows us here to reduce the investigations into the differences
between the previously studied and the new Hopf-cyclic cohomology to the analysis of categories
of modules over two differential graded algebras (DGAs). Namely, in the notation of the paper,
we have an algebra D̂(H) whose modules are the anti-Yetter–Drinfeld contramodules, we have
a DGA D̂(H)[θ] with dθ = σ−1 that yields the new mixed anti-Yetter–Drinfeld contramodules,
and we have a DGA D̂(H)/(σ − 1)[θ] with dθ = 0 that yields the previously studied setting,
i.e., the mixed complexes in stable anti-Yetter–Drinfeld contramodules. Thus, it suffices for
our purposes to compare the DG categories of modules over these two DGAs. We concentrate
on finite dimensional Hopf algebras H. We show that if the square of the antipode is trivial,
i.e., S2 = Id then the DG categories coincide (Proposition 2.5):
Result 1.1. Let H be a finite dimensional Hopf algebra such that the square of the antipode
is equal to the identity, i.e., S2 = Id. Then the categories of mixed complexes in stable aY D-
contramodules and mixed aY D-contramodules are DG-equivalent.
On the other hand, if we consider Sweedler’s Hopf algebra T2(−1) (the smallest case of
S2 6= Id) then they do not (Proposition 3.6):
Result 1.2. Let H = T2(−1), then the mixed complexes in the category of stable anti-Yetter–
Drinfeld contramodules are not DG-equivalent to the category of mixed anti-Yetter–Drinfeld
contramodules.
Conventions. All algebras A in monoidal categories are assumed to be unital associative.
Our H is a Hopf algebra over some fixed algebraically closed field k, of characteristic 0, and Vec
denotes the category of k-vector spaces. For the purposes of this paper we are only interested in
finite dimensional Hopf algebras. We use the following version of Sweedler’s notation: For h ∈ H
we denote the coproduct ∆(h) ∈ H ⊗H by h1 ⊗ h2. The letter S denotes the antipode of H.
The number p is prime. Finally, DG stands for differential graded.
2 Twisted Drinfeld double
Let H be a Hopf algebra. From [20] we see that the study of the Hochschild and cyclic homologies
of HM, the monoidal category of H-modules, reduces to the study of modules over a certain
monad on HM. Recall that the consideration of Hochschild and cyclic homologies of monoidal
categories is motivated by their recently discovered role [20] in the understanding of Hopf-cyclic
theory coefficients.
Briefly, we have the monad (see [20] for more details):
Homk(H,−) : HM→ HM (2.1)
with the H-module structure on Homk(H,V ):
x · ϕ = x2ϕ
(
S
(
x3
)
(−)x1
)
,
Mixed vs Stable Anti-Yetter–Drinfeld Contramodules 3
for x ∈ H and ϕ ∈ Homk(H,V ). The unit 1V : Id(V )→ Homk(H,V ) is
1V (v)(h) = ε(h)v
and a crucial central element σV : Id(V )→ Homk(H,V ) is
σV (v)(h) = hv.
The anti-Yetter–Drinfeld contramodules then coincide with modules over this monad (shown
in [20]), while the stable ones consist of those for which the action of σ agrees with that of 1, and
the mixed ones introduced in [20] are the homotopic version of this on the nose requirement.
Recall the mixed complexes of [12]. These are complexes of vector spaces (V •, d) with
a homotopy h such that dh + hd = 0. We can replace vector spaces with R-modules for
some ring R. Note that as observed in [12], the DG-category of mixed complexes in R-modules
is isomorphic to the DG-category of DG-modules over R[θ], where θ is a freely and centrally
adjoined degree −1 graded commutative variable (naturally R itself is placed in degree 0, so
that R[θ] = R→ R as a complex) and d = 0 on R[θ]. The action of θ gives the homotopy h.
We can generalize the considerations of [12] so as to apply to our particular situation. Namely,
let z ∈ Z(R), i.e., zr = rz for all r ∈ R. Define a DG-algebra R[θ] by placing R in degree 0
and θ in degree −1. Let θ commute with R and itself, so in particular θ2 = 0. So far it is as
above. Now define the differential to be 0 on R and dθ = z. This is well defined and unique by
the Leibniz rule. We observe that the category of DG-modules over R[θ] consists of complexes
of R-modules equipped with a homotopy h such that dh+ hd = z.
Now recall from [20]:
Definition 2.1. We say that (M•, d, h) is a mixed anti-Yetter–Drinfeld contramodule if (M•, d)
is a complex of contramodules, i.e., modules over the monad (2.1), and h is a homotopy annihi-
lating σ − 1. More precisely,
dh+ hd = σ − 1.
In this section we will define an explicit DG-algebra that will yield the mixed anti-Yetter–
Drinfeld (aYD) contramodules (for H finite dimensional) as its DG-modules. The construction
of the twisted convolution algebra below is analogous to the classical Drinfeld double D(H)
and its anti-version Da(H) [9] (we review these in Appendix A, where we expand upon this
comparison).
Definition 2.2. Let H be a Hopf algebra with an invertible antipode S, define a twisted
double D̂(H) as follows. The multiplication on D̂(H) := End(H) is
(f ? g)(h) = f
(
h1
)2
g
(
S
(
f
(
h1
)3)
h2f
(
h1
)1)
,
thus the multiplicative identity, which we denote by 1 is ε(−)1, and the central element σ(h) = h
is invertible with inverse S−1.
Definition 2.2 is extracted from the monad (2.1) with the sole purpose consisting of making
the following lemma a tautology.
Lemma 2.3. Let H be a finite dimensional Hopf algebra.
� The category of anti-Yetter–Drinfeld contramodules over H is isomorphic to D̂(H)-mo-
dules.
� The category of stable anti-Yetter–Drinfeld contramodules is isomorphic to modules over
A := D̂(H)/(σ − 1).
4 I. Shapiro
� The DG-category of mixed anti-Yetter–Drinfeld contramodules is isomorphic to DG-mo-
dules over the DG algebra
B := D̂(H)[θ],
where θ is a freely adjoined degree −1 graded commutative variable and dθ = σ − 1, with
d|
D̂(H)
= 0.
Proof. In the finite dimensional case, as vector spaces, D̂(H) = End(H) ' H∗ ⊗H. Further-
more, as an algebra, D̂(H) is the quotient of the free product algebra, generated by H∗ and H,
by the relation:
hχ = χ
(
S
(
h3
)
(−)h1
)
h2, (2.2)
where h ∈ H, χ ∈ H∗. Thus, modules over the algebra are both H-modules and H-contra-
modules (same as H∗-modules for H finite dimensional). The two actions satisfy the requisite
compatibility condition for contramodules, as specified in [5], and ensured by (2.2). Clearly,
modules over A consist of the full subcategory of objects on which σ acts by identity, these are
exactly the stable contramodules. Finally, a DG-module over B is just a complex of D̂(H)-
modules with a homotopy given by the action of θ. The condition dθ = σ − 1 ensures that
dh+ hd = σ − 1 on M•. �
Our main goal is to compare the category of mixed aYD contramodules to the category
of mixed complexes of stable aYD contramodules. By the preceding lemma this means determin-
ing when, and more interestingly when not, the category of DG-modules over B is DG-equivalent
to A[θ]-modules (with θ of degree −1 and d = 0). The study of Hopf-cyclic cohomology has
thus far only concerned itself with the latter.
The following simple lemma takes care of a lot of cases.
Lemma 2.4. Let H be a finite dimensional Hopf algebra and suppose that the action of σ − 1
on D̂(H) is diagonalizable. Then the categories of mixed complexes in stable aY D-contramodules
and mixed aY D-contramodules are DG-equivalent.
Proof. Since the action of the central element σ− 1 on D̂(H) is diagonalizable we may decom-
pose D̂(H) as a product of algebras D̂(H)0 ⊕ D̂(H)+ with D̂(H)0 being the 0-eigenspace
and D̂(H)+ all the other eigenspaces. We have an inclusion of DGAs: D̂(H)0[θ] → B that
induces an isomorphism on cohomology. Namely, as complexes B is D̂(H)
σ−1→ D̂(H) whereas
D̂(H)0[θ] is D̂(H)0
0→ D̂(H)0 and σ − 1 is invertible on D̂(H)+. Note that D̂(H)0 ' A and we
are done. �
Proposition 2.5. Let H be a finite dimensional Hopf algebra such that the square of the antipode
is identity, i.e., S2 = Id. Then the categories of mixed complexes in stable aY D-contramodules
and mixed aY D-contramodules are DG-equivalent.
Proof. We need characteristic 0 here. Since S2 = Id, so H is semi-simple [14, 15], so D(H)
(its Drinfeld double) is semi-simple [17]. By Lemma A.3, we know that it follows that Da(H)
is semi-simple and thus by [19] so is D̂(H). Thus, by Schur’s lemma, the action of the central
element σ − 1 is diagonalizable and we are done by Lemma 2.4. �
In light of the above we need to consider an example of H with S2 6= Id. It turns out that
the smallest, dimension wise, such example suffices.
Mixed vs Stable Anti-Yetter–Drinfeld Contramodules 5
3 Taft Hopf algebras
We fix a prime p and a primitive pth root of unity ξ ∈ k in the following. The Taft Hopf algebra
Tp(ξ) [21] is generated as a k-algebra by g and x with the relations
gp = 1, xp = 0,
gx = ξxg.
It is sometimes called the quantum sl2 Borel algebra. It is p2 dimensional over k. Furthermore,
the coalgebra structure is
∆(g) = g ⊗ g, ∆(x) = x⊗ 1 + g ⊗ x
with ε(g) = 1, ε(x) = 0, and thus S(g) = g−1, while S(x) = −g−1x. Note that
S2(x) = ξ−1x 6= x,
making T2(−1) the smallest Hopf algebra with S2 6= Id. The Taft algebra T2(−1) is somewhat
different from the other Tp(ξ) and has its own name: Sweedler’s Hopf algebra.
3.1 The identification with the dual
The Taft algebra is isomorphic to its dual. We need some explicit formulas establishing the
isomorphism Tp(ξ) ' Tp(ξ)∗ and its inverse. Suppose that ω is a pth root of unity, let
(n)ω = 1 + · · ·+ ωn−1
and
(n)ω! = (n)ω · · · (1)ω.
The verification of the following is left to the reader; key details can be found in [18]. The
lemma itself can be obtained from [16].
Lemma 3.1. As Hopf algebras
Tp(ξ)
∗ ' Tp(ξ).
Proof. Consider a basis of Tp(ξ):
{
gixj
}p−1
i,j=0
so that
{(
gixj
)∗}
denotes the dual basis of Tp(ξ)
∗.
Then the isomorphism of Hopf algebras and its inverse are given by
gixj 7→ (j)ξ−1 !
∑
l
ξi(j+l)
(
glxj
)∗
and
(gixj)∗ 7→ 1
p(j)ξ−1 !
∑
l
ξ−l(i+j)glxj .
�
Corollary 3.2. The twisted double D̂(Tp(ξ)) is a quotient of k 〈x, x′, g, g′〉 .
� The relations are
xp = x′p = gp − 1 = g′p − 1 = 0,
gg′ = g′g, gx = ξxg, g′x′ = ξx′g′, gx′ = ξ−1x′g, g′x = ξ−1xg′,
xx′ − ξ−1x′x = 1− ξ−1g′−1g.
6 I. Shapiro
� The actions of g′ and g on a D̂(Tp(ξ))-module V , yields a (Z/p)2-grading on V by their
eigenspaces, i.e., g′, g act on Vij by ξi, ξj, respectively. Thus x and x′ have degrees (−1, 1)
and (1,−1), respectively.
� The S1-action of σ on Vij is
p−1∑
l=0
ξ(i−l)(j+l)
(l)ξ−1 !
x′lxl. (3.1)
Proof. We use the identification of vector spaces D̂(Tp(ξ)) = (Tp(ξ))
∗⊗Tp(ξ) as in Lemma 2.3
followed by (Tp(ξ))
∗ ⊗ Tp(ξ) ' Tp(ξ)⊗ Tp(ξ) from Lemma 3.1. We let
x′ = x⊗ 1, x = 1⊗ x, g′ = g ⊗ 1, g = 1⊗ g
in the latter. To derive the rest of the relations we apply (2.2). The action of σ =
∑
ij
(
gixj
)∗⊗
gixj is computed on the graded components directly. �
Observe that it follows from Corollary 3.2 that gg′ ∈ D̂(Tp(ξ)) is central. Since we see that
(gg′)p = 1 so its action on D̂(Tp(ξ)) is diagonalizable with eigenvalues ξs, s ∈ Z/p. Thus as
an algebra
D̂(Tp(ξ)) =
⊕
s
D̂(Tp(ξ))/(gg
′ − ξs) (3.2)
so that it suffices to understand D̂(Tp(ξ))/(gg
′ − ξs). There are two cases: p = 2 and p > 2.
We will begin by briefly discussing the latter (though without addressing the S1-action), and
then concentrate our attention on the former (with examining the S1-action) to achieve the goal
set out in the abstract.
3.2 The case of p > 2
Let p > 2, then there exists a primitive pth root of unity q ∈ k such that
q2 = ξ−1.
We then have
D̂(Tp(ξ))/(gg
′ − ξs) ' uq(sl2). (3.3)
More precisely, let
E =
qs+1
q − q−1
x′, F = xg′, and K = qs+1g,
so that D̂(Tp(ξ))/(gg
′ − ξs) is generated by E, F , K subject to
Ep = F p = Kp − 1 = 0,
[E,F ] =
K −K−1
q − q−1
, KEK−1 = q2E, and KFK−1 = q−2F.
This shows that:
D̂(Tp(ξ)) ' uq(sl2)⊗OZ/p ' uq(sl2)⊗ kZ/p. (3.4)
As we will see below the case of p = 2 is very different, in particular as s varies, the algebra will
change significantly whereas here it does not (3.3).
Mixed vs Stable Anti-Yetter–Drinfeld Contramodules 7
Remark 3.3. See [13], where the Taft algebra is called the quantum sl2 Borel algebra, and
its Drinfeld double is computed. The result obtained is identical to ours in (3.4), though we
compute the twisted double. This is not surprising as we see from Appendix A that our analysis
of D̂(H) can be interpreted as that of D(H), with σ being a new ingredient. Note that more
generally, a comparison between Drinfeld doubles of Nichols algebras and quantized universal
enveloping algebras can be found in [2].
3.3 The case of p = 2
We need to describe the algebra D̂(T2(−1)) in greater detail, paying particular attention to the
element σ.
By (3.2) the category of DG-modules over D̂(T2(−1))[θ] is a product of categories, C0 × C1,
corresponding to the cases s = 0 and s = 1. We will deal with both separately. More precisely,
for a D̂(T2(−1))[θ]-module V , we decompose
V = (V00 ⊕ V11)⊕ (V01 ⊕ V10).
Observe that by the Corollary 3.2 we have that
xx′ + x′x = 1 + (−1)s,
so that
(x′x)2 = (1 + (−1)s)x′x. (3.5)
We see from (3.5) that the minimal polynomial of x′x depends only on s; this is exclusive to
p = 2 and makes this case tractable. Note that by (3.1):
σ|V00 = 1− x′x, σ|V11 = −1 + x′x, and σ|V01 = σ|V10 = 1 + x′x. (3.6)
Let
Ds = D̂(T2(−1))/(gg′ − (−1)s).
We begin with s = 0: The category of D0-modules consists of Z/2-graded vector spaces
(V = V00 ⊕ V11) equipped with degree changing operators x and x′ subject to the relation
xx′ + x′x = 2. The action of σ − 1 on V00 is −x′x and on V11 it is x′x− 2.
Lemma 3.4. The category C0 consists of the mixed complexes of [12].
Proof. Note that D0/(σ−1)-modules are just vector spaces. Indeed, let y = x′/2. For improved
clarity, denote by xi and yi the action of x and y, respectively, that originates at Vii. After
modding out by σ − 1 we have by (3.6) that y1x0 = 0 =⇒ x1y0 = 1 and y0x1 = 1. So
y0 : V00 ' V11 : x1.
Furthermore, x2 = y2 = 0 so that both x0 and y1 are the 0 maps. Thus
V = V00 ⊕ V11 7→ V00
establishes an equivalence of categories between D0/(σ − 1)-modules and Vec.
The action of σ − 1 on D0 is diagonalizable by (3.5) and so by the proof of Lemma 2.4 the
algebras D0[θ] (dθ = σ − 1) and D0/(σ − 1)[θ] (dθ = 0) are quasi-isomorphic. Thus, C0, being
DG-equivalent to D0/(σ− 1)[θ]-modules, is by the above discussion, equivalent to k[θ]-modules.
These are just the mixed complexes of [12]. �
8 I. Shapiro
Remark 3.5. Note that not only does C0 consist of the usual mixed complexes but it also does
not provide any evidence of the need for the mixed aY D-contramodules (see the proof above).
Moving on to s = 1 we find that things change for the better. Recall that D1 is generated by
x, x′, g
subject to
x2 = x′2 = g2 − 1 = 0, xx′ = −x′x, gx = −xg, gx′ = −x′g.
Furthermore, σ − 1 acts as x′x by (3.6).
Proposition 3.6. Let H = T2(−1), then the mixed complexes in the category of stable anti-
Yetter–Drinfeld contramodules are not DG-equivalent to the category of mixed anti-Yetter–
Drinfeld contramodules.
Proof. By the preceding discussion, i.e., the decomposition of the category into a product,
it suffices to show that the categories D1[θ]-mod (where dθ = x′x) and D1/(x
′x)[θ]-mod (where
dθ = 0) are not DG-equivalent.
Recall that the Hochschild cohomology HH i(C•) of a DG algebra C• is an invariant of its
DG category of modules [22]. We will compute HH−1. In our simple case of a DG algebra
C = C−1
d→ C0 concentrated in two degrees, we have
HH−1(C) = ker
(
C−1
α→ C0 ⊕Hom
(
C0, C−1
))
,
where α(x) = (dx, [x,−]).
The key observation here is that the center of D1 is spanned by 1, xx′, xx′g, while that
of D1/(x
′x) is spanned by 1. Thus, in the first case we get that HH−1 is spanned by xx′
and xx′g. In the second case it is spanned by 1. �
A Appendix
Our purpose in this section is to compare D̂(H) of Definition 2.2 to the more familiar Drinfeld
double D(H) in the case of a finite dimensional Hopf algebra H. Though not original, see [7]
for Definition A.1 and [9] for Definition A.2, since our conventions differ from the usual ones we
spell out the definitions again below (for H finite dimensional):
Definition A.1. The algebra D(H) is generated by H and H∗ subject to the relations:
χh = h2χ
(
h3(−)S−1
(
h1
))
,
for χ ∈ H∗ and h ∈ H. Thus D(H) = H ⊗H∗ as vector spaces.
Definition A.2. The algebra Da(H) is generated by H and H∗ subject to the relations:
χh = h2χ
(
h3(−)S
(
h1
))
,
for χ ∈ H∗ and h ∈ H. Thus Da(H) = H ⊗ H∗ as vector spaces. The central element is
σ = ei ⊗ ei, where ei is any basis of H and ei is its dual basis of H∗.
Mixed vs Stable Anti-Yetter–Drinfeld Contramodules 9
Note that modules over Da(H) as specified in Definition A.2 can be identified with what
is usually called left-right anti-Yetter–Drinfeld modules, i.e., left modules and right comodu-
les [9]. Recall that if H is finite dimensional then we have an S1-equivariant equivalence between
D̂(H)-modules and Da(H)-modules [19]. We will thus focus on the comparison between Da(H)
and D(H). It is known that in general they give very different categories of modules [10]. It is
immediate that if S2 = Id then the algebras in fact coincide, since the only difference between
them is S in Definition A.2 and S−1 in Definition A.1. Below we extend that observation slightly
so as to cover our case of Taft algebras where we do not have S2 = Id, but instead we get
S2(h) = uhu−1
for some group-like element u ∈ H, i.e., ∆(u) = u ⊗ u. For Tp(ξ) we have S2(a) = g−1ag with
∆g−1 = g−1 ⊗ g−1, so that u = g−1.
Recall that Hopf algebras possessing such an element u as above are called pivotal, see [1, 3]
for example. Their categories of representations are thus pivotal as well, i.e., equipped with
a monoidal isomorphism from the identity functor to the double dual functor. This natural
transformation is given by the action of u ∈ H; it is monoidal since u is group-like and mapping
to the double dual since S2(h) = uhu−1.
The following lemma is a straightforward computation but can be obtained from [10]:
Lemma A.3. Let H be a finite dimensional Hopf algebra and suppose that there exists a u ∈ H
with ∆(u) = u⊗ u such that S2(h) = uhu−1 for all h ∈ H. Then
D(H)→ Da(H),
h⊗ χ 7→ h⊗ χ((−)u)
is an isomorphism of algebras.
Thus for Taft algebras, Drinfeld doubles can play the role of D̂(H), as long as we are careful
to remember about the crucial central element σ.
Acknowledgements
This research was supported in part by the NSERC Discovery Grant number 406709. The author
wishes to thank the referees for their many helpful suggestions.
References
[1] Andruskiewitsch N., Angiono I., Garćıa Iglesias A., Torrecillas B., Vay C., From Hopf algebras to tensor cate-
gories, in Conformal Field Theories and Tensor Categories, Math. Lect. Peking Univ., Springer, Heidelberg,
2014, 1–31, arXiv:1204.5807.
[2] Andruskiewitsch N., Radford D., Schneider H.J., Complete reducibility theorems for modules over pointed
Hopf algebras, J. Algebra 324 (2010), 2932–2970, arXiv:1001.3977.
[3] Barrett J.W., Westbury B.W., Spherical categories, Adv. Math. 143 (1999), 357–375, arXiv:hep-th/9310164.
[4] Ben-Zvi D., Francis J., Nadler D., Integral transforms and Drinfeld centers in derived algebraic geometry,
J. Amer. Math. Soc. 23 (2010), 909–966, arXiv:0805.0157.
[5] Brzeziński T., Hopf-cyclic homology with contramodule coefficients, in Quantum groups and noncommuta-
tive spaces, Aspects Math., Vol. E41, Vieweg + Teubner, Wiesbaden, 2011, 1–8, arXiv:0806.0389.
[6] Connes A., Moscovici H., Cyclic cohomology and Hopf algebras, Lett. Math. Phys. 48 (1999), 97–108,
arXiv:math.QA/9904154.
[7] Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2
(Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798–820.
https://doi.org/10.1007/978-3-642-39383-9_1
https://arxiv.org/abs/1204.5807
https://doi.org/10.1016/j.jalgebra.2010.06.002
https://arxiv.org/abs/1001.3977
https://doi.org/10.1006/aima.1998.1800
https://arxiv.org/abs/hep-th/9310164
https://doi.org/10.1090/S0894-0347-10-00669-7
https://arxiv.org/abs/0805.0157
https://doi.org/10.1007/978-3-8348-9831-9_1
https://arxiv.org/abs/0806.0389
https://doi.org/10.1023/A:1007527510226
https://arxiv.org/abs/math.QA/9904154
10 I. Shapiro
[8] Hajac P.M., Khalkhali M., Rangipour B., Sommerhäuser Y., Hopf-cyclic homology and cohomology with
coefficients, C. R. Math. Acad. Sci. Paris 338 (2004), 667–672, arXiv:math.KT/0306288.
[9] Hajac P.M., Khalkhali M., Rangipour B., Sommerhäuser Y., Stable anti-Yetter–Drinfeld modules,
C. R. Math. Acad. Sci. Paris 338 (2004), 587–590, arXiv:math.QA/0405005.
[10] Halbig S., Generalised Taft algebras and pairs in involution, arXiv:1908.10750.
[11] Jara P., Ştefan D., Hopf-cyclic homology and relative cyclic homology of Hopf–Galois extensions, Proc.
London Math. Soc. 93 (2006), 138–174.
[12] Kassel C., Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), 195–216.
[13] Kerler T., Mapping class group actions on quantum doubles, Comm. Math. Phys. 168 (1995), 353–388,
arXiv:hep-th/9402017.
[14] Larson R.G., Radford D.E., Finite-dimensional cosemisimple Hopf algebras in characteristic 0 are semisim-
ple, J. Algebra 117 (1988), 267–289.
[15] Larson R.G., Radford D.E., Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), 187–195.
[16] Nenciu A., Quasitriangular pointed Hopf algebras constructed by Ore extensions, Algebr. Represent. Theory
7 (2004), 159–172.
[17] Radford D.E., Minimal quasitriangular Hopf algebras, J. Algebra 157 (1993), 285–315.
[18] Radford D.E., Westreich S., Trace-like functionals on the double of the Taft Hopf algebra, J. Algebra 301
(2006), 1–34.
[19] Shapiro I., On the anti-Yetter–Drinfeld module-contramodule correspondence, J. Noncommut. Geom. 13
(2019), 473–497, arXiv:1704.06552.
[20] Shapiro I., Categorified Chern character and cyclic cohomology, arXiv:1904.04230.
[21] Taft E.J., The order of the antipode of finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. USA 68
(1971), 2631–2633.
[22] Toën B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–
667, arXiv:math.AG/0408337.
https://doi.org/10.1016/j.crma.2003.11.036
https://arxiv.org/abs/math.KT/0306288
https://doi.org/10.1016/j.crma.2003.11.037
https://arxiv.org/abs/math.QA/0405005
https://arxiv.org/abs/1908.10750
https://doi.org/10.1017/S0024611506015772
https://doi.org/10.1017/S0024611506015772
https://doi.org/10.1016/0021-8693(87)90086-X
https://doi.org/10.1007/BF02101554
https://arxiv.org/abs/hep-th/9402017
https://doi.org/10.1016/0021-8693(88)90107-X
https://doi.org/10.2307/2374545
https://doi.org/10.1023/B:ALGE.0000026785.03997.60
https://doi.org/10.1006/jabr.1993.1102
https://doi.org/10.1016/j.jalgebra.2004.04.023
https://doi.org/10.4171/JNCG/327
https://arxiv.org/abs/1704.06552
https://arxiv.org/abs/1904.04230
https://doi.org/10.1073/pnas.68.11.2631
https://doi.org/10.1007/s00222-006-0025-y
https://arxiv.org/abs/math.AG/0408337
1 Introduction
2 Twisted Drinfeld double
3 Taft Hopf algebras
3.1 The identification with the dual
3.2 The case of p>2
3.3 The case of p=2
A Appendix
References
|
| id | nasplib_isofts_kiev_ua-123456789-211323 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T23:43:22Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Shapiro, Ilya 2025-12-29T11:11:58Z 2021 Mixed vs Stable Anti-Yetter-Drinfeld Contramodules. Ilya Shapiro. SIGMA 17 (2021), 026, 10 pages 1815-0659 2020 Mathematics Subject Classification: 16E35; 16T05; 18G90; 19D55 arXiv:2010.02768 https://nasplib.isofts.kiev.ua/handle/123456789/211323 https://doi.org/10.3842/SIGMA.2021.026 We examine the cyclic homology of the monoidal category of modules over a finite-dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter-Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter-Drinfeld contramodules (recently introduced). This research was supported in part by the NSERC Discovery Grant number 406709. The author wishes to thank the referees for their many helpful suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Mixed vs Stable Anti-Yetter-Drinfeld Contramodules Article published earlier |
| spellingShingle | Mixed vs Stable Anti-Yetter-Drinfeld Contramodules Shapiro, Ilya |
| title | Mixed vs Stable Anti-Yetter-Drinfeld Contramodules |
| title_full | Mixed vs Stable Anti-Yetter-Drinfeld Contramodules |
| title_fullStr | Mixed vs Stable Anti-Yetter-Drinfeld Contramodules |
| title_full_unstemmed | Mixed vs Stable Anti-Yetter-Drinfeld Contramodules |
| title_short | Mixed vs Stable Anti-Yetter-Drinfeld Contramodules |
| title_sort | mixed vs stable anti-yetter-drinfeld contramodules |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211323 |
| work_keys_str_mv | AT shapiroilya mixedvsstableantiyetterdrinfeldcontramodules |