All Questions
Tagged with rt.representation-theory hopf-algebras
120 questions
2
votes
1
answer
141
views
Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring.
There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
- 542
5
votes
0
answers
92
views
$\text{Rep}(D_4)$ and its three fiber functors
It is well-known that the fusion category $\text{Rep}(D_4)$ of representations of the dihedral group $D_4$ of order 8 admits three distinct fiber functors. Therefore, there are three different Hopf ...
3
votes
0
answers
60
views
$G$-crossed (braided) fusion categories and Tannaka duality
Many important concepts in tensor category theory have their counterpart in Hopf algebra theory under Tannaka duality. They have the general form: let $A$ be an XX-algebra, and let Rep$(A)$ denote the ...
- 727
5
votes
1
answer
80
views
Weakly involutive $R$-matrices and representations of the symmetric group $S_N$ in restricted subspaces of $V^{\otimes N}$
An $R$-matrix is a matrix $R\in\operatorname{End}(V\otimes V)$ (where $V$ is a finite dimensional vector space) that solves the Yang–Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
where ...
- 727
0
votes
0
answers
55
views
Weakly symmetric Hopf algebras
Let $A$ be a finite dimensional Hopf algebra over a field $K$ that is weakly symmetric (meaning $soc P = top P$ for each indecomposable projective $A$-module P).
Question: Is $A$ then automatically ...
- 26.5k
2
votes
0
answers
69
views
Is anything known about the center of the Fomin-Kirillov algebra?
Let $\mathcal{B}_{\mathbb{S}_m}$ be the quotient of the Fomin-Kirillov algebra so that its pairing becomes certainly nondegenerate. This algebra is conjecturally isomorphic to the Fomin-Kirillov ...
- 1,066
4
votes
0
answers
168
views
Representations of $C\left(SO_q(n)\right)$
A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
- 73
4
votes
0
answers
56
views
When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?
Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...
- 143
7
votes
1
answer
365
views
Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?
What is the smallest group $G$ such that $\operatorname{Rep}(G)$ has a non-symmetric braiding (or just an easy example)?
I seem to remember a result classifying all universal $R$-matrices of $\mathbb ...
- 301
1
vote
0
answers
90
views
Brauer trees that are Hopf algebras
Let $T$ be a Brauer tree with associated Brauer tree algebra $KT$ for some field $K$.
Question 1: For which Brauer trees does there exist a field $K$ such that $KT$ is a Hopf algebra (or more ...
- 26.5k
3
votes
1
answer
219
views
Classification of periodic Hopf algebras
Let $A$ be a finite dimensional algebra over a field $K$.
$A$ is called periodic if $A$ as an $A$-bimodule is a periodic module, that is $\Omega^n(A) \cong A$ for some $n \geq 1$. Being periodic ...
- 26.5k
5
votes
1
answer
318
views
Up to date summary on semisimple Hopf algebra over $\mathbb{C}$
Question: Is there an up to date summary of results on the classification of semisimple Hopf algebras over $\mathbb{C}$ (or a field of characteristic 0)?
Here are some questions I wonder about:
...
- 26.5k
3
votes
0
answers
151
views
Is there a classical version of Yetter-Drinfeld modules?
One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$.
If we think ...
- 3,446
9
votes
0
answers
325
views
Equivalence of Yetter-Drinfeld modules to Drinfeld center: is there a purely categorical proof?
Let $H$ be an Hopf algebra over a field $k$, and let $\mathcal{C}$ be the monoidal category of left $H$-modules.
It is known that the Drinfeld center of $\mathcal{C}$ is equivalent (as a braided ...
- 3,446
2
votes
0
answers
62
views
Quiver and relations for Hopf algebras associated to quiver algebras
Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$.
$A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via
$[x,y]=xy-yx$ and $x^{p}=x^p$. ...
- 26.5k
2
votes
0
answers
84
views
Representation finite Hopf algebras up to stable equivalence
It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra.
Question: Is it true that every representation-finite Hopf algebra is stable ...
- 26.5k
1
vote
1
answer
204
views
Equivariant description of indecomposable elements in shuffle algebra
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Tor{Tor}$Let's suppose $V$ is a $k$-vector space equipped with its standard (left) $\GL (V)$-action. The shuffle algebra is the graded dual of the ...
- 553
1
vote
0
answers
112
views
When are Brauer tree algebras Hopf algebras?
Question 1: Which Brauer tree algebras are Hopf algebras?
For example every representation-finite group algebra is a Brauer tree algebra and thus a Hopf algebra, but not every Brauer tree algebra ...
- 26.5k
6
votes
0
answers
148
views
What about Hopf algebra and fusion structures for intertwiner algebras?
Let $G$ be a complex, reductive group and let $V_1, \dotsc, V_r$ be a collection of finite dimensional, irreducible complex
representations of $G$. Let $\mathcal{A} = \mathrm{End}_G(V_1 \otimes \dotsb ...
- 2,137
1
vote
0
answers
44
views
Representation-finite blocks of Hopf algebras up to derived equivalence
Question: Which representation-finite selfinjective algebra is derived equivalent to a block of a (finite dimensional) Hopf algebra?
Famous examples are all Brauer tree algebras. Are there more ...
- 26.5k
3
votes
0
answers
59
views
Is there a condition such that the $A$ action of a $A \rtimes H$-module is a restriction of the $H$-action?
Let $H$ be a Hopf algebra and $A$ a subalgebra of $H$ such that $A$ is a left coideal of $H$ (that is $\Delta(A) \subset H \otimes A$) and $A$ is preserved by the adjoint action of $H$.
Consider now ...
- 437
7
votes
1
answer
254
views
Group-like elements in quotients of group rings
$\DeclareMathOperator\Gr{Gr}$Let $R$ be a local ring, let $A$ be a finite abelian group, and let $I$ be a Hopf ideal of the ring $R[A]$. The quotient $R[A]\twoheadrightarrow R[A]/I$ induces a map on ...
- 308
4
votes
2
answers
136
views
If $\operatorname{Hom}(\delta_V, \delta_W) = 0$, then $\mathfrak{C}(\delta_V) \cap \mathfrak{C}(\delta_W) = 0.$
Let $(A, \Delta)$ be a Hopf $^*$-algebra and $\delta_V: V \to V \otimes A$ and $\delta_W: W \to W \otimes A$ be two corepresentations of $(A, \Delta).$
Assume that the space of intertwiners $\...
- 175
4
votes
3
answers
540
views
Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule
A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion.
Now every Hopf algebra $H$ admits a one-dimensional ...
- 1,144
8
votes
1
answer
537
views
Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig
"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
- 18.4k
6
votes
1
answer
591
views
Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$
$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between
$$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$
The proof in "Tensor Categories ...
- 91
4
votes
1
answer
101
views
Non-cosemisimple duals of pointed Hopf algebras
I take the following quote from an answer to this question
A Hopf algebra is called pointed if all its simple left (or right)
comodules are one-dimensional. The quantized enveloping algebras and
...
- 145
0
votes
0
answers
106
views
Hopf algebra antipodes and right left comodule equivalences
Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
- 1,144
5
votes
1
answer
211
views
Simple quotients of a triple tensor product
Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let also $V_1, V_2, V_3$ finite-dimensional simple modules over $\mathcal{H}$ and $Q$ be a simple quotient of $V_1\otimes V_2\otimes V_3$. Is it ...
- 353
4
votes
0
answers
118
views
Examples of semisimple Hopf algebras where the category of representations has certain properties
I wish to find an example of a semisimple (hence finite dimensional) Hopf algebra $H$ with the following properties:
$H$ is nontrivial (i.e. not a group algebra or the dual of one);
The category of ...
- 41
2
votes
2
answers
496
views
Hopf algebra structure on Frobenius algebras
It was shown by Abrams (see https://www.sciencedirect.com/science/article/pii/S0021869399979012 ) that every Frobenius algebra has a canonical coalgebra structure.
Question 1: Has it been studied ...
- 26.5k
4
votes
3
answers
344
views
Coinvariants of tensor products of Hopf algebras
Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
3
votes
1
answer
104
views
Irreducibility of product bicomodules
Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right,
$H$-comodule respectively. The tensor product
$$
V \otimes W
$$
has an obvious $H$-$H$-bicomodule structure.
If $V$ and $W$ are ...
- 1,144
5
votes
1
answer
215
views
Classification of $\operatorname{Rep}D(H)$
Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
- 5,230
2
votes
1
answer
98
views
A weaker version of strongly graded algebras
Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that
$$...
5
votes
2
answers
680
views
Characters on Hopf algebras
For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
8
votes
3
answers
528
views
Classification of $\operatorname{Rep} D(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
- 5,230
1
vote
0
answers
139
views
Submodules of $V\otimes V^*$
Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...
- 353
5
votes
2
answers
462
views
Subfunctor of internal Hom
Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{...
- 353
6
votes
0
answers
338
views
Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?
Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...
- 63.9k
1
vote
1
answer
165
views
"Nice" bases for finite dimensional semisimple Hopf algebras
Let $H$ be a finite dimensional semisimple Hopf algebra over $\mathbb{C}$. Can one choose a basis $\{h_1, \dots, h_n \}$ of $H$, where $h_1 = 1$, such that if we write
$$
\Delta(h_i) = \sum_{1 \leq j,...
- 13
1
vote
0
answers
249
views
Images and Kernels of tensor products of homomorphisms of modules
Let $\mathcal{H}$ be a Hopf algebra and let $M_1,M_2,N_1,N_2$ modules over $\mathcal{H}$. If $f:M_1\rightarrow N_1$ and $g:M_2\rightarrow N_2$ are homomorphisms of modules, then are the following ...
- 353
6
votes
1
answer
294
views
What properties of a finite group fibre functor give its endomorphisms a hopf algebra structure?
Tannaka duality for a finite group lets us recover the group algebra $\mathbb{C}[G]$ as the endomorphisms of the forgetful functor $F:RepG\rightarrow Vect$, and taking the monoidal automorphisms ...
- 1,949
2
votes
1
answer
97
views
When is $N^{*} \otimes_K M$ projective for a local Hopf algebra?
Given a finite dimensional local Hopf algebra $A$ over a field $K$ and two finite dimensional indecomposable modules $N$ and $M$. Is it known when the module $N^{*} \otimes_K M$ is projective? Can ...
- 26.5k
4
votes
1
answer
367
views
Examples of basic coalgebras
For an algebraically closed field $k$, let $C$ be a $k$-coalgebra. Given a minimal injective cogenerator $E$, there is a so-called basic coalgebra $B_C=coend^C(E)$, s.t. the comodule categories $Mod^C$...
- 1,813
11
votes
4
answers
2k
views
The tensor product of two monoidal categories
Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation ...
- 835
5
votes
2
answers
403
views
Indecomposable, non-simple, modules of quantum groups at roots of unity
Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...
- 7,746
5
votes
0
answers
218
views
Lusztig's completion for universal enveloping algebra
In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
- 677
12
votes
3
answers
849
views
Subalgebra of a group algebra
Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra.
Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$.
Question: Is there any ...
- 5,230
4
votes
1
answer
215
views
Explicit examples of finite dimensional, involutive Hopf algebras with traceless antipode?
$\require{AMScd}$
In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $H$ where the antipode $S:H \to H$ is traceless.
Unfortunately, the method of proof in [...
- 1,231