Questions tagged [hopf-algebras]

A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a structure exists on the group algebra $k G$ of a finite group $G$.

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14 votes
1 answer
652 views

Quantum braid group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...
3 votes
0 answers
35 views

Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?

Let $H_1$ and $H_2$ be finite dimensional Hopf algebras that are twist equivalent, i.e. $H_2$ is obtained from $H_1$ using a Drinfeld twist. My question is: are the Drinfeld doubles $D(H_1)$ and $D(...
1 vote
0 answers
120 views

Can't parse a statement in an article on coalgebras and umbral calculus

This question is cross-posted from MSE. I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", ...
4 votes
1 answer
129 views

Existence of an element in a Hopf algebra that satisfies a 'flip' property

Consider a Hopf algebra $H$ where $S$ is the antipode and $\Delta$ is the coproduct. Given $a \in H$, I want to know if there always exists an element $b \in H$ satisfying the 'flip' property: $$\...
3 votes
0 answers
75 views

Yetter-Drinfeld modules for Hopf monads

1. Context. 1.1. Classical Yetter-Drinfeld modules. Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...
6 votes
1 answer
171 views

Hopf monads in categorical probability theory

1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
8 votes
1 answer
744 views

A "concrete" example of a one-sided Hopf algebra

I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition. To be more ...
4 votes
0 answers
159 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 ...
3 votes
0 answers
114 views

Tannaka duality for Hopf algebroids

Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional ...
3 votes
0 answers
124 views

Generalized wreath products of commutative algebras with Hopf algebras

Fix $k$ a commutative ring (or, if more convenient, assume it’s a field or even an algebraically closed field of characteristic 0, which is the case I’m mainly interested in). Let $A$ be a unital ...
9 votes
0 answers
256 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 ...
7 votes
0 answers
329 views

An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra. There already exists a generalization of Cauchy theorem using exponent, see [KSZ06]. We are interesting in an alternative ...
3 votes
0 answers
43 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 ...
2 votes
1 answer
220 views

Completely contractive Banach algebra structure on the dual of a Hopf $C^*$-algebra

Let $(A, \Delta)$ be a Hopf $C^*$-algebra, i.e. $A$ is a $C^*$-algebra, and $\Delta: A \to M(A\otimes A)$ is an injective non-degenerate $*$-homomorphism that is coassociative: $$(\iota \otimes \Delta)...
6 votes
0 answers
112 views

How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?

Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity. Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...
5 votes
2 answers
215 views

An algebra map between Hopf algebras that does not commute with the counit

Let $(H,\Delta,\epsilon,S)$ be a Hopf algebra. Can there exist an algebra map $\phi:H \to H$ such that $$ \epsilon(\phi(g)) \neq \epsilon(g), ~~~~~ \textrm{ for some } g \in H? $$ Does the anti-pode ...
8 votes
1 answer
208 views

How does the Tannaka duality work for weak Hopf algebras and fusion categories?

I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...
2 votes
0 answers
22 views

Examples of (weak-)bialgebras/Hopf algebras with a finite dimensional unitary representation and corepresentation and polynomial growth rate

I need examples (the more the better, even better if there is a systematic way of construction) of (weak-)bialgebras or (weak-)Hopf algebras $H$ with a finite dimensional representation $\rho$ and a ...
3 votes
1 answer
300 views

Is there a definition of Heisenberg double for quasi-Hopf algebras?

$\newcommand{\dmod}{\text{-}\mathrm{mod}}$Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $k$ and $A\dmod$ be the category of finite dimensional left $A$-modules. Since $A\dmod$ is a ...
3 votes
1 answer
192 views

Is the preimage of a Hopf subalgebra a Hopf subalgebra?

I asked this a week ago at MSE, but nobody replied. Could anybody enlighten me if the following is true? Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in ...
4 votes
1 answer
276 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: ...
7 votes
1 answer
311 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 ...
4 votes
2 answers
571 views

An algebra with more than one Frobenius algebra structure

Can a (finite dimenaional) $\mathbb{K}$-algebra $A$ be equipped with more than one Frobenius structure $\lambda:A \to \mathbb{K}$? Of course we identify two structures $\lambda$ and $\lambda'$ if they ...
2 votes
1 answer
114 views

Associated graded algebras and symmetric Frobenius algebras

Let $A$ be a filtered algebra and let $G$ be its associated graded algebra. As discussed in this question, if $G$ is Frobenius, then $A$ is also Frobenius. If $G$ is a symmetric Frobenius algebra, ...
0 votes
0 answers
88 views

Concrete examples of quantum duality principle

Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
0 votes
1 answer
185 views

Milnor and Moore paper "On the structure of Hopf algebra" proposition 1.7 , filtration and grading

In the Milnor and Moore paper, "On the structure of Hopf algebras" proposition 1.7 said the following: $A$ a connected $K$-algebra. $N$ a left $A$ module that is connected as a $K$-graded ...
0 votes
0 answers
133 views

Is $[n]_q!$ invertible in $\mathbb C [[h]]\ $?

Consider the Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F, H$ and relations $:$ $$[H, E] = 2 E,\ \ [H, F] = - 2 F,\ \ [E, F] = \frac {q^H - q^{-H}} {q - q^{-...
1 vote
0 answers
101 views

How does $R \equiv 1\ (\text {mod}\ h)\ $?

Definition $:$ Let $H$ be a Hopf algebra. An invertible element $R \in H \otimes H$ is called a coboundary structure on $H$ if $(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$ $(2)$ $R_{21} = R^{-1},$ $(...
4 votes
1 answer
154 views

Drinfeld-Jimbo quantum groups for $q=0$

In the Wikipedia page of Drinfeld--Jimbo quantum groups the values of $q=0,1$ are excluded so as to avoid dividing by zero. The $q=1$ case is discussed in this old question. What about the $q=0$ case? ...
0 votes
0 answers
89 views

How to show that quantum $sl_2 (\mathbb C)$ is a Hopf algebra deformation of $U (sl_2 (\mathbb C))\ $?

The quantum $sl_2 (\mathbb C)$ is the non-commutative, non-cocommutative Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F$ and $H$ with the relations $:$ $$[H, ...
11 votes
2 answers
1k views

Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has ...
2 votes
0 answers
82 views

Subcoalgebras of symmetric algebra

Consider the symmetric algebra $S(V)$, with its coalgebra structure: $\Delta(x)=1\otimes x+x\otimes1$ on $V$, extended multiplicatively. What are its subcoalgebras? In some vague sense, they seem to ...
4 votes
1 answer
204 views

Hopf algebra and coideal question

Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put $$B^+:= B\cap \ker(\epsilon).$$ It can be shown that $B^+$ is a two-...
0 votes
1 answer
166 views

Hopf algebra of representative k-valued functions of an abstract group

Let $G$ be an abstract group. If we can embed $G$ into a group $H$, in a way that we had $G$ and $H$ of the same Hopf algebra of representative $k$-valued functions ($R(G)\sim R(H)$ as Hopf algebras). ...
5 votes
1 answer
267 views

Malcev completion of free groups

Let $K$ be a field with $\operatorname{char} K=0$, $\hat{L}_n$ the complete free Lie algebra of $n$ variables $x_1,\dotsc,x_n$ and $\exp(\hat{L}_n)$ its associated group with the product given by BCH ...
1 vote
0 answers
36 views

Stable Picard group of the tensor product of two Hopf algebras

Suppose we know the stable Picard groups (=Picard group of the stable module category) of two cocommunicative Hopf $k$-algebras $G$ and $H$. Is it possible to deduce the stable Picard group of $G\...
0 votes
0 answers
63 views

A name for "anti-symmetric" Frobenius algebras?

Let $V$ be an $N$-dim vector space and denote its exterior algebra by $\Lambda^{\ast}(V)$. The algebra $\Lambda$ has an obvious Frobenius algebra structure $B(-,-)$ given by wedgeing and then ...
0 votes
0 answers
99 views

Comodules category

Let $G$ be an abstract group, under which conditions we may have equivalent (resp. isomorphic) categories $Mod_{G}$ and $Comod_{R(G)}$, of $G-$modules and $R(G)-$comodules, where, $R(G)$ stands for ...
1 vote
0 answers
83 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 ...
3 votes
1 answer
213 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 ...
19 votes
2 answers
1k views

Hopf algebra reference

I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf ...
4 votes
1 answer
182 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 vote
0 answers
114 views

Classifying of low-dimensional Frobenius algebras

Does there exist a classification of finite-dimensional Frobenius algebras in low dimensional cases. This question is motivated by the following analogous question for Hopf algebras.
14 votes
2 answers
1k views

Is a bialgebra with all group-like elements invertible a Hopf algebra?

We know that in a Hopf algebra all group-like elements are invertible. Is the converse also true? Here is the precise formulation of my question : Let $B$ be a bialgebra and $GLE$ = { $g \in B ~|~ g \...
9 votes
1 answer
461 views

Is a biideal of a Noetherian Hopf algebra automatically Hopf?

Let $H$ be a Hopf algebra over a field $k$, and $I$ be a biideal of $H$. I am looking for conditions that guarantee that $I$ is a Hopf ideal (that means $S\left(I\right)\subseteq I$). One condition ...
9 votes
1 answer
426 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
2 votes
0 answers
75 views

Is there a coalgebraic definition of filtered algebras?

If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define $$...
4 votes
2 answers
187 views

Bialgebras with rigid representation theory

Repost from math.SE since no answer after two months, but feel free to close if not appropriate: Everything is finite-dimensional over a field $k$. Let $B$ be a bialgebra with $B\text{-mod}$ its ...
20 votes
10 answers
4k views

Hopf algebras examples

Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course. Some of the examples I know are well-...
3 votes
0 answers
130 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 ...

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