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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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Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"

In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent ...
Lagrenge's user avatar
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6 votes
2 answers
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Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra always have finite order?

Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let $B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
Lagrenge's user avatar
  • 577
5 votes
0 answers
97 views

The Balmer spectrum and the thick tensor ideals of the derived category of a Hopf algebra

Given a Hopf algebra $H$ over a field $\mathbb{k}$, the category of finite-dimensional left-$H$-modules naturally becomes a rigid monoidal category with exact monoidal product. Thus clearly the ...
Jannik Pitt's user avatar
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50 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 ...
Mare's user avatar
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3 votes
1 answer
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quantum invariants, ribbon Tannakian duality and classification of ribbon Hopf algebras

In a nutshell, my question is: Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction? I will now make it more precise. One could define a ...
Léo S.'s user avatar
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2 votes
0 answers
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Is the Drinfeld element of a semisimple quasitriangular Hopf algebra invariant under the Drinfeld twist?

Let $A$ be a finite dimensional semisimple quasitriangular Hopf algebra over $\mathbb{C}$ with universal $R$-matrix denoted by $\mathcal{R}\in A\otimes A$. The Drinfeld element of $A$ is defined as $$...
Lagrenge's user avatar
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6 votes
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If a strong monoidal functor $F$ has an ambidextrous adjoint, then how close is the adjoint to being strong monoidal?

Let $F : C \to D$ be a strong (symmetric, say) monoidal functor. Suppose that $G : D \to C$ is both left and right adjoint to $F$ (an ambidextrous adjunction). Then by doctrinal adjunction $G$ is both ...
Tim Campion's user avatar
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3 votes
1 answer
400 views

Is Malcev completion an embedding?

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ...
Qwert Otto's user avatar
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Action of Hopf algebra of identity supported distributions on a Lie group

The Hopf algebra of identity supported distributions on a lie group is cocommutative. It is well known that it is a group object in the category of cocommutative coalgebras. Is there a canonical ...
Lefevres's user avatar
2 votes
1 answer
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Germs of left invariant differential operators on a group

Are there germs at the identity of linear differential operators on a group which are not germs at the identity of left invariant differential operators? I feel like the answer is no but the statement ...
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1 answer
217 views

Hopf algebras actions

Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions? There must be a common core, if the same term is ...
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2 votes
0 answers
59 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 ...
Christoph Mark's user avatar
3 votes
2 answers
115 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(...
Lagrenge's user avatar
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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", ...
Daigaku no Baku's user avatar
4 votes
1 answer
146 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: $$\...
pyroscepter's user avatar
3 votes
0 answers
91 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 $(V,\rho,\Delta)$ ...
Max Demirdilek's user avatar
6 votes
1 answer
192 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 ...
Max Demirdilek's user avatar
4 votes
0 answers
162 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 ...
Surajit's user avatar
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3 votes
0 answers
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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 ...
Max Demirdilek's user avatar
3 votes
0 answers
131 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 ...
David Gao's user avatar
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4 votes
0 answers
51 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 ...
szantag's user avatar
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6 votes
0 answers
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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 ...
matha's user avatar
  • 183
2 votes
1 answer
227 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)...
Andromeda's user avatar
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2 votes
0 answers
26 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 ...
Lagrenge's user avatar
  • 577
8 votes
1 answer
248 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 ...
Lagrenge's user avatar
  • 577
3 votes
1 answer
195 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 ...
Sergei Akbarov's user avatar
4 votes
2 answers
594 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 ...
Béla Fürdőház 's user avatar
2 votes
1 answer
120 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, ...
Béla Fürdőház 's user avatar
0 votes
0 answers
99 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 ...
yohei ohta's user avatar
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1 answer
189 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 ...
IrbidMath's user avatar
  • 103
7 votes
1 answer
339 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 ...
shin chan's user avatar
  • 301
0 votes
0 answers
136 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^{-...
Anacardium's user avatar
1 vote
0 answers
103 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 $(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$ $(2)$ $R_{21} = R^{-1},$ $(...
Anacardium's user avatar
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0 answers
92 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, ...
Anacardium's user avatar
4 votes
1 answer
165 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? ...
Jake Wetlock's user avatar
  • 1,154
2 votes
0 answers
88 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 ...
grok's user avatar
  • 2,499
5 votes
1 answer
298 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 ...
Qwert Otto's user avatar
5 votes
2 answers
231 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 ...
Lorenzo Del Vecchiopontopolos's user avatar
1 vote
0 answers
39 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\...
Syu Gau's user avatar
  • 435
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 ...
Didier de Montblazon's user avatar
4 votes
1 answer
212 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-...
Andromeda's user avatar
  • 169
1 vote
0 answers
87 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 ...
Mare's user avatar
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3 votes
1 answer
218 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 ...
Mare's user avatar
  • 26.3k
0 votes
1 answer
174 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). ...
user502786's user avatar
4 votes
1 answer
289 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: ...
Mare's user avatar
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1 vote
0 answers
121 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.
Didier de Montblazon's user avatar
2 votes
0 answers
78 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 $$...
Chris's user avatar
  • 264
3 votes
0 answers
145 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 ...
Antoine Labelle's user avatar
9 votes
0 answers
293 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 ...
Antoine Labelle's user avatar
5 votes
1 answer
201 views

States "absorbed" by a Haar idempotent on a compact quantum group

Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when $$a\bullet b=b=b\bullet a?$$ Can we say that $b$ absorbs $a$? Can we say ...
JP McCarthy's user avatar

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