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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 $G$.

32 questions from the last 365 days
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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[...
Justin Bloom's user avatar
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 ...
Alonso Perez-Lona's user avatar
13 votes
1 answer
280 views

Finiteness of the number of Hopf subalgebras

Let $H$ be a finite-dimensional Hopf algebra over the complex field. Question: Does $ H $ have a finite number of Hopf subalgebras? In the case where $ H $ is semisimple, the answer is yes. According ...
Sebastien Palcoux's user avatar
3 votes
0 answers
87 views

Functorial relationships between Hopf algebras and rough paths

In rough paths theory, the signature of a path $x: [0, T] \to \mathbb{R}^d$ is an element in the tensor algebra $T((\mathbb{R}^d))$. These signatures reside within the group-like elements and ...
Furdzik Zbignew's user avatar
13 votes
0 answers
332 views

Lie theory for quantum groups?

$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives: Compact quantum groups in the sense of Woronowicz. Deformation of the universal enveloping algebra of a Lie algebra in ...
user82261's user avatar
  • 357
5 votes
1 answer
179 views

Semisimplicity of algebras in fusion categories

Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
AffSch's user avatar
  • 61
6 votes
0 answers
349 views

Quantum Hilbert's fifth problem

Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case. The definition of a quantum ...
Sebastien Palcoux's user avatar
3 votes
0 answers
65 views

A combinatorial Dyson-Schwinger equation, tree diagrams, and compositional inversion of a Laurent series

In "Tree hook length formulae, Feynman rules and B-series", Bradley Jones and Karen Yeats state on pg. 9: Combinatorial Dyson-Schwinger equations are functional equations with solutions in $...
Tom Copeland's user avatar
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0 votes
0 answers
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Categorical duals for Yetter-Drinfeld modules [duplicate]

Yetter-Drinfeld (YD) modules appear naturally in the theory of Hopf algebras. They are both modules and comodules at the same time, satisfying a certain compatibility condition, as presented here. The ...
Yilmaz Caddesi's user avatar
0 votes
0 answers
124 views

Do the following two notions of quantum groups sometimes coincide?

On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
Raoul's user avatar
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3 votes
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$G$-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 ...
Zhiyuan Wang's user avatar
2 votes
1 answer
78 views

Does there exist a nontrivial triangular weak Hopf algebra?

Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair ($H,\mathcal{R}$) where $H$ is a WHA and $\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(...
Zhiyuan Wang's user avatar
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 ...
Zhiyuan Wang's user avatar
4 votes
0 answers
68 views

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 ...
Zhiyuan Wang's user avatar
7 votes
2 answers
201 views

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 $B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
Zhiyuan Wang's user avatar
5 votes
0 answers
130 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
  • 1,474
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 ...
Mare's user avatar
  • 26.5k
3 votes
1 answer
165 views

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
  • 213
4 votes
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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 $$...
Zhiyuan Wang's user avatar
6 votes
0 answers
122 views

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
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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
0 votes
0 answers
51 views

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
90 views

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 ...
user avatar
0 votes
1 answer
294 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 ...
user avatar
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 ...
Christoph Mark's user avatar
3 votes
2 answers
135 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(...
Zhiyuan Wang's user avatar
1 vote
0 answers
130 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", ...
Daigaku no Baku's user avatar
4 votes
1 answer
161 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
98 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
207 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
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 ...
Surajit's user avatar
  • 73
3 votes
0 answers
133 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 ...
Max Demirdilek's user avatar