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Results tagged with hopf-algebras
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user 118681
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 $G$.
2
votes
0
answers
27
views
Examples of (weak-)bialgebras/Hopf algebras with a finite dimensional unitary representation...
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 f …
- 727
7
votes
2
answers
199
views
Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra a...
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}) \} …
- 727
4
votes
Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra a...
As mentioned in David's answer, the smallest such $n$ is called the exponent of the Hopf algebra $H$. It was first conjectured by Kashina that that the exponent of a finite dimensional semisimple Hop …
- 727
4
votes
0
answers
76
views
Is the Drinfeld element of a semisimple quasitriangular Hopf algebra invariant under the Dri...
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
$$ …
- 727
8
votes
1
answer
265
views
How does the Tannaka duality work for weak Hopf algebras and fusion categories?
I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works i …
- 727
3
votes
2
answers
121
views
Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?
Let $H_1$ and $H_2$ be finite dimensional Hopf algebras that are twist equivalent, i.e. $H_2$ is obtained from $H_1$ using a Drinfeld twist. My question is: are the Drinfeld doubles $D(H_1)$ and $D(H_ …
- 727
0
votes
Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?
Actually, Proposition 3.2 of Etingof & Gelaki 99' gives an explicit construction of the Drinfeld twist $J$ relating $D(H_1)$ and $D(H_2)$ and is quite simple: let $H_2=H_1^J$. Since $H_1$ is a Hopf s …
- 727
4
votes
0
answers
60
views
Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoreti...
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent stateme …
- 727
5
votes
1
answer
74
views
Weakly involutive $R$-matrices and representations of the symmetric group $S_N$ in restricte...
An $R$-matrix is a matrix $R\in\operatorname{End}(V\otimes V)$ (where $V$ is a finite dimensional vector space) that solves the Yang–Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
where bo …
- 727
6
votes
2
answers
293
views
Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
I'm interested in solutions to the Yang-Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the sym …
- 727
1
vote
Accepted
Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
Here is a family of examples indexed by an integer $m\geq 3$.
Let
\begin{eqnarray}
R^{ij}_{kl}=\lambda_{ij}c_{kl}-\delta_{ik}\delta_{jl},\tag{1}
\end{eqnarray}
where $\lambda, c$ are $m\times m$ m …
- 727
3
votes
0
answers
54
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
2
votes
1
answer
74
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( …
- 727