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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Generalized antipode of bialgebras with respect to a given homomorphism
Let $k$ be a field, say $k=\mathbb C$. If $(A,\Delta_A,\varepsilon_A)$ is a coalgebra and $(B,m_B,1_B)$ is an algebra, then $\mathrm{Hom}_k(A,B)$ is an algebra under the convolution product
$$
(f*g)(...
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Universal Hopf symmetry of an algebra
Let $A$ be an algebra and $H$ be a Hopf algebra. We say that $H$ is a Hopf symmetry of $A$ if $A$ is an $H$-module algebra, i.e., if $H$ has an action on $A$ that respects the multiplication and unit ...
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Does a fiber functor of the Drinfeld center $Z(\mathcal{C})$ always imply a fiber functor of $\mathcal{C}$?
Let $\mathcal{C}$ be a unitary fusion category, and let $Z(\mathcal{C})$ be its Drinfeld center. Suppose $Z(\mathcal{C})$ has a fiber functor $F: Z(\mathcal{C})\to \mathrm{Vec}$. Question: must $\...
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Hopf pairing on the Drinfeld quantum double
I've often encountered (even on this forum) the claim that if $A$ and $B$ are Hopf algebras together with a Hopf pairing $A \otimes B \rightarrow \mathbb{k}$, then the Drinfeld quantum double $D = A \...
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Restricting domains of coalgebra duals
Take a Hopf algebra $H$ and a subalgebra $A$, then look at the Hopf dual $H^{\circ}$ of $H$ and the dual coalgebra $A^{\circ}$ of the algebra $A$. We have a map $\rho:H^{\circ} \to A^{\circ}$ given ...
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Relation between completions of $U_q(\mathfrak{sl}_2)$
Consider the quantum group $U_q(\mathfrak{sl}_2)$ for generic $q$, which is a $\mathbb{C}$-algebra. There are two "completions" of this quantum group that one can consider to accommodate a ...
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On the quantum SL(2) group
I'm trying to get a better understanding of seemingly different concepts of quantum groups, especially in the case of the $SL(2)$ group.
(1) Drinfeld-Jimbo deformation
I'm following "Quantum ...
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Non-Unitary comodules of Hopf *-algebras
Let $H$ be a Hopf algebra endowed with a $*$-structure, making it a Hopf $*$-algebra. Take a finite-dimensional $H$-comodule $(V,\Delta_V)$ and choose a basis $e_i$. In terms of this basis we have ...
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Lie group homomorphisms and Hopf algebras homomorphisms correspondence
I was reading this post: Is it possible to construct a formal group law from a Lie group without choosing coordinates?.
My question is how the correspondence between Lie groups and formal groups can ...
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Up to what order are finite-dimensional Hopf algebras classified?
Up to what order are finite-dimensional Hopf $\mathbb{C}$-algebras classified? Is there a table of this classification available somewhere?
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linear poisson structure on dual of universal enveloping algebra
Given a Lie algebra $\mathfrak{g}$ which has a universal enveloping algebra $U(\mathfrak{g})$ and $\mathfrak{g}^{*}$ is a linear poisson structure. Is there a canonical linear poisson structure on $U(...
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linear poisson algebra and universal enveloping algebra
I am intrested in the concrete case of a Lie algebra $\mathfrak{g}$ which has a universal enveloping algebra $U(\mathfrak{g})$ and $\mathfrak{g}^{*}$ is a linear poisson structure. Is the Sweedler ...
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non-quasitriangular Hopf algebras whose category of modules is lax braided
A Hopf algebra $H$ is called quasitriangular if it has an invertible $R$-matrix. In this case, the category of left $H$-modules becomes a braided monoidal category. I known that Sweedler’s 4-...
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Module Algebra and Smash Product
$\newcommand\hash{\mathbin\#}$I am reading the paper "E. Kirkman, J. Kuzmanovich, and J. J. Zhang. “Gorenstein Subrings of Invariants under Hopf Algebra
Actions”. In: Journal of Algebra 322 (2009)...
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The Drinfeld-center of the category of Hopf algebra (co)modules
Consider $\mathcal{M}_A$ the monoidal category of right $A$-modules, for some Hopf algebra $A$, where the tensor product of two $A$-modules is defined in the usual way via the coproduct of $A$.
In the ...
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