# Questions tagged [quantum-groups]

Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.

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### Classification of $\operatorname{Rep} D(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
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### Submodules of $V\otimes V^*$

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...
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### Embedding problems on quantum groups？

We work over the field of complex numbers. We have known that Lie algebra of type $A_2$is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
383 views

### Comultiplication of elements of partition of unity

Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra). Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
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### F-symbols for compact Lie groups

Consider a Unitary Modular Tensor Category constructed from the quantum group of some compact, simple and simply-connected Lie group $U_q(G)$ at $q=e^{2\pi i/(k+h)}$ for some integer $k$. In general, ...
301 views

### Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?

Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...
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### Tannaka-Krein duality in Chari-Pressley's book

I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here. V.Chari and A.N.Pressley in their "Guide to Quantum ...
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### Some quantities associated to finite dimensional Hopf algebras

let $(H,\Delta,m,s)$ be a Hopf algebra. To this Hopf algebra one can associate two obvious linear maps $T_H, S_H: H \to H$ with $T_H=m\circ \Delta,\quad S_H=s$. Are there two finite dimensional ...
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