All Questions
Tagged with hopf-algebras representation-theory
91
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On the orthogonality relations for quantum Clebsch-Gordan coefficients
I am currently reading Quantum groups by Christian Kassel, more precisely, section VII.7 which is about quantum Clebsch-Gordan coefficients.
To make this question self-contained, let me introduce the ...
- 4,232
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29
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$\mathbb{C}G$-modules and $\mathbb{C}^{G}$-comodules
I know that representations of a group $G$ are essentially $\mathbb{C}G$-modules. How is it that every $\mathbb{C}G-$module is also equivalent to a $\mathbb{C}^{G}-$comodule. I have not found this ...
- 659
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How does base change affect group schemes?
I am reading through Waterhouse's Introduction to Affine Group Schemes and am having trouble understanding how I should think of base change as it relates to the way a groups scheme "looks".
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1
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1
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34
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Corepresentations of quantum subgroups
If $H$ is a Hopf subalgebra of the Hopf algebra $G$ and $\alpha:V\longrightarrow V\otimes H$ is right corepresentation of $H$ on the vector space $V$ (finite--dimensional), is there a way to induce a ...
- 139
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24
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About the regular representation of weak hopf algebra
In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now ...
- 11
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Definition of equivalent representation of Hopf algebras
I have a question about equivalent representation of Hopf algebras because I am not unfamiliar with Hopf algebras.
Here is my question:
If $\rho_1$ and $\rho_2$ are two representations of a Hopf ...
- 1,159
4
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38
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Where does the unitarity structure of $U_q(\mathfrak{sl}_2)$ come from?
It is known that when $q$ is the root of unity, the representation of the quantum group $U_q(\mathfrak{sl}_2)$ is a unitary modular tensor category. However, if we want it to have the dagger structure,...
- 41
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Regular functions on torsors
Let $X \rightarrow Y$ be a torsor for a linear algebraic group $G$ (i.e. $X$ is a principal $G$-bundle over $Y$). Assume also that both $X$, and $Y$ are affine. What can be said about $\mathbb{C}[X]$ ...
- 135
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69
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When is an image of a Hopf algebra a Hopf algebra?
Suppose $R\subseteq S$ is a flat extension of rings and $A/R$, $B/S$ are flat Hopf algebras. Let $\varphi:A\otimes_R S\to B$ be a surjective $S$-Hopf algebra homomorphism. When is it the case that $\...
- 593
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How to construct irreducible representations of a Hopf algebra based on a finite non-Abelian group (beyond quantum double)?
I'm ultimately trying to construct a Hopf algebra based on a finite non-abelian group $G$ such that its irreducible representations:
are labeled by $g\in G$ and $\rho\in\text{irr}(G)$
have the fusion ...
- 153
6
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142
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Is this a valid q-deformation of $SU(2)$ via the $SU(3)$ Lie algebra?
In my physics research, I stumbled upon a rather simple deformation of the $SU(2)$ algebra, and I was wondering whether it qualifies as a `$q$-deformed Lie algebra' (a notion which is unfamiliar to me)...
- 1,436
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$T(z) e^{-\partial_z} $ for Yangian is a Manin matrix
Let $T(u)$ be the generating matrix of the Yangian $Y(\mathfrak{gl}_n)$ of $\mathfrak{gl}_n$. So we have the identity $[T_{ij}(u),T_{kl}(v)]=\frac{1}{u-v}(T_{kj}(u)T_{il}(v)-T_{kj}(v)T_{il}(u))$.
We ...
- 135
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1
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68
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Question about Hopf algebra and actions on endomorphisms rings
Let $H$ be a Hopf algebra and $M, N$ two left $H$-modules, then we can define a left $H$-module structure on $\mathrm{Hom}(M,N)$ given by $(h * f)(m) = h_1 f(S(h_2) m)$, where $S$ is the antipode. My ...
- 1,684
3
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118
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Proof of Tannaka recognition theorem
I am trying to prove the theorem 5.12.7 in the book “Tensor categories” by Etingof, Gelaki, Nikshych and Ostrik.
The statement is as follows:
The assignments
$(\mathcal{C},F) \mapsto H = \mathrm{End}(...
- 31
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For two modules over a Hopf algebra $H$, are the module homomorphisms the same as the $H$-invariant linear maps?
Let $H$ be a Hopf algebra over a field $k$ and $V, W$ two $H$-modules. The antipode and comultiplication on $H$ allow us to turn $\mathrm{Hom}_k(V, W)$ into a $H$-module by setting
$$
(h \cdot f)(v) = ...
- 3,607