# Questions tagged [quantum-groups]

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure.

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### Relation between integrable representations and highest weight representations.

Let $g$ be a simple Lie algebra and $U_q(g)$ the corresponding quantum group. What are the relation between integrable representations and highest weight representations of $U_q(g)$? Are all highest ...
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### Von Neumann algebraic Quantum Object is direct sum of type I factors

I am looking at the non-standard quantum projective spaces $A:=\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. Now I want to show that if I take the von Neumann algebra ...
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### Endomorphism ring of an irreducible comodule

Given a Hopf algebra over $\mathbb{C}$, and a irreducible comodule $V$, I can show that Mor$(V,V) \simeq \mathbb{C}$, where Mor$(V,V)$ is the space of comodule maps from $V$ to $V$. My proof uses that ...
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### Quantum group notation

I was jumping into the deep end and reading a few papers and lectures on quantum groups. My knowledge on Lie algebras is a bit thin but I was just wondering the notation used in the starting of this ...
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### Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...
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### A question of the book “a guide to quantum groups”

I am reading this book "a guide to quantum groups" written by V.C. and A.P. But the proof of propersition 4.2.3 on page 121 confused me. Just this place " Applying $id \bigotimes S \bigotimes S^2$ to ...
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### Why the coproduct of quantum groups are defined in this way?

Let $U_q(g)$ be a quantum group generated by $e_i, f_i, k_{\lambda}$, $\lambda \in Q$, $Q$ is the weight lattice of the Lie algebra $g$. The coproduct of $U_q(g)$ is defined as follows (I only write ...
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### An equation about the Sklyanin bracket

I am reading the lecture http://www.math.uiuc.edu/~ruiloja/Poisson2014/EtingofLectures.pdf. I have a question on page 25. I do not know how to calculate the equation (3.2). Thank you very much for any ...
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### How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed?

I am reading the book a guide to quantum groups. I have a question on page 18. How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed? Any help will be greatly appreciated!...
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### How calculate the cartan matrix of the twisted quantum affine algebras?

the cartan matrix of the type $A_{2}^{(2)}$, $A_{2r-1}^{(2)}$, $A_{2r-1}^{(2)}$, $D_{r+1}^{(2)}$, $E_{6}^{(2)}$. I know the cartan matrix of the type $A_{2}^{(2)}$ is \begin{align} \left( \begin{...
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### How to compute Casimir elements of $g \otimes g$?

Let $g$ be a Lie algebra. How to compute Casimir elements of $g \otimes g$? I am asking this question because in the book a guide to quantum groups, page 80, there is an equation $r_{12} + r_{21}=t$, ...
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### Relations between Lie algebras and Lie coalgebras.

Let $g^*$ be the dual vector space of a vector space $g$. Suppose that $g^*$ is a Lie algebra and $[,]_{g^*}: \Lambda^2 g^* \to g^*$ satisfies the Jacobi identity. Let $\delta: g \to \Lambda^2 g$ be ...
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### Notation in quantum groups.

The quantum group $U_q(sl_3)$ is generated by $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$ subject to some relations. I read some papers and there is a notation $K_{\lambda}$, where $\lambda$ is ...
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### The functor from $sl_2-mod$ to $U_q(sl_2)-mod$.

Let $sl_2-mod$ be the category of all finite dimensional $sl_2$-modules and let $U_q(sl_2)-mod$ be the category of all finite dimensional $U_q(sl_2)$-modules, $q$ is not a root of unity. It is said ...
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### Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?

Let $g$ be a complex simple Lie algebra and $U_q(g)$ the corresponding quantum group. Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional ...
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### Relation between quantum affine algebra $U_q(\widehat{sl_2})$ and the affine Lie algebra $\widehat{sl_2}$?

The relation between the definition of quantum group and correpsonding lie algebra is discribed here. Are there some similar relation between $U_q(\widehat{sl_2})$ and $\widehat{sl_2}$? There are two ...
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### Why does $\operatorname{Ad}_h((S\otimes 1)(Q))=\epsilon(h)(S\otimes 1)(Q)$ in a quasi-triangular Hopf algebra?

I'm reading a proof that in a quasi-triangular Hopf algebra $H$, $(S\otimes 1)Q$ is $\operatorname{Ad}$-invariant. Here $Q=\tau(R)R$, where $R$ is the invertible element in $H\otimes H$ satisfying all ...
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### Is this system of inequalities (and equality) tractable?

I have some real parameters here. The $\mu_i$ - for $i=1,2,3,4,5$ - are 'convex coefficents' in that $\mu_i\geq 0$ and $\sum_{i}\mu_i=1$. The $x$ and $z$ are such that $x^2+z^2\leq 1$. The ...
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### The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional ...