In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. (Def: http://en.m.wikipedia.org/wiki/Quantum_group)

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How to show that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding?

Let $H$ be a bialgebra and ${}_H^H YD$ the category of Yetter-Drinfeld modules over $H$. It is said that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding. ...
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States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
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What are the applications of quantum shuffle multiplication?

I am writing a research paper on quantum shuffle multiplication, and there is just one piece that I'm missing: I want to give one or several examples of its applications, and so far I have not been ...
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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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A question of the paper“Crystallizing the q -Analogue of Universal Enveloping Algebras”?

I'm reading the paper "Crystallizing the q -Analogue of Universal Enveloping Algebras" written by Masaki Kashiwara. But there is something I don't know. Can anyone tell me how to use the ...
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Compatibility of Yetter-Drinfeld modules.

Let $H$ be a Hopf algebra. A Yetter-Drinfeld module over $H$ is a triple $(V, \cdot, \delta)$, where $\cdot : H \otimes V \to V$ , $\delta : V \to H \otimes V$ are actions and coactions respectively, $...
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How $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$

My question is how $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$ if we set $t=q^h$ and $q$ tends to 1.
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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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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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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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How to show that antipode is anti-Poisson and counit is Poisson?

I am reading the book Algebras of Functions on Quantum Groups: Part I by Leonid I. Korogodski and Yan S. Soibelman. I have a question about the proof of that antipode is anti-Poisson and counit is ...
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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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quantum matrices, quantum determinant

Homework, except I'm completely clueless, so if someone could potentially point me to similar worked examples or help explain this one step at a time it would be much appreciated. Could you explain ...
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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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Alternative Cotensor Definition

Let $H$ be a Hopf algebra, and $(M,\rho_r)$ and $(N,\rho_l)$ right and left $H$-comodules respectively. As usual, we define their cotensor product to be $$ M \square_H N := \text{ker}\{(\rho_r \...
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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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Representations of $SU_q(n)$

I am searching for a classification of all irreducibel representations of the quantum group $SU_q(n)$ for general $n$. Can someone give referenced or some statements about this? Moreover does one has ...
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Examples of Algebraic quantum groups

I am reading articles about algebraic quantum groups, which are defined (see A. Van Dael) as a regular multiplier Hopf algebra $(A,\Delta)$ for which there exists a non-zero functional $\varphi$ on $A$...
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What is $q^{\frac{1}{2}h \otimes h}$?

One can find in some sources (e.g. Pavel Etingof, Lectures on representation theory and KZ equation, page 91) formula for $R$ matrix (for $\mathfrak{sl}_2$ case) $$R = q^{\frac{1}{2} h \otimes h} \...
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The inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$.

I am reading the book. On page 80, there is a concept the inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$. Here $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra $...
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Why do we need the condition “cocommutative” in the definition of a coPoisson Hopf algebra?

In this paper, page 5, Section 3.6, in the definition of a coPoisson Hopf algebra $H$, it is said that: a coPoisson Hopf algebra is a cocommutative Hopf algebra $A$ with a map $\delta: A \to A \otimes ...
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Relation between Poisson brackets and Poisson bivectors.

I am reading the book a guide to quantum groups. I have some questions about the relation between Poisson brackets and Poisson bivectors. In the end of page 21 and in the beginning of page 22, it is ...
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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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How to compute $df(e)$ explicitly?

I am reading the book. On page 244, the formula (9.2.3.4). I would like to compute the bracket on g^* induced from the Poisson bracket on C[G] explicitly in the example of $G=SL_2$. The formula is: \...
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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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GNS construction of a weight

In the theory of quantum groups in the operator algebraic setting, one deals with weights (instead of positive linear functionals). Definition: A weight is a function $\phi $ : $A^+ \rightarrow [0, \...
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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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How to determining the roots and step operators of a L(SO(3)

I've come across a question in my "Lie Group and Lie Algebras for Physicists" course that asks me to determine the a basis for the Cartan subalgebra of $L(SO(3))$ and "hence find the roots and write ...
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Center of $U_q(g)$.

Let $g$ be a complex simple Lie algebra and let $U_q(g)$ be the corresponding quantum group. Is it true that the invariants of $U_q(g)$ under the adjoint action is the center of $U_q(g)$? It seems ...
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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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Homomorphisms of quantum spaces

Suppose we have a look at the Hopf $*$-algebra $U_q(sl(2))$ and the Hopf $*$-algebra $A_q(\widetilde{S}^3)$ introduced in the paper: http://arxiv.org/pdf/q-alg/9605017v1.pdf. I want to find a relation ...
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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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Trace of Product of Powers of $A$ and $A^\ast$

Let $n$ be odd, $\displaystyle v=1,...,\frac{n-1}{2}$ and $\displaystyle \zeta=e^{2\pi i/n}$. Define the following matrices: $$A(0,v)=\left(\begin{array}{cc}1+\zeta^{-v} & \zeta^v+\zeta^{2v}\\ \...
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Graphical calculus for braided Hopf algebras

I am trying to understand the graphical calculus presented in Ohtsuki's book Quantum Invariants. I think if I understand these first few examples it will help me greatly. Let $(A,m,i,\Delta,\epsilon,...
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Relation between Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$.

What is the relation between the definitions of Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$? There are two definitions of $U_q(\widehat{sl_2})$. The following is Jimbo ...
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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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On proposition I.3.2 of 'Quantum groups' by Kassel.

I am reading the book Quantum groups by Kassel. In proposition I.3.2 at the very beginning the reader is asked to show that under the identifications made, the maps $\Delta,\varepsilon$ and $S$ ...
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Relations in $U_q(sl(3))$ by induction

I am looking to the quantum group $U_q(sl(3))$ with generators $E_1,E_2,F_1,F_2,K_1$ and $K_2$. I want to find out what the elements to to the elements of the form $F_2^mF_1^nv$ where v is a vector ...
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Basis of a given $U_q(\mathfrak{sl}(n+1))$-module

I have a question. Lets have a look to the quantum group $U_q(\mathfrak{sl} (n+1))$ with the well-known relations. We can construct a $U_q(\mathfrak{sl} (n+1))$ module $N_0$ by: $$U_q(\mathfrak{sl}(n+...
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Inner product on quantized enveloping algebra

I have a question about a procedure, described in section $2.1.5$ of "Quantum bounded symmetric domains". Here the author describes how to introduce an inner product on $U_q(\mathfrak{g})$. Therefore ...
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Notation $A_q^{2|0}$ and $A_q^{0|2}$ in Manin's book.

In Manin's book: quantum groups and non-commutative geometry, there are two notations $A_q^{2|0}$ and $A_q^{0|2}$. Here $$ A_q^{2|0} = k<x,y>/(xy-q^{-1}yx), \\ A_q^{0|2} = k<\xi,\eta>/(\xi^...
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Manin triples: Invariant scalar product?

In Drinfel'd's article `Quantum Groups' he talks about Manin Triples as being a Lie algebra $\mathfrak{g}$ with an invariant scalar product (with some other stuff), and I don't quite understand what ...
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Quantum planes and quantum matrices.

Let $A = \mathbb{C}_q<x,y>/(xy-qyx)$ be a quantum plane. Let $M = \left( \begin{matrix} a & b \\ c & d \end{matrix} \right)$. If we require that $x'y' = qy'x'$, where $\left( \begin{...
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How to show that $J$ is a coideal of $C$ if and only if $J^{\perp}$ is a subalgebra of $C^*$?

Let $C$ be a coalgebra and $J$ a subvector space of $C$. How to show that $J$ is a coideal of $C$ if and only if $J^{\perp}$ is a subalgebra of $C^*$? Here $J^{\perp} = \{f\in C^*: \langle f, v \...
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Questions about $Hom_k(V, W) \cong W \otimes V^*$.

I am reading the book. I have some questions on page 111 about $Hom_k(V, W) \cong W \otimes V^*$. Let $W, V$ be $A$-modules. It is said that the vector space isomorphism $\varphi: Hom_k(V, W) \...
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What is the natural map $U_{A} \otimes_A \mathbb{Q}(q) \to U_q$ and what are the elements in $U_{\epsilon}$?

I am reading the book. I am trying to understand the natural map $U_{A} \otimes_A \mathbb{Q}(q) \to U_q$ and the algebra $U_{\epsilon}$ on page 288 (Section 9.2). What is the natural map $U_{A} \...
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What are quantum symmetric matrices and quantum exterior matrices?

What are quantum symmetric matrices and quantum exterior matrices? I searched google but didn't find the definitions. Thank you very much. Edit: "algebras of quantum symmetric and quantum exterior ...