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What Scott's comment is getting at is that you need to have an abstract definition of "quantum Lie group" if you want to have a classification result. As the theory of quantized enveloping algebras a …

For the first question, I would use the dual pairing with $U_q(\mathfrak{sl}_N)$. The $u_i^j$'s are defined to be matrix coefficients of the vector representation of $U_q(\mathfrak{sl}_N)$ with respe …

I don't have the text of Drinfeld's address near to hand, but the standard way I know to do this is to take the subalgebra of the (finite) dual generated by the matrix coefficients of the irreducible …

Background When constructing the exterior algebra of a (finite-dimensional, complex) vector space $V$, there are two equivalent pictures. The first is the quotient picture. First you define the ten …

For your first question, the answer is yes, as Casteels pointed out in the comments. The reason is that, for $\mathfrak{sl}_N$, every finite-dimensional irreducible representation appears as a subrep …

The coefficients of $R$ are essentially the coefficients of the braiding of the vector representation of $U_q(\mathfrak{g})$. So, more or less, you are asking for a general formula in terms of Cartan …

I don't think that you really need to learn much more algebra before you start on Hopf algebras. As long as you know about groups, rings, etc, you should be fine. An abstract perspective on these th …

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where …

A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and antipo …

If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with deg …

(1) I think Podles only introduced the quantum 2-spheres. His paper is linked from MathSciNet, so you should be able to get it if you have access to ams.org. I think the higher-dimensional spheres …

2-cocycle twists of Hopf algebras Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map $$ f: H \otimes H \to k$$ such that $$ f(x_{(1)},y_{(1)})f(x_{(2)} y_{( …

I do not know of a general method for quantizing the group algebra of a finite group. However, there is a way to do it for Coxeter groups (finite or not): the result is called an Iwahori-Hecke algebr …

This is shown in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. The result you ask for is Proposition 63 in Chapter 11. I'd expand more upon this but I have to give a ta …

I don't know if you still care, but I think I found the answer to your question. Look at Proposition 1 in Chapter 14 of Quantum Groups and Their Representations by Klimyk and Schmudgen. It shows tha …