2
votes
1answer
49 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
1
vote
1answer
47 views

How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
1
vote
1answer
22 views

Questions about root operators.

I am reading the notes. On line 13 in the section Root operators, it is said that The operator $f_1$ maps from the space $V(\mu)$ to $V(\mu-(1,-1,0))$. I don't know why. We have $$ f_i V (\mu) ...
4
votes
0answers
47 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
6
votes
0answers
107 views

A little bit of Intuition for Corepresentations from Representations

Hi folks I am trying to prove what I think should be a straightforward enough result but I am having to make a somewhat unnatural definition to do it. This unnatural definition is hinted at in a paper ...
2
votes
0answers
44 views

Skew polynomial algebra and deformation

Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x ...
5
votes
1answer
118 views

Towards a Quantum Peter Weyl Theorem

This is taken from Timmermann's Invitation to Quantum Groups and Duality. Let $(A,\Delta)$ be a *-Hopf algebra and let $\chi:V\rightarrow V\otimes A$ be a corepresentation of $(A,\Delta)$ on a vector ...
1
vote
1answer
70 views

$C(G) \otimes C(G) \simeq C(G \times G)$

Let be $G$ a compact group and $C(G)$ the vector space of continue complex functions defined over $G$, $C(G):=\{f:G \to \mathbb{C}\}$, where $f$ is a continue function. How can I prove that $$ C(G) ...
0
votes
1answer
47 views

A Tensor Product Identification

Hi folks just looking at Timmermann's "Introduction to Quantum Groups and Duality" and looking at the algebra of representative functions on a compact topological group $G$. I take off in Example ...
6
votes
0answers
135 views

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 ...
2
votes
1answer
91 views

elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
1
vote
1answer
45 views

Kernel of a Comodule Map is a Sub-Comodule

Let $H$ be a Hopf algebra, $(V,\Delta_R)$ a right $H$-comodule map, and $f:V \to V$ a right $H$-comodule map. Since by definition we must have, for all $v \in V$, that $$ \Delta_R(f(v)) = \sum ...
2
votes
2answers
208 views

Is there some kind of character theory for representations of finite dimensional algebras?

We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with ...
5
votes
0answers
124 views

Tensor product decomposion in Lie algebras

As a chemist, I do this all the time...for symmetry groups. Which are finite, luckily :-) For knot theory purposes, I'd like to have a complete list of Lie group irreps R with the property that the ...
0
votes
1answer
140 views

Compare of coefficients of two formal power series

Define $$ \Lambda_{i}(u) = \sum_{r=0}^{\infty} \Lambda_{i,r}u^{r}, \Psi_{i}(u) = \sum_{m=0}^{\infty} \psi_{i,m}u^{m}=k_i\frac{\Lambda_{i}(uq_i^{-1})}{\Lambda_{i}(uq_i)}. $$ How can we show that $$ ...
1
vote
1answer
92 views

questions about the paper: Affine quivers and canonical bases

I am reading the paper Affine quivers and canonical bases. I have a question on page 114 of the paper. In the proof of property (b), line 6 of page 114, why "for each $\gamma \neq 1$, $tr(\gamma, ...
4
votes
1answer
120 views

How to compute the Gel'fand Models for a (quantum) Lie Algebra

Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean $\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and ...