1
vote
0answers
26 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 ...
1
vote
1answer
35 views

Construction of a Manin triple

Let $\mathfrak{g}$ be a Lie bialgebra and $\mathfrak{g}^*$ be its dual. My question is how to construct a bracket on the direct sum $\mathfrak{g}\oplus\mathfrak{g}^*$ such that we obtain a Manin ...
5
votes
0answers
87 views

Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra

As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
4
votes
1answer
317 views

The longest word in Weyl group and positive roots.

How to write down a reduced decomposition of the longest word in a Weyl group? For example, how to write down a reduced decomposition of the longest word in type B3 Weyl group? For a decomposition of ...
2
votes
2answers
176 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 ...
2
votes
1answer
159 views

Linear independency of a set of functions.

Let $m\in \mathbb{Z}, \mu_{m}^{j}\in \mathbb{C}, \lambda_{m'}^{j}\in \mathbb{C}, \Psi_{i,r}^{+}\in \mathbb{C}$. $$\lambda^{j}(z)=\sum_{m'\in \mathbb{Z}}\lambda_{m'}^{j}z^{m'}$$ ...
4
votes
1answer
111 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 ...