Questions tagged [lie-bialgebras]
5 questions
3
votes
1 answer
What is meant by saying that $\theta$ is orthogonal?
I am reading the materials discussed in lecture $5$ from the lecture notes on quantum groups about Belavin-Drinfeld classification theorem written by Pavel Etingof and Oliver Schiffmann.
In the first half of this lecture the authors proved that any…
ACB
- 3,068
2
votes
1 answer
What is the meaning of $(\mu \otimes \mu) ({\rm id}\otimes \tau \otimes {\rm id})$?
Here is one of the figures whose commutativity express the fact that $\Delta$ is a morphism of algebra:
$\require{AMScd}$
$$\begin{CD}H\otimes H @>\Delta\otimes\Delta>> (H\otimes H)\otimes (H\otimes H)\\
@V \mu V V @VV (\mu\otimes\mu)({\rm…
Emptymind
- 2,217
1
vote
1 answer
Commutative diagrams in the definition of bialgebras, what do they mean?
I am reading the definition of Bialgebras over a field $\mathbb{K}$. The definition is the following:
A bialgebra over a field $\mathbb{K}$ is a vector space $B$ over $\mathbb{K}$ equipped with $\mathbb{K}$-linear maps (multiplication) $\nabla : B…
Saikat
- 1,687
0
votes
0 answers
Intuition about Lie Bialgebra Structure
I am reading Etingof's Lectures on Quantum Groups, Chapter 2. There, the authors define a Lie bialgebra structure on $T_eG$ for $G$ a Lie-Poisson group. Questions:
First, they define a Lie algebra structure on $T_x^*M$ for any Poisson manifold $M$…
user1104937
0
votes
1 answer
Why are Lie bialgebras classified up to automorphisms?
Lie algebras are usually classified modulo isomorphisms of the underlying vector space, i.e. invertible changes of basis, which are $GL(N)$ maps of the generators $X'^i = M^i{}_j \, X^j$. For example, there are only two classes of 2D Lie algebras:…
Spinoro
- 145