All Questions
Tagged with hopf-algebras lie-groups
16
questions
6
votes
1
answer
273
views
What is the group of group-like elements of a quantum group?
A quantum group is not a group.
For example, the Drinfeld-Jimbo "quantum doubles" are Hopf algebras obtained by deforming the universal enveloping algebras of Lie algebras.
But in every Hopf algebra,...
- 555
3
votes
0
answers
70
views
$U(\mathfrak{g})$ is identified as Hopf algebra to $\mathbb{C}[[G]]^*$?
I was reading section 4.5.1 of https://arxiv.org/pdf/1801.00123.pdf but I got stuck at the following because I don't know much about Hopf algebras.
Let $G$ be a complex Lie group and let $\mathbb{C}[...
- 2,672
1
vote
0
answers
42
views
Computing the ribbon element of $\operatorname{U_q}(\mathfrak{sl}_2)$
I'm trying to understand the computation of the ribbon element of $\operatorname{U_q}(\mathfrak{sl}_2)$. I've laid out an argument by Andre Henriques below. I don't understand how the conclusion (...
- 1,472
4
votes
0
answers
140
views
Lie algebra of a compact Lie group
Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
- 2,872
3
votes
1
answer
101
views
Relations between representations and corepresentations of dually paired Hopf algebras
It is well known that if two Hopf algebras $A, B$ are dually paired and $\phi$ is a corepresentation of $A$ then it canonically induces a representation $R_\phi$ of $B$. I have a few "converse" ...
- 1,717
3
votes
1
answer
419
views
Universal enveloping algebra vs algebra of continuous functions
I'm a physicist studying quantum groups, but this question is about the usual classical Lie groups (though it is related to the language that is used to describe the deformation to quantum groups, ...
- 555
2
votes
1
answer
857
views
Intuition behind the relation of commutative Hopf algebra and Groups
I've heard that a commutative Hopf algebra can be thought an algebra construction over the space of functions on a group to the ground field. The product should be the pointwise multiplication, ...
- 9,254
1
vote
0
answers
61
views
From a compact topological group to a commutative Hopf algebra
As we know we can associate a commutative Hopf algebra to any compact topological group as follows:
Let $G$ be a compact topological group. Consider the space of continuous
functions on $G$ denoted by ...
- 71
1
vote
1
answer
291
views
Dual Jacobi identity for Lie bialgebra
I'm studying Lie bialgebras: https://en.wikipedia.org/wiki/Lie_bialgebra.
I'm a bit confused about the way of writing the so called "dual Jacobi identity".
On Majid's book "a Quantum group Primer" ...
- 522
1
vote
0
answers
52
views
Dual Pairing between $U_q(\mathfrak{su}(n))$ and $\mathcal{O}(SU_q(n))$
In page 327 of Klymik and Schmudgen`s book Quantum Groups and their representations, theorem 18 reads:
There exists unique dual pairings of the pairs:
$U_q(gl_n)$ and $\mathcal{O}(GL_q(n))$,
SL_q(n)....
- 1,717
9
votes
2
answers
975
views
Duality between universal enveloping algebra and algebras of functions
There is a well known duality (of Hopf algebras) between universal enveloping algebra $U(\mathfrak{g})$ of a complex Lie algebra $\mathfrak{g}$ of a compact group $G$ and the algebra of continuous ...
- 1,717
4
votes
1
answer
503
views
Is the special orthogonal group really rationally homotopy commutative?
It is a classical result that the rational cohomology of $SO(n)$ is given by:
$$H^*(SO(2m); \mathbb{Q}) = \begin{cases}
S(\beta_1, \dots, \beta_{m-1}, \alpha_{2m-1}) & n = 2m \\
S(\beta_1, \dots, ...
- 54.7k
3
votes
1
answer
226
views
Relation of $G$-invariants and $g$ -invariants.
Let $G$ be a Lie group and $g$ its Lie algebra. Let $H$ a Hopf algebra and $M$ an $H$-module. By definition, $m \in M$ is called invariant if $x.m=\epsilon(x)m$, $\forall x \in H$, where $\epsilon: H \...
- 14.6k
1
vote
1
answer
908
views
Understanding tensor-products in the commutative diagram of a k-Algebra
I'm having trouble getting to grips with the commutative diagram for an algebra over a field $k$.
The main problem is that my understanding of the tensor product is weak.
I have seen $V \otimes W$ ...
- 166
2
votes
1
answer
73
views
Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?
Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
- 14.6k