All Questions
Tagged with hopf-algebras quantum-groups
112
questions
0
votes
0
answers
36
views
On the orthogonality relations for quantum Clebsch-Gordan coefficients
I am currently reading Quantum groups by Christian Kassel, more precisely, section VII.7 which is about quantum Clebsch-Gordan coefficients.
To make this question self-contained, let me introduce the ...
- 4,232
1
vote
1
answer
34
views
Corepresentations of quantum subgroups
If $H$ is a Hopf subalgebra of the Hopf algebra $G$ and $\alpha:V\longrightarrow V\otimes H$ is right corepresentation of $H$ on the vector space $V$ (finite--dimensional), is there a way to induce a ...
- 139
1
vote
0
answers
44
views
Morphisms between modules over a Hopf algebra
Let $U$ be a Hopf algebra over a field $K$ and $M,N$ be $U$-modules. Then the set of all $K$-linear maps $\textrm{Hom}_{K}(M,N)$ has a canonical $U$-module structure given by
$$(u.f)(m) = \sum_{(u)} ...
- 126
0
votes
0
answers
34
views
Solution of the Yang-Baxter equation not coming from quasi-triangular structure
Let $A$ be an associative, unital algebra over a field $\Bbbk$, and let $R \in A \otimes A$ be an invertible element which is a solution of the Yang-Bater equation in $A \otimes A \otimes A$ $$R_{12}...
- 1,562
1
vote
1
answer
74
views
Haar integral of a finite dimensional Hopf algebra: an explicit expression
Let $\mathcal{H}$ be a finite dimensional Hopf algebra. A nonzero element $\Omega\in \mathcal{H}$ is called an integral in $\mathcal{H}$ if $$x~\Omega=\epsilon(x)\Omega,~~\forall x\in \mathcal{H}.\tag{...
- 836
2
votes
0
answers
106
views
Is the preimage of a Hopf subalgebra a Hopf subalgebra?
The following must be simple, but I have no intuition here, so excuse me.
Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces ...
- 2,463
1
vote
0
answers
47
views
If $H$ is a Hopf algebra deformation of $U (\mathfrak g)$ then $H/h H \simeq U (\mathfrak g)$ as a Hopf algebra.
Question $:$ If $H$ is Hopf algebra deformation of the universal enveloping algebra $U (\mathfrak g)$ for some Lie algebra $\mathfrak g$ according to the above definition then how to show that $H/h H \...
- 2,612
0
votes
1
answer
19
views
Compute $S(E^i F^j K^l)$ in $U_q$
Here is the question I am trying to solve:
Compute $S(E^i F^j K^l)$ in $U_q.$
Here is my thoughts:
Definition: We define $U_q = U_q(\mathfrak{sl}(2))$ as the algebra generated by the four variables $E ...
- 2,087
0
votes
2
answers
326
views
Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$
Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$ (as defined in the previous exercise; assume that the field $k$ is of characteristic zero.)
Here is the exercise before it:
Let $L$ ...
- 2,087
3
votes
2
answers
230
views
Determining the grouplike elements of a Hopf algebra
Here is the question I am trying to solve:
For any Lie algebra $L$ determine the group of grouplike elements of the Hopf algebra $U(L).$
Some definitions:
1-To any Lie Algebra $L$ we assign an (...
- 2,087
0
votes
1
answer
135
views
Proving that a symmetric bilinear form on $L$ is invariant
Let $L$ be a Lie algebra and $\rho: L \to \mathfrak{gl}(V)$ a finite-dimensional representation of L. Define a symmetric bilinear form on $L$ by $$\langle x, y \rangle_{\rho} = tr (\rho(x) \rho(y))$$
...
user965463
0
votes
0
answers
46
views
Why are the maps $\eta, \mu, \Delta, \varepsilon$ linear?
Here is the question I am trying to solve:
(Divided powers) Consider the vector space $C = k[t]$ of polynomials in one variable. Prove that there exists a unique coalgebra structure $(C, \Delta, \...
user965463
3
votes
1
answer
117
views
Proving the uniqueness of a map
Here is the question I am trying to solve:
(Tensor product of coalgebras) Let $(C, \Delta, \varepsilon)$ and $(C', \Delta ', \varepsilon ')$ be coalgebras. Show that the linear maps $\pi: C \otimes C' ...
user965463
2
votes
1
answer
88
views
What is the meaning of $(\mu \otimes \mu) ({\rm id}\otimes \tau \otimes {\rm id})$?
Here is one of the figures whose commutativity express the fact that $\Delta$ is a morphism of algebra:
$\require{AMScd}$
$$\begin{CD}H\otimes H @>\Delta\otimes\Delta>> (H\otimes H)\otimes (H\...
- 2,087
0
votes
1
answer
60
views
Why we need an orthonormal basis?
Here is the question I am trying to solve:
Prove that $\lambda$ is injective.
Here is the definition of the linear map $\lambda$:
Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their ...
- 2,087