All Questions
32 questions
17
votes
2
answers
2k
views
Examples of representations of quantum groups
I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
- 21.8k
13
votes
1
answer
411
views
Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators
$\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E_\pm, H$ with commutation relations
$$
[H,E_{\pm}]=\pm 2 E_\pm,\quad [E_+,E_-]=H
$$
admits the well-known representation on $\mathbb{C}[x]$ with
$$
...
- 133
11
votes
3
answers
663
views
Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules
In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra:
The space of ...
- 219
8
votes
1
answer
390
views
What is the ring structure on Lusztig's integral form of quantum $\mathfrak{sl}(2)$?
Consider the quantum group $U_q(\mathfrak{sl}_2)$, with generators $E,F,K$ such that $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$. Write $[n]=\frac{q^n-q^{-n}}{q-q^{-1}}$, and $[n]!=[n][n-1]\dotsm[1]$.
In ...
- 492
7
votes
0
answers
183
views
Relationship between R-matrix and Casimir element?
Given a simple Lie algebra $\mathfrak{g}$, is there any relation between its Casimir element and the $R$-matrix of the related Yangian $Y(\mathfrak{g})$?
- 671
6
votes
5
answers
1k
views
Is there a version of Temperley-Lieb using sl(3) rather than sl(2)?
This question is a spin-off from Sammy Black's question on super Temperley-Lieb. Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the ...
- 54.6k
5
votes
2
answers
403
views
Indecomposable, non-simple, modules of quantum groups at roots of unity
Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...
- 7,746
5
votes
1
answer
429
views
Crystal basis for quantum groups and Lie algebras
Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\...
- 6,201
5
votes
0
answers
128
views
Classification of connected finite affine type A crystals
In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...
- 163
5
votes
0
answers
218
views
Lusztig's completion for universal enveloping algebra
In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
- 677
4
votes
2
answers
498
views
Classifications of Lie bialgebras
What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets $\...
- 6,201
4
votes
1
answer
101
views
Non-cosemisimple duals of pointed Hopf algebras
I take the following quote from an answer to this question
A Hopf algebra is called pointed if all its simple left (or right)
comodules are one-dimensional. The quantized enveloping algebras and
...
- 145
4
votes
0
answers
280
views
Why does Kashiwara define $U_q(\mathfrak{g})$ over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$?
When defining crystal bases, why do we typically view $U_q(\mathfrak{g})$ as an algebra over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$? In Kashiwara's original paper introducing crystal bases, he ...
- 264
4
votes
0
answers
203
views
The Killing form on quantized enveloping algebras and reduction to the classical case
Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
- 3,637
3
votes
1
answer
367
views
The adjoint representation of $U_q({\frak sl}_2)$ on itself
Let $U_q(\frak{sl}_2)$ denote the quantum universal enveloping algebra of $\frak{sl}_2$, and consider the adjoint action
$$
\mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(...
- 1,144
3
votes
1
answer
151
views
Are all the Lie bialgebra structure on $sl_n$ coboundary?
In the case of $sl_2$, there are three Lie bialgebra structures. We have three cobrackets $\delta: sl_2 \to \Lambda^2 sl_2$. Each $\delta$ can be written as $\delta=d r$ for some matrix $r$. Therefore ...
- 6,201
3
votes
1
answer
591
views
"Quantum Littlewood-Richardson" Rule?
Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...
3
votes
1
answer
386
views
Modules which are direct sum of weight spaces.
For a semisimple Lie algebra $\mathfrak{g}$, a highest weight module $V(\lambda)$ with highest weight weight $\lambda$ has the property that every submodule $W$ of $V(\lambda)$ is a direct sum of the ...
- 6,201
3
votes
1
answer
492
views
Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]
Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as
$\operatorname{Rep}(G)$ ...
- 309
3
votes
2
answers
288
views
Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$
By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...
- 6,201
3
votes
0
answers
135
views
How to understand extremal vector?
Extremal vectors are defined in Kashiwara's paper. The definition is as follows.
Simple reflections in the Weyl group of $\mathfrak{g}$ acts on the crystal basis of integrable $U_q(\mathfrak{g})$-...
- 6,201
3
votes
0
answers
156
views
Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)
Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows.
$$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...
- 2,216
2
votes
0
answers
132
views
A question about q-binomials at roots of unity
I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization&...
- 137
2
votes
0
answers
49
views
Action of Lusztig braid group operators on locally finite part
Let $\mathbf{U}_q(\mathfrak{g})$
be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we ...
- 141
2
votes
0
answers
78
views
Quantum Schubert cell algebra and quantum odd-dimensional euclidean space
De Concini, Kac, Procesi introduced quantum Schubert cell algebra associated to a complex Lie algebra $\mathfrak{g}$ which is denoted by $\mathcal{U}^{w}_{\epsilon}$ where $w$ is an element of Weyl ...
2
votes
0
answers
102
views
Category O for (Yangian) toroidal Lie algebras?
Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
- 1,516
2
votes
0
answers
163
views
Quantum invariant: The canonical $2$-tensor
In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
- 395
2
votes
0
answers
71
views
Comodules of the $B,C$ and $D$ series quantum groups
In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...
- 101
1
vote
0
answers
139
views
Submodules of $V\otimes V^*$
Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...
- 353
1
vote
0
answers
71
views
Low-dimensional classical r-matrices
Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, r_{...
- 6,201
1
vote
0
answers
216
views
polynomial representation of $sl_{2}(k)$
Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
\...
0
votes
0
answers
148
views
Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$
Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) \...
- 6,201