All Questions
Tagged with quantum-groups lie-algebras
72 questions
13
votes
0
answers
332
views
Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
4
votes
0
answers
183
views
Does a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?
Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:
\begin{align*}
& ac = e^{-h}ca, \quad bd = e^{-h}...
- 255
1
vote
1
answer
64
views
What is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right] _{\varepsilon _i}$ in the restricted specialization in QUE algebras?
I have a question about the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. It comes from section $9.3$ on page $300$ of this book
In Section 9.1, the authors define ...
- 137
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
0
votes
0
answers
124
views
Do the following two notions of quantum groups sometimes coincide?
On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
- 163
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
120
views
Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?
Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})...
- 255
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:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
- 1,516
3
votes
1
answer
180
views
A filtration on Drinfeld-Jimbo quantum enveloping algebras
For the universal enveloping algebra $U(\frak{g})$ of a Lie algebra $\frak{g}$, one can define in a natural way an increasing $\mathbb{N}_{0}$-filtration. By the Poincaré-Birkhoff–Witt theorem, the ...
0
votes
0
answers
105
views
Concrete examples of quantum duality principle
Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
- 255
0
votes
0
answers
138
views
Is $[n]_q!$ invertible in $\mathbb C [[h]]\ $?
Consider the Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F, H$ and relations $:$
$$[H, E] = 2 E,\ \ [H, F] = - 2 F,\ \ [E, F] = \frac {q^H - q^{-H}} {q - q^{-...
- 41
1
vote
0
answers
106
views
How does $R \equiv 1\ (\text {mod}\ h)\ $?
Definition $:$ Let $H$ be a Hopf algebra. An invertible element $R \in H \otimes H$ is called a coboundary structure on $H$ if
$(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$
$(2)$ $R_{21} = R^{-1},$
$(...
- 41
0
votes
0
answers
99
views
How to show that quantum $sl_2 (\mathbb C)$ is a Hopf algebra deformation of $U (sl_2 (\mathbb C))\ $?
The quantum $sl_2 (\mathbb C)$ is the non-commutative, non-cocommutative Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F$ and $H$ with the relations $:$
$$[H, ...
- 41
4
votes
1
answer
175
views
Drinfeld-Jimbo quantum groups for $q=0$
In the Wikipedia page of Drinfeld--Jimbo quantum groups the values of $q=0,1$ are excluded so as to avoid dividing by zero. The $q=1$ case is discussed in this old question. What about the $q=0$ case? ...
- 1,144
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
2
votes
0
answers
80
views
The dual of elements $E$, $F$, and $H$ of $U_h(\mathfrak{sl}_2)$ corresponds to which element of $F_h(\mathrm{SL}_2)$ by isomorphism?
$\newcommand{\sl}{\mathfrak{sl}}\DeclareMathOperator\SL{SL}$Let $U_h(\sl_2)$ be the quantized universal enveloping algebra of $\sl_2(\mathbb{C})$ and $F_h(\SL_2)$ be the quantized function algebra of $...
- 255
1
vote
1
answer
105
views
Problem in understanding a fact about Belavin-Drinfeld triple
A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental ...
- 199
0
votes
0
answers
70
views
Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\ $?
Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
- 199
1
vote
0
answers
75
views
Problem in understanding Theorem $6.2.9$ from Chari and Pressley
The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...
- 199
5
votes
0
answers
123
views
Product of $U^+_q(\mathfrak{sl}_2)_i$ in $U_q(\mathfrak{g})$ according to some reduced expression
Let $\mathfrak{g}$ be some simple Lie algebra, $\alpha_1,\alpha_2,\cdots,\alpha_n$ be its simple roots. Let $U^+_q(\mathfrak{sl}_2)_i$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $E_i,K_i$. ...
- 91
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
4
votes
1
answer
237
views
Exponential map and Lie correspondence within a Hopf algebra setting
The Cartier-Konstant-Milnor-Moore (et al.) theorem for Hopf algebras states that a cocommutative Hopf algebra over $\mathbb{C}$ is isomorphic to a smash product of a universal enveloping algebra of a ...
- 41
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
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
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
9
votes
2
answers
479
views
Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?
In a Lie algebra $\mathfrak{g}$ the Jacobi identity $\newcommand{\bracket}[2]{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$ holds. ...
- 1,161
9
votes
1
answer
332
views
The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$
I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
- 141
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
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
3
votes
0
answers
91
views
Hopf algebras structure and quantum affine algebras
I'm looking for some information about the Hopf algebras structure and the quantum groups.
In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...
- 31
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
2
votes
0
answers
70
views
Embedding problems on quantum groups？
We work over the field of complex numbers.
We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
- 165
10
votes
3
answers
1k
views
Hopf structure on the universal enveloping of a super Lie algebra
The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
- 835
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
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
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
6
votes
0
answers
118
views
Yangians as unique deformation
In Drinfeld's paper "Hopf algebras and the quantum Yang-Baxter equation" there is a statement (Theorem 2) that Yangian is a unique quantization of the corresponding Lie bialgebra.
My question is ...
- 99
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
5
votes
1
answer
579
views
The Ungraded Milnor-Moore Theorem
Let $k$ be a field of characteristic $0$.
There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\...
user30211
3
votes
1
answer
295
views
When is this map of Hopf algebras Surjective?
I'm reading Akhil Mathew's blog post on Formal Lie Theory in Characteristic Zero.
Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ ...
user30211
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
10
votes
1
answer
191
views
Exceptional Quantum Groups as FRT-Algebras
Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...
- 459
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
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
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
2
answers
997
views
Can one define quantized universal enveloping algebras in a basis-free way?
(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&...
- 32.5k
5
votes
1
answer
270
views
Is there any work on quantization of distributions?
Let $G$ be a Lie group and consider the space $C_c^\infty(G)$ of compactly supported complex-valued smooth functions on $G$ and $D'(G) = (C_c^\infty(G))'$ the topological dual linear space of $C_c^\...
- 561
4
votes
0
answers
626
views
Lusztig's definition of quantum groups
In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form $U^{\mathbb{Q}(q)}_q$ that is rather different from the usual approach (see [1,Ch.9.1] for ...
- 1,813
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