All Questions
Tagged with lie-algebras qa.quantum-algebra
68 questions
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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 ...
- 1,846
4
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0
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183
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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}...
- 12.9k
2
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0
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132
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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&...
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2
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0
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48
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Proof of redundancy for defining relation in current algebra $J$ presentation
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The $\mathfrak{g}$-current (Lie) algebra is $\mathfrak{g} \otimes \mathbb{C}[t]$, with Lie bracket given by $[a \otimes t^m, b \otimes t^n]...
1
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0
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50
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Simple highest weight modules of quantum affine algebras
Let $U_q(\hat{\mathfrak g})$ be a quantum affine algebra, and let $L$ be an integrable simple highest weight module of $U_q(\hat{\mathfrak g})$. In [Lu], Lusztig proved that the limit of $L$ when $q\...
- 11
2
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1
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312
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Reasons about the difference between twisted affine algebras of $A_{2l}$ and other types
I am reading about Kac–Moody algebras. I have a question about the construction of twisted affine algebras.
Let $\sigma$ be a graph automorphism of the simple Lie algebra $\mathfrak{g}$.
When $\...
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0
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120
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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
3
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1
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180
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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 ...
4
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1
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175
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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? ...
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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
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80
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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
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0
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151
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Is there a classical version of Yetter-Drinfeld modules?
One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$.
If we think ...
- 3,446
5
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1
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143
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PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$
I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig.
Alternatively I ...
- 8,866
2
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72
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Dimensions of centralizers in quantum Lie algebras associated to $\mathfrak{sl}_n$
Following ideas of Woronowicz, Lyubaschenko and Sudbery defined in Quantum Lie algebras of type $A_n$ the notion of a quantum Lie algebra $\mathfrak{sl}_n$. Let me focus on the case where $q$ is not a ...
- 12.9k
5
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0
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123
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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
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1
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492
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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
$\operatorname{Rep}(G)$ ...
30
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4
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3k
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A mysterious Heisenberg algebra identity from Sylvester, 1867
I am trying to understand two papers by James Joseph Sylvester:
P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of ...
- 10.5k
4
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1
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237
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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 ...
- 1,144
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353
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Homotopy transfer of cyclic L-infinity algebras
Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...
- 35.5k
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185
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Exact sequence of L-infinity-algebras
We call a sequence of $L_\infty$-algebras (weak) maps
$$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$
is exact if it is exact on the the underlying chain complexes level.
Thought I don't know ...
- 35.5k
3
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1
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367
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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
$$
\mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(...
- 8,524
13
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1
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411
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Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators
$\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E_\pm, H$ with commutation relations
$$
[H,E_{\pm}]=\pm 2 E_\pm,\quad [E_+,E_-]=H
$$
admits the well-known representation on $\mathbb{C}[x]$ with
$$
...
- 7,746
9
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2
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479
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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. ...
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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 ...
- 7,746
4
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1
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101
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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
...
- 12.3k
3
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0
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91
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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 ...
- 63.9k
4
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1
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170
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Quantum Hamiltonian reduction and tensor products
Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras.
Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and ...
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70
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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 ...
- 63.9k
4
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278
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Are vertex operator algebras ever conspiratorial?
I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
- 3,486
5
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2
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403
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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 ...
- 3,377
11
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3
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1k
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Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)?
We have known that any finite dim Lie algebra can be embeded into it's enveloping algebra $U(\mathfrak{g})$, my question is: is there any "quantum Lie algebra" embeded into the quantum enveloping ...
- 63.9k
12
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1
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840
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Comparing two similar procedures for quantizing a Casimir Lie algebra
My primary reference for this question is the very good book Quantum Groups and Knot Invariants by C. Kassel, M. Rosso, and V. Turaev. I'm also drawing from P. Etingof and O. Schiffmann, Lectures on ...
- 12.9k
13
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2
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999
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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&...
- 31.5k
11
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3
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663
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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 ...
- 7,746
6
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118
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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
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224
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Universal Enveloping algebra of a L$_\infty$ algebra
In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras ...
- 183
4
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1
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755
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Commutators for quantum Lie algebras
Can the usual definition of a Lie algebra via commutators be simply adapted
to quantum Lie algebras? Graphically you have the IHX scheme, with the X
being a virtual crossing (so to say). Does it ...
CommunityBot
- 1
4
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1
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265
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Framing dependence of HOMFLY polynomial
I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...
- 1,430
10
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1
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191
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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 ...
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2
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2k
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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 ...
7
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183
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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
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627
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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
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0
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71
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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
5
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1
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429
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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(\...
- 15.5k
12
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1
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688
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Comparing a Chevalley basis with the canonical basis of the adjoint module?
First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} \...
CommunityBot
- 1
9
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1
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338
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Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$?
Let $n\in\mathbb N$. Let $k$ be a commutative ring in which $1,2,3,\ldots$ are invertible. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $...
CommunityBot
- 1
2
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0
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159
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The role of the Vandermonde determinant in representations of affine Lie algebras
I am reading a paper 'Yangians and R-matrices' by Chari & Pressley (1990) and to classify representations for particular quantum groups, they define a "quantum Vandermonde determinant". They also ...
- 21
4
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0
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220
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What does "control of a deformation problem" mean?
Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
- 6,210
7
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0
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182
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Deformation of Noether's first theorem
Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...
- 6,210
2
votes
0
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193
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How to write BRST-BV for dg-Lie?
The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?
- 6,210