All Questions
Tagged with qa.quantum-algebra rt.representation-theory
149 questions
8
votes
1
answer
702
views
Central extensions of loop groups
Let $LG=\operatorname{Maps}(S^1,G)$ be the loop group of a compact Lie group $G$. I should add some adjectives to $G$, but for sake of simplicity let's just take $G=SU(2)$.
There is a central ...
- 18.7k
32
votes
1
answer
2k
views
Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$
Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
- 18.7k
9
votes
2
answers
362
views
Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups
For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
- 219
18
votes
3
answers
2k
views
Hopf dual of the Hopf dual
Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some ...
- 835
4
votes
0
answers
145
views
Indecomposable modules for the big quantum group
I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
- 41
5
votes
0
answers
144
views
Are the integral forms of quantized coordinate algebras always Noetherian?
Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected ...
- 126
2
votes
0
answers
56
views
Quantum Lie algebra formalism that doesn't violate P symmetry
begin tl;dr: I just read this paper which gives the equations for the structure constants, braiding operators etc. for a generic quantum Lie algebra. I always found it very annoying that in the ...
- 4,829
7
votes
2
answers
378
views
Hopf Subalgebras of Quantized Algebras
As is well known, quantized enveloping algebras $U_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets ...
- 219
6
votes
1
answer
243
views
Jones-Wenzl-type projectors for Brauer algebras
Jones-Wenzl projectors are an explicit combinatorial description of the central projectors of the Temperley-Lieb algebra.
They also describe very explicitly the failure of certain representations to ...
- 2,533
5
votes
1
answer
183
views
Schur's Lemma for Quantized Universal Enveloping Algebra
Let $U_q(\mathfrak{g})$ (defined over $\mathbb{C}(q)$) be the quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$. Let $M$ a finite-dimensional simple left $U_q(\mathfrak{g})$...
- 671
9
votes
1
answer
766
views
The difference between $q$-deformations and $h$-deformations
What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras?
In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping ...
- 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
4
votes
1
answer
192
views
Why are the divided difference operators of the nil Hecke ring only of degree 1?
In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups I" (arXiv, journal, MSN), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator ...
- 185
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
11
votes
1
answer
356
views
What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically?
Background. As we know from Fred Cohen's Thesis, taking homology of the little 2-discs operad $\mathcal{D}_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\...
- 1,981
2
votes
0
answers
87
views
Modules over quantum complete intersections
Let $a_i \geq 2$ be natural numbers and $q_{ij}$ field elements of the field $k$ for $i>j$.
A quantum complete intersection is the algebra $A:=k<x_1,...,x_n>/(x_i^{a_i},x_i x_j - q_{ij} x_j ...
- 26.5k
7
votes
0
answers
248
views
Trace on a KLR algebra
The cyclotomic KLR algebra is isomorphic to the Ariki-Koike algebra over a field and so admits a trace (this is used in Hu-Mathas' paper to define bases for the KLR algebra corresponding to Murphy and ...
- 1,413
20
votes
1
answer
586
views
$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
The following formula of astonishing beauty and power (imho):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 24.9k
7
votes
0
answers
432
views
What is the endomorphism cooperad?
In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...
- 2,074
3
votes
0
answers
220
views
Generalisation of the quantum exterior algebra
One might generalise the classical exterior algebra as follows to the quantum exterior algebra:
$K<x_1,...x_n>/(x_i^2,x_i x_j + q_{i,j}x_j x_i)$ with nonzero field elements $q_{i,j}$ for $i<j$...
- 26.5k
1
vote
0
answers
88
views
Reference request: Nichols algebras of a braided vector space with a diagonal braiding
Are there some references of the proof of the following result?
Let $(V, c)$ be a braided vector space over a field $k$ with a basis $x_1, \ldots, x_n$, where $c$ is a diagonal braiding given by
\...
- 6,201
6
votes
1
answer
392
views
Corepresentations of Tensor Products of Hopf Algebras
Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure ...
- 159
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
7
votes
0
answers
161
views
Are the weight spaces of indecomposable $U_q\mathfrak{sl}(2)$-modules at most 2-dimensional?
This is a follow up of this question.
Let $U_q\mathfrak{sl}(2)$ be Lusztig's integral form of the quantized enveloping algebra of $\mathfrak{sl}_2$, specialised at $q$ a root of unity. This is an ...
- 43.2k
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
3
votes
0
answers
97
views
Simple modules of quantum toroidal algebras
Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials.
...
- 6,201
22
votes
0
answers
481
views
What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?
Let $\mathfrak g=\mathfrak{sl}(2)$.
Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness.
Let $U_q\mathfrak g$ be Lusztig's integral form of the ...
- 43.2k
6
votes
1
answer
272
views
Bialgebraic structure of Sklyanin algebra
Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...
- 367
5
votes
0
answers
191
views
Modular double of elliptic quantum group
By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as $\mathfrak{...
- 367
7
votes
1
answer
417
views
Bounding $p$-adic characters and Jacquet-Langlands transfert
I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_\pi$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on ...
- 5,424
7
votes
2
answers
405
views
The Irreducible Representations of the Sekine Quantum Groups
Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
$$\mathcal{B}_n=\mathbb{Z}_n\times\mathbb{Z}_n=\{(i,...
- 1,037
3
votes
2
answers
985
views
Appropriate Recursion relations for Wigner 3j Symbols
I am attempting to code the Cosmic Microwave Lensed Temperature and Polarisation power spectra from first principles and have been told to code the relevant Wigner 3j symbols using recursion rather ...
- 31
2
votes
0
answers
159
views
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
53
votes
4
answers
5k
views
Intuition behind the definition of quantum groups
Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to ...
- 21.8k
2
votes
1
answer
176
views
How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module?
Context
Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the ...
- 357
5
votes
0
answers
123
views
Signs associated to self-dual simple objects in a fusion category
Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element:
The square of $a$ equals Muger's "squared dimension" of $X$, an ...
- 1,821
5
votes
1
answer
600
views
exceptional cases in Kazhdan-Lusztig
The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine).
What's special about those cases?
- 43.2k
1
vote
1
answer
437
views
The coproducts $\mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ and $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$
A coproduct $\varphi: \mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ is given by: $x \mapsto 1 \otimes x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}_q[U]$. There is a ...
- 6,201
4
votes
1
answer
959
views
Jones polynomial of tangles using Temperley-Lieb algbra?
The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...
- 803
1
vote
0
answers
228
views
Clebsch Gordan coefficients of compact quantum groups
Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
- 11
11
votes
1
answer
629
views
$q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?
Define n-quantum vector space to be the algebra
$$
{\mathbb C}_q^n := \mathbb{C}\left< x_i \mid i =1, \ldots, N\right>/\left<x_i x_j = q x_j x_i \mid i<j\right>.
$$
For $q=1$, we get ...
- 465
10
votes
2
answers
833
views
Update on list of open problems for Cherednik/Symplectic Reflection Algebras
Background:
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
- 1,066
1
vote
1
answer
314
views
classification of irreducible finite dimensional representation of affine hecke algebra of type A
Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity.
The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...
- 1,457
5
votes
2
answers
554
views
Jones polynomial of the concatenation of two braids
Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings.
Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$?
Here, $J_L(q)...
- 103
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 ...
4
votes
1
answer
147
views
Equivalence of star products on two differents Poisson algebras?
Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
- 53
3
votes
1
answer
469
views
A little bit of Intuition for Corepresentations from Representations
I asked this question over on Math.Stack --- where it has a bounty --- but I didn't really get a helpful response so I am asking the question here. One commenter suggests that I am confusing left- ...
- 1,037
4
votes
1
answer
755
views
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 ...
- 4,829
15
votes
2
answers
2k
views
When are Jones-Wenzl projectors defined?
(I am hoping that someone well-versed in the literature of Temperley-Lieb algebras or of quantum groups at roots of unity can answer my question. Fingers crossed.)
Consider the Temperley-Lieb algebra ...
- 483
9
votes
0
answers
627
views
Quantum Drinfeld-Sokolov reduction for a module
There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...
- 390