All Questions
Tagged with rt.representation-theory qa.quantum-algebra
149 questions
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
4
votes
1
answer
280
views
Vertex embeddings of quantum groups via quivers
Let Uq be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of Uq, we see that for each vertex of the finite Dynkin diagram, there is an inclusion Uq($\hat{sl_2}$)...
- 8,866
4
votes
0
answers
56
views
When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?
Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...
- 143
4
votes
0
answers
76
views
On the order of the head of product of two simple modules over Quiver Hecke Algebras
My question is:
We assume the underlying quiver is a Dynkin quiver. Let $L(\lambda)$ and $L(\mu)$ be two simple modules over Quiver Hecke algebra $R$ where $\lambda$ and $\mu$ are two Konstant ...
- 71
4
votes
0
answers
350
views
Interpolated simple integral fusion categories of Lie type
$\DeclareMathOperator\PSL{PSL} \DeclareMathOperator\Rep{Rep}$The idea motivating this post is that there should exist a global understanding of the unitary fusion categories $\Rep(G(q))$, with $G(q)$ ...
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
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
4
votes
0
answers
110
views
Is there a good reference for how ribbon structures change when one switches coproducts?
I'm just going assume readers are familiar with the notions of R-matrix and ribbon categories.
Given a quasi-triangular Hopf algebra $A$ with $R$-matrix $R$, one can construct the co-opposite Hopf ...
- 44.7k
4
votes
0
answers
241
views
Analogy between canonical basis of U(n_-) and Schur functors, each under restriction
.1. For any category $\mathcal C$, possibly enriched over schemes, define $Rep({\mathcal C})$ to be the functor category ${\mathcal C} \to {\bf Vec}$ with direct sum inherited from $\bf Vec$. (If $\...
- 27.8k
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
$$
\mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(...
- 1,144
3
votes
1
answer
591
views
"Quantum Littlewood-Richardson" Rule?
Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...
3
votes
2
answers
985
views
Appropriate Recursion relations for Wigner 3j Symbols
I am attempting to code the Cosmic Microwave Lensed Temperature and Polarisation power spectra from first principles and have been told to code the relevant Wigner 3j symbols using recursion rather ...
- 31
3
votes
1
answer
492
views
Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]
Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as
$\operatorname{Rep}(G)$ ...
- 309
3
votes
1
answer
104
views
Irreducibility of product bicomodules
Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right,
$H$-comodule respectively. The tensor product
$$
V \otimes W
$$
has an obvious $H$-$H$-bicomodule structure.
If $V$ and $W$ are ...
- 1,144
3
votes
1
answer
190
views
Quantum group associated to a reductive group
In most of the classical references about quantum groups, these objects are defined as a one-parameter deformation of the universal enveloping algebra. However, I have read in several papers that it ...
- 31
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- ...
- 1,027
3
votes
0
answers
151
views
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
3
votes
0
answers
134
views
Relative strength of Jones and colored Jones polynomials
this is my first post here.
I've been studying some Knot Theory and I came to a question concerning invariants.
We know that the Jones polynomial is related to the RT-invariant associated to the two-...
- 31
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
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
3
votes
0
answers
323
views
Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types
Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible finite-...
- 1,453
3
votes
0
answers
515
views
What happens geometrically when you take associated-graded (or complete, ...) of a group ring at its augmentation ideal?
I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
- 54.6k
2
votes
4
answers
650
views
Role of fiber functor monoidal structure in Tannakian bialgebra reconstruction
Shahn Majid presents in section 9.4 of "Foundatations of Quantum Group Theory" a Tannaka-type reconstruction theorem for producing a $k$-bialgebra out of its $k$-linear monoidal category of modules $\...
- 1,368
2
votes
1
answer
476
views
Are there other algebra structures on the regular representation of a group?
Let $G$ be a (discrete, say) group and $\mathbb K$ a field. The regular representation $G^{\mathbb K}$ is the vector space of all functions $G \to \mathbb K$. It is a (left, say) $G$-module: given $...
- 54.6k
2
votes
1
answer
312
views
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 $\...
- 137
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
2
votes
1
answer
287
views
On the definition of the Cherednik algebra of a variety with a finite group action
Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...
- 541
2
votes
1
answer
160
views
Highest-$\ell$-weight tensor products and diagram subalgebras
Let $U_q(\mathcal{L}({\mathfrak{g}}))$ be a quantum loop algebra and $I$ the set of indexes of Dynking diagram of $\mathfrak{g}$. Consider $J\subset I$ a connected subdiagram, so that $U_q(\mathcal{L}(...
- 353
2
votes
1
answer
228
views
A question on Lusztig's `graph with automorphism' construction?
Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix $D=\mathrm{diag}(d_1,\ldots,...
- 1,472
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
72
views
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 ...
- 7,300
2
votes
0
answers
108
views
Why are the quantum Fock spaces in FLOTW the same as Uglov's?
Theorem 2.5 in the well-known FLOTW paper [1] and Theorem 2.1 in Uglov's paper [2] both refer to the original JMMO paper [3]
to define quantum Fock spaces, i.e. Fock spaces for $U_q(\widehat{\...
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
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
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
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
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
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
1
vote
1
answer
168
views
Decomposition of C' Kazhdan-Lusztig basis element associated to longest word in S_n
I'm trying to decompose the Kazhdan-Lusztig C' basis element associated to the longest word in $S_n$, $C'_{w_0}$ into products and sums of elements $C'_w$ where $w < w_0$ in the Bruhat order. For ...
- 115
1
vote
0
answers
125
views
Tangle hypothesis and ribbon category
The tangle hypothesis, when specialized to ordinary framed tangles, says that the framed tangles form the free braided category with all duals (i.e. considered as a 3-category, all the 1- and 2-...
- 1,263
1
vote
0
answers
50
views
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
1
vote
0
answers
166
views
How to understand a definition in KLR algebra in the setting of quantum affine algebras?
I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra:
$$
X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1)
$$
This ...
- 6,201
1
vote
0
answers
88
views
Reference request: Nichols algebras of a braided vector space with a diagonal braiding
Are there some references of the proof of the following result?
Let $(V, c)$ be a braided vector space over a field $k$ with a basis $x_1, \ldots, x_n$, where $c$ is a diagonal braiding given by
\...
- 6,201
1
vote
0
answers
228
views
Clebsch Gordan coefficients of compact quantum groups
Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
- 11
1
vote
0
answers
216
views
polynomial representation of $sl_{2}(k)$
Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
\...
1
vote
0
answers
121
views
Product knot invariants
Let $I_1(L)$ be a Reshetikhine-Turaev link invariant coming from the (quantum Lie) group $G_1$ and representation $\lambda_1$ having the S matrix $S_1$. Let $I_2(L)$ be a Reshetikhine-Turaev link ...
- 4,829
1
vote
0
answers
247
views
Annulator of Tensor Power in a Quantum Group
There is a little question haunted me for few days. I will be grateful to anyone who can give me any clue how to solve it.
Let $V$ be a nontrivial module of $\mathrm{U}_q(\mathfrak{g})$ (the ...
- 153
0
votes
2
answers
203
views
Structure of Homomorphisms of commutative C^* algebra
Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$.
Let ${\cal P}$ be the $...
- 45
0
votes
0
answers
106
views
Hopf algebra antipodes and right left comodule equivalences
Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
- 1,144