Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are ...

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12
votes
1answer
433 views

$(q,x)$-analog of $n!$

While doing some work in geometric representation theory I have come across the following sequence of polynomials in two variables $(q,x)$ which I would like to denote by $n!_{q,x}$. For small $n$ ...
47
votes
4answers
3k views

Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
2
votes
1answer
352 views

Why is the quantum Lorentz group not connected?

Podles and Woronowicz' construct the quantum Lorentz group, by which they mean $SL_q(2,\mathbb{C})$, as a quantum double of the compact quantum group $SU_q(2)$. More precisely, it is a bicrossed ...
10
votes
0answers
369 views

canonical basis via Gelfand-Tsetlin basis

Do there exist explicit formulas for the action of Lusztig's canonical basis of $U_q(\mathfrak n_+)$ in the Gelfand-Tsetlin basis of a Verma module for $sl(n)$?
9
votes
0answers
466 views

Representations of quantum groups at roots of unity

I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
6
votes
1answer
318 views

Quantum coordinate ring at root of unity

Noah Snyder gave a great answer to this question about different versions of a quantum group $U_q(\mathfrak g)$ when $q$ is a root of unity. I want to ask about forms of the deformed coordinate ring ...
3
votes
1answer
512 views

Alternative Definition of the Quantum Determinant?

Let $M_q(n)$ be the standard quantum matrices (over the complex numbers) with generators $u^i_j$ for $i,j = 1, \ldots ,N$, and reations $$ u^i_ju^k_j = qu^k_ju^i_j, \text{ for } i < k, ~~~~~~~ ...
5
votes
1answer
248 views

What vector space does the Kauffman bracket skein algebra of FxI act on?

The Kauffman bracket skein module $K_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). ...
11
votes
0answers
371 views

Is the quantum dilogarithm related in any way to cohomology of quantum groups?

Is the quantum dilogarithm related in any way to cohomology of quantum groups? This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain ...
7
votes
1answer
308 views

Are the Drinfeld compact quantum groups simply connected ?

To fix notations : let G be simply connected simple compact group, and $U_q(\mathcal{G})$ the Drinfeld-Jimbo universal algebra quantization of its complexified algebra defined as usual, with q not ...
12
votes
2answers
581 views

Projective modules over quantum groups

My question is short: How can one calculate $\operatorname{Tor}_{U_q(\mathfrak g)}(k,k)$? ($k$ is the ground field of characteristic zero). If we had a regular universal enveloping algebra ...
3
votes
2answers
243 views

Does there exist a canonical “degree” filtration on quantum groups?

For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of ...
10
votes
1answer
443 views

R-matrices, crystal bases, and the limit as q -> 1

I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of ...
2
votes
2answers
261 views

2-tangles and quantum groups and 2-groups

Turaev developed the notion of a quantum group by considering the category of tangles (thought of with objects as collections of 2$n$ points and with morphisms being braids between them with cups and ...
11
votes
1answer
497 views

A left inverse for the comultiplication on a Hopf von Neumann algebra

Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments. $\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following ...
7
votes
0answers
321 views

Quantum Braid Group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...
5
votes
1answer
695 views

Kontsevich Integral without associators?

Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding ...
1
vote
1answer
143 views

Fixed points of quantised enveloping algebra for affine $\mathfrak{sl}_n$

Consider the automorphism of the algebra $U_q(\widehat{\mathfrak{sl}}_n)$ induced by the obvious diagram automorphism of the extended type A Dynkin diagram. More precisely, if the vertices of the ...
6
votes
0answers
141 views

Embedding quantum affine algebras at different levels

For $l$ a positive integer, an affine Lie algebra $\widehat{\mathfrak{g}}$ has a level $l$ embedding $\phi_l: \widehat{\mathfrak{g}} \longrightarrow \widehat{\mathfrak{g}}$ which takes $x\otimes t^k$ ...
9
votes
3answers
590 views

Is the nc torus a quantum group?

The non-commutative n-torus appears in many applications of non-commutative geometry. To stay in the setting $n=2$: it is a C$^\ast$-algebra generated by unitaries $u$ and $v$, satisfying $u v = e^{i ...
0
votes
1answer
322 views

Affine quantum groups of type A

I have a general question regarding quantum groups. It seems to me that the representation theory of the algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}}_{e-1})$ has many parallels with the ...
1
vote
0answers
148 views

Braidings for Comodules of Co-quasi-triangular Hopf algebra

Let $V$ be a (right-)$H$ comodule wrt a coaction $\Delta_R$, where $H$ is a co-quasi-triangular Hopf algebra with co-quasi-triangular Hopf algebra structure $R$. It is well-known that $V$ has a ...
2
votes
0answers
203 views

dimension of induced comodule

Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...
2
votes
0answers
99 views

Haar Functionals and Coquasi-triangular Structures

In this question it is mentioned that the coordinate algebra $C_q[G]$ Drinfeld--Jimbo algebras, for $G$ a compact semi-simple Lie group, admit a unique positive definite Haar functional. I was ...
2
votes
1answer
410 views

How to interpret sections over the $SU(2)$ character variety as sections over the $SL(2,\mathbb{C})$ character variety?

The motivation for this question comes from the Volume Conjecture of Kashaev-Murakami-Murakami. The Jones family of invariants of knots and $3$-manifolds can all be defined using $SU(2)$ and some ...
3
votes
0answers
300 views

What is the state in the WRT TQFT associated to a handlebody?

Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the ...
7
votes
1answer
599 views

Yetter--Drinfeld Modules and Braidings

Let $H$ be a Hopf algebra, $M$ a right $H$-comodule for a coaction $\Delta_R$, and $\triangleleft$ a right $H$-action on $M$ such that $M$ is a Yetter--Drindfeld module. We know from general theory ...
7
votes
1answer
313 views

What are the primitives of Lusztig's twisted bialgebra $\mathbf{f}$?

I'm trying to understand the coproduct on Lusztig's $\mathbf{f}$ and, apart from the Chevalley generators, I don't know of any more primitive elements. In the $q=1$ case, the Weyl group acts as a Hopf ...
0
votes
1answer
328 views

pointed Hopf algebra

Is that true that any pointed Hopf algebra is quantum group? Thanks!
6
votes
0answers
424 views

Where can I find tables of dual canonical basis vectors?

Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra. Now presumably this algorithm has been implemented ...
5
votes
1answer
589 views

Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
1
vote
0answers
333 views

Complement to the Kernel of a Hopf Algebra Map

Given two Hopf algebras $A,B$ over a field $k$, and a Hopf algebra map $\pi:A \to B$, are there any general tricks for finding a complement to the kernel of $\pi$. That is, how can one find a subspace ...
11
votes
3answers
1k views

How to calculate the Witten-Reshetikhin-Turaev invariants from a triangulation?

I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, ...
15
votes
2answers
2k views

How many definitions are there of the Jones polynomial?

Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
6
votes
4answers
967 views

Compact Quantum Groups from Hopf Algebras

For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way ...
8
votes
2answers
721 views

Quantization of conjugacy classes in a Lie group

Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\\!/G$. Then let ...
7
votes
2answers
571 views

Drinfeld's equivalence of quantized function algebras and quantized universal enveloping algebras

In his 1986 ICM address, Drinfeld discusses a way of producing a quantized function algebra (or more precisely a quantized formal series Hopf algebra) from a quantized universal enveloping algebra -- ...
5
votes
2answers
620 views

infinite group that maps onto dihedral group

The group is generated by $y_i$, $i=0, ...,p-1$ with relations $y_0y_1=y_1y_2=...=y_{p-1}y_0$ $y_0y_2=y_1y_3=...=y_{p-1}y_1$ $\vdots$ $y_0y_{p-1}=y_1y_0=...y_{p-1}y_{p-2}$ I have run into this ...
6
votes
2answers
831 views

Drinfeld't map, centre of quantum group, representation category of quantum group

My question is about the Drinfeld't map between $Rep(U_q(\mathfrak{g}))$ and $Z(U_q(\mathfrak{g}))$. I have heard the reference 1989 paper by Drinfeld't "Almost cocommutative Hopf algebras" - but this ...
2
votes
1answer
484 views

notation in Lusztig's book: introduction to quantum groups.

What are the relations between the notation in Lusztig's book introduction to quantum groups and the usual notation about quantum groups. For example, $v$ in Lusztig's book corresponds to the usual ...
16
votes
3answers
5k views

Quantum mathematics?

"Quantum" as a term/prefix used to be genuinely physical: what was supposed to be physically continuous turned out to be physically quantized. What sense does this distinction make inside ...
1
vote
0answers
204 views

On the decomposition of two representations of the Iwahori-Hecke algebra of type A_n

Consider the Iwahori Hecke algebra $H_q(n)$ of the symmetric group $S(n)$, the $\Bbb{Z}[q^{1/2},q^{-1/2}]$ algebra with generators $T_i$, $1 \leq i \leq n-1$, subject to the braid relations and the ...
31
votes
5answers
3k views

Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
4
votes
1answer
360 views

History of the Odd Dimensional Quantum Spheres

After reading this question, I began to wonder about the history of quantum $(2N-1)$-spheres. Basically I have two questions: (1) Who first introduced the $(2N-1)$-spheres, and who first introduced ...
1
vote
1answer
234 views

Ore Extensions and the Construction of the Quantum General Linear Group

In the usual (fomal) construction of the quantum general linear group $GL_q(N)$, an Ore extension is used. See for example Kassel. Why is this necessary? Surely one can just augment the set of ...
1
vote
1answer
204 views

Dual of a Basis for a Hopf Algebra Conatined in all Dually Paired Hopf Algebras

For an infinite dimensional Hopf algebra $H$, a non-degenerate dually pairing Hopf algebra $H'$, and a choice of basis $e_i$ of $H$, is the dual basis $e^i$ (defined of course by $e^i(e_j) = ...
11
votes
2answers
490 views

Relationship between “different” quantum deformations

This is a generic question, a good answer to it may be a reference to a corresponding paper\textbook, but any useful comments would be okay too. Let $\mathfrak{g}$ be a (simple) Lie algebra and ...
2
votes
0answers
137 views

Outer automorphism for $U_q(\mathfrak{su}(2|2))$

It is known that Lie superalgebra $\mathfrak{su}(2|2)$ (and only this one, not arbitrary $\mathfrak{su}(n|n)$) has the nontrivial central extension which forms an $\mathfrak{sl}_2$ triplet, let's call ...
10
votes
1answer
464 views

Generators of the Odd Dimensional Quantum Spheres

As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the ...
1
vote
1answer
167 views

Generators of the Augmentation Ideal (Counit Kernel)

For the Hopf algebra $SL_q(N)$ it is clear that the kernel of the counit contains the ideal generated by the elements $(u^i_i-1)$ and $u^i_j$, for $i \neq j$. However, I cannot seem to arrive at at ...