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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4
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4answers
294 views

Non-Drinfeld--Jimbo Deformations and Finite Quantum Groups

As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called ...
7
votes
1answer
169 views

Real forms of Drinfeld-Jimbo quantum groups

A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and ...
4
votes
2answers
485 views

$q$-Deforming Woronowicz's Leibniz Rule

The Woronowicz definition of a differential calculus over an algebra consists of a pair $(\Omega,$d$)$, where $\Omega$ is an $A-A$-bimodule, and $$ \text{d}:A \to \Omega, $$ is a bimodule map, ...
4
votes
1answer
141 views

Question about unusual highest weight modules for $U_q(sl(2))$

Background Let $U_q(sl(2))$ be the quantum group associated with $sl(2)$ i.e. the associative algebra with 1 over $Q(q)$ generated by $x^+,x^-,K,K^{-1}$ with relations $$KK^{-1}=K^{-1}K=1$$ ...
1
vote
0answers
229 views

Why are there two Hopf algebra structures on a Kac--Moody Algebra.

For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by $$ \Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ...
1
vote
1answer
110 views

Zero Sums in a $q$-Deformation Remain Zero for $q=1$

Looking at previous question, I begun to think and came upon the following more solid question: Let $SU_q(N)$ be the usual quantized coordinated algebra of $SU(N)$. Now consider, for some fixed value ...
7
votes
0answers
154 views

Does the braid group act faithfully on the quantized enveloping algebra?

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where ...
0
votes
1answer
111 views

Deformations and Dimensions: $q$-Deform Finite to Infinite?

Let $A$ be a (complex) (finitely generated) algebra with some type of $q$-deformation $A_q$, where $A_1 = A$. Moreover, let $V_q$ be a vector subspace of $A_q$, such that $V_1$ is a ...
11
votes
7answers
1k views

Open problems in the theory of compact quantum groups

What are the important open problems in the theory of compact quantum groups? Or conjectures? Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules ...
6
votes
1answer
143 views

Matrix model or cocycle twist construction for q-deformations of compact simple Lie groups in $q=-1$?

S. Zakrzewski constructed compact or locally compact quantum groups with deformation parapmeter $q=-1$ for subgroups of $GL(2;C)$, e.g. $SU_{-1}(2)$, $SU_{-1}(1,1)$, $SL_{-1}(2,R)$, as algebras of ...
3
votes
1answer
189 views

Action of left $\mathbb{C}_q[SL_2]$-crossed modules

Shahn Majin and Xavier Gomez say in the beginig of their article (Noncommutative cohomology and electromagnetism on $\mathbb{C}_q [SL_2]$ at roots of unity) that tha action of left $\mathbb{C}_q ...
17
votes
9answers
1k views

expository papers related to quantum groups

Hello all, I know basic representation theory(finite groups, lie groups&lie algebras) and I want to get a flavor of quantum groups (why they are useful, important results etc) and other related ...
2
votes
1answer
275 views

Explicit Descriptions of $q-SO(2)$, and $q-Sp(2)$?

When ever I hear noncommutative geometers talking about quantum groups, it is usually $q-SU(2)$ that they are discussing. As a result there are many good and explicit generator and relation ...
4
votes
1answer
206 views

Convex PBW bases

Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, ...
2
votes
1answer
89 views

Zero Actions on a Hopf Module Preserved Under the AntiPode?

Let $H$ be a Hopf algebra, and $(M,\triangleleft)$ a $H$-module. Now for $m \in M$, and $h \in H$, then is it true in general that $$ m \triangleleft h = 0 ~~~ \implies ~~~~m \triangleleft S(h) = 0? ...
2
votes
1answer
163 views

Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...
12
votes
3answers
658 views

Isomorphisms of quantum planes

Let $k$ be a field and $q\in k^{*}$. The quantum plane $k_{q}[x,y]$ is the algebra $k\langle x,y\rangle/\langle xy=qyx \rangle$ (i.e. the quotient of the free non-commutative $k$-algebra on two ...
21
votes
2answers
670 views

quantum groups… not via presentations

Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a_{ij}$, the quantum group $U_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K_i$, $E_i$, $F_i$ (the ...
4
votes
0answers
139 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 ...
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vote
1answer
133 views

Skein Modules on closed 3-manifolds

Hi all! Why are skein modules 1-dimensional on closed 3-manifolds? The result seems clear on closed manifolds with vanishing first Betti number (e.g. $S^3$), but I don't see how to prove it for, say, ...
2
votes
1answer
212 views

Non-Faithfully Flat Quantum Homogeneous Spaces

Let $G$ and $H$ be two Hopf algebras, and $\pi: G \to H$ a Hopf algebra map. We will call an algebra of the form $$ M:= \lbrace m \in G ~ | ~ m_{(1)} \otimes \pi(m_{(2)}) = m \otimes 1 \rbrace $$ a ...
1
vote
1answer
102 views

$H$-Hopf modules equal the tensor products of their coinvariants with H

In a comment for this old question, it was said that > There is a theorem that Hopf modules are (up to isomorphism) the tensor products of their coinvariants with H. (This theorem is usually worded ...
4
votes
1answer
232 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 ...
13
votes
1answer
416 views

Categorifying the equality of product and coproduct of symmetric functions

Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...
4
votes
3answers
1k views

Mistake in Wikipedia Entry “Coalgebra”

Consider the following quote from the Wikipedia entry Coalgebra: The kernel of every coalgebra morphism $f : C_1 \to C_2$ is a coideal in $C_1$, and the image is a subcoalgebra of $C_2$. I can't ...
3
votes
2answers
448 views

Fourier Transform on Locally compact quantum groups

I Have read some articles on locally compact quantum groups and the Fourier transform on them. I wonder yet why we define the fourier transform as an operator valued functions from $L^1(\mathbb{G})$ ...
1
vote
1answer
202 views

Trivial Hopf Coinvariant Subspace Example

For $G,H$ Hopf algebras, and $\pi:G \to H$ a Hopf algebra map, can some-one give me an example of a (right) $H$-comodule $(V,\Delta_R)$, such that $$ (G \otimes V)^{\text{co}H} = \lbrace g_{(1)} ...
0
votes
1answer
109 views

Coinvariant Subalgebras of Hopf Comodules and Quotients

For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta_R(W) \subseteq W \otimes H$, and note that this implies that ...
0
votes
2answers
165 views

alpha derivations

I am curious. Is there a "slick" way of showing that given an arbitrary algebra $A$ with generating set $X$, an algebra endomorphism $\alpha : A\to A$ and a function (satisfying some conditions) $d : ...
5
votes
3answers
548 views

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 ...
3
votes
1answer
178 views

Modules which are direct sum of weight spaces.

For a semisimple Lie algebra $\mathfrak{g}$, a highest weight module $V(\lambda)$ with highest weight weight $\lambda$ has the property that every submodule $W$ of $V(\lambda)$ is a direct sum of the ...
9
votes
1answer
397 views

Unitary representations of Quantum Groups

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...
3
votes
3answers
449 views

Hopf structure of Uq(sl(2))

Given the universal enveloping algebra, $U(\mathfrak{sl}(2))$ the coalgebra structure is defined such that the generators $X,Y$ and $H$ are primitive elements. From this, is there a "nice" way to ...
1
vote
0answers
116 views

Hopf Comodule Decompositions and Coinvariant Elements

For $H$ an Hopf algebra, and $V$ a right $H$-comodule with coaction $\Delta_R$, for which there exists a vector space decomposition $V=V_+ \oplus V_-$. Are the following two statements equivalent? ...
9
votes
3answers
747 views

Quantum group Uq(sl(2))

I'm looking at motivating the standard deformation of $U(\mathfrak{sl}(2))$. As an algebra $U(\mathfrak{sl}(2))$ is generated by $X,Y$ and $H$ and subject to the relations $[X,Y] = H$, $[H,X] = 2X$ ...
8
votes
1answer
529 views

Where does the canonical basis differ from the KLR basis?

The question implicitly asked in Ben Webster's question is: Does the canonical basis of Uq(n+) agree with the basis coming from categorification via Khovanov-Lauda-Rouqier algebras? Thanks to ...
3
votes
0answers
168 views

Fundamental theorem of coalgebras

Hi, Is there an "equivalent" to the fundamental theorem of coalgebras (any element of a coalgebra is contained in a finite dimensional sub-coalgebra) in the theory of algebraic quantum groups ...
12
votes
1answer
414 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 $n$ ...
44
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
347 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 ...
9
votes
0answers
338 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
431 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
307 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
472 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, ~~~~~~~ ...
4
votes
1answer
224 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
341 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 ...
6
votes
2answers
302 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
572 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
218 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
419 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 ...