**7**

votes

**1**answer

184 views

### $q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?

Define n-quantum vector space to be the algebra
$$
{\mathbb C}_q^n := \mathbb{C}< x_i ~ | ~ i =1, \ldots, N>/<x_i x_j = q x_j x_i ~ | ~ i< j>.
$$
For $q=1$, we get the usual polynomial ...

**15**

votes

**0**answers

373 views

### Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...

**11**

votes

**0**answers

210 views

### Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...

**11**

votes

**0**answers

336 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 ...

**10**

votes

**0**answers

151 views

### Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...

**9**

votes

**0**answers

335 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

**0**answers

423 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 ...

**7**

votes

**0**answers

125 views

### When is Rep(U_q(g)) invariant under q -> -q and why?

Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...

**7**

votes

**0**answers

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 ...

**7**

votes

**0**answers

293 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 ...

**6**

votes

**0**answers

138 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 $x\otimes t^k$ ...

**6**

votes

**0**answers

406 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 ...

**6**

votes

**0**answers

170 views

### What is the q-analogue of the Lefschetz decomposition?

The representation theory behind the Lefschetz decomposition in Kahler geometry was summarised very neatly by Victor Protsak in his answer to
29907
Let $W$ be a $2n$-dimensional symplectic vector ...

**6**

votes

**0**answers

219 views

### Explicit Braid Group Reps from quantum SO(N) at roots of unity

This question is related to this one (and indeed the goals are similar).
Let $N$ be odd and consider the braided fusion category $\mathcal{C}$ (actually modular) obtained from $U_q\mathfrak{so}_N$ ...

**4**

votes

**0**answers

87 views

### Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras

As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...

**4**

votes

**0**answers

138 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 ...

**4**

votes

**0**answers

198 views

### q-deformation of the unitary group integral

There is a well-known orthogonality property of $U(N)$ group characters
$$
\int d U \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim_\mu}
$$
where the integral is ...

**4**

votes

**0**answers

286 views

### Working with quadratic Lie algebras

A quadratic Lie algebra is a Lie algebra with an invariant inner product and the main examples are semisimple Lie algebras. This definition then makes sense in any linear symmetric monoidal category. ...

**3**

votes

**0**answers

78 views

### States and extremal states of quantum SU(2) and the Podleś sphere

Is there any description (preferably somehow related to the original generators) for the state space (as in C*-algebras) of quantum SU(2) and the Podleś sphere? If so (this is pushing my luck) are the ...

**3**

votes

**0**answers

68 views

### Is multiplication continuous in quantum-SU(2) with respect to the $L^2$-norm

For the Hopf $\ast$-algebra ${\cal O}_q(SU(2)$, with its unique Haar measure $h$, we have an inner product, and a norm, on ${\cal O}_q(SU(2))$ defined by
$$
<x,y> : = h(xy^*), ...

**3**

votes

**0**answers

119 views

### How a unitary corepresentation of a Hopf C*-algebra, deals with the antipode?

Let $\mathcal{A}$ be a Kac algebra (Hopf C*-algebra), with comultiplication $\delta$, counit $\epsilon$ and antipode $S$.
The following definition comes from this paper (p51-52) of ...

**3**

votes

**0**answers

217 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 ...

**3**

votes

**0**answers

111 views

### Hopf Algebra Pairings and Module-Comodule-Equivalences

Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...

**3**

votes

**0**answers

165 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 ...

**3**

votes

**0**answers

285 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 ...

**3**

votes

**0**answers

131 views

### The Tangent Spaces of the Bicovariant Calculi over Quantum SU(3)

In Woronowicz's approach to differential calculi on Hopf algebras, a calculus over an algebra $A$ can be specified by a finite dimensional subspace of the dual of $A$. The best known example is his ...

**3**

votes

**0**answers

122 views

### Restricting Differential Calculi to Quotients

Let $A$ be an algebra, $B \subset A$ a subalgebra, and $(\Omega^1(A),d)$ a first order differential calculus over $A$. Now, as is well known, the subalgebra of $\Omega^1(A)$ consisting of elements of ...

**2**

votes

**0**answers

84 views

### Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$

I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167:
Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra ...

**2**

votes

**0**answers

145 views

### Canonical basis of quantum groups

I am trying to understand the canonical basis of quantum groups and different ways to construct the canonical basis of quantum groups. In the comments of Lusztig's papers, the paper [92], CANONICAL ...

**2**

votes

**0**answers

116 views

### quantum deformations of tensor category

I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...

**2**

votes

**0**answers

187 views

### dimension of induced comodule

Let $\pi : G --> H$ 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

**0**answers

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

**0**answers

134 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 ...

**1**

vote

**0**answers

84 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 ...

**1**

vote

**0**answers

95 views

### The convolution on the finite dimensional weak Hopf $C^*$-algebras

Let $\mathbb{A}$ be a finite dimensional weak Hopf $C^*$-algebra, and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{A}}$ and the convolution $a * b ...

**1**

vote

**0**answers

64 views

### Reference request: Lusztig's symmetries

Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$,
\begin{align}
...

**1**

vote

**0**answers

83 views

### Quantum Algebras — Crystal Basis/Graph

Suppose I have a finite-dimensional irreducible $U_q(sl_2)$-module say $V$, and (L,B) is its crystal basis.
How do you find the crystal basis of the evaluation $U'$-module $V_{x=1}$? And is there a ...

**1**

vote

**0**answers

69 views

### Classifying Equivariant Maps Between Fin-Dim Irreducible Modules

Let $G$ be a compact semi-simple Lie group, (or to be even more concrete let $G = SL(N)$), and let $V$ and $W$ be finite dimensional irreducible representations of $G$. Surely it is very well-known ...

**1**

vote

**0**answers

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

**0**answers

115 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?
...

**1**

vote

**0**answers

142 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 ...

**1**

vote

**0**answers

331 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 ...

**1**

vote

**0**answers

199 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 ...

**1**

vote

**0**answers

185 views

### Bicovariant Calculi on the Quantum Unitary Groups

The bicovariant differential calculi on quantum-$SU(n)$ have been classified (by Schmudgen I think) and have been shown to have non-classical dimension. My question is whether or not the bicovariant ...

**0**

votes

**0**answers

41 views

### Does the standard Podlés sphere have a quasitriangular Hopf algebra structure? Do quantum homogeneous spaces have one in general?

Function spaces on (classical) homogeneous spaces can have a bialgebra structure:
Take $S^2$ to be the unital, associative algebra generated by $x, y, z$ with the relation $x^2 + y^2 + z^2 = 1$ and ...

**0**

votes

**0**answers

178 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\\
...

**0**

votes

**0**answers

130 views

### $h$-adic Completion of $U_q(\frak{sl}_2)$?

Consider the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as ...