**18**

votes

**0**answers

509 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 ...

**17**

votes

**0**answers

156 views

### Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?

The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...

**11**

votes

**0**answers

253 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

377 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

181 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 ...

**10**

votes

**0**answers

384 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

480 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

138 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

164 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

326 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

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 $x\otimes t^k$ ...

**6**

votes

**0**answers

428 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

175 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

225 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 $U_q\mathfrak{so}_N$ ...

**5**

votes

**0**answers

42 views

### What's the relation between half-twists and star structures on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...

**4**

votes

**0**answers

146 views

### Number of Irreducible Representations of $U_q(n)$ of Dimension $n$?

For quantum group $U_q(n)$, is it true that it has exactly two non-isomorphic irreducible corepresentations with dimension $n$, and that one is the dual of the other? I know result is in the Chapter ...

**4**

votes

**0**answers

95 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

145 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

188 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 ...

**4**

votes

**0**answers

209 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

305 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

81 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

147 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

230 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

126 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

303 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

137 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

124 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

42 views

### Co-quasitriangular Hopf algebra - notation

In one article I found the following statement :
If $A$ is a Hopf algebra over $k$ with co-quasitriangular structure $r$, then one can define map $r_{21}*r:A\otimes A\rightarrow k \ \ $ (...).
...

**2**

votes

**0**answers

88 views

### Is there an E8 symmetry in the zero-field Ising model?

In the paper On classification of modular tensor categories by Rowell, Stong and Wang, they list the Ising modular category $I$ as having 3 objects $1$, $\sigma$ and $\psi$, with fusion rules ...

**2**

votes

**0**answers

98 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

205 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

127 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

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

**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

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 ...

**1**

vote

**0**answers

91 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 ...

**1**

vote

**0**answers

129 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

130 views

### The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...

**1**

vote

**0**answers

88 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

71 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

243 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

122 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

150 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

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 ...

**1**

vote

**0**answers

206 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

192 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

127 views

### Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) ...

**0**

votes

**0**answers

58 views

### Semisimple Coquasitriangular Hopf Algebras

Let $(G,R)$ be a coquasitriangular Hopf algebra, which we furthermore assume to be cosemisimple. Generalising classical Peter-Weyl, cosemisimplicity of $G$ is well-known to imply an isomorphism
$$
G ...

**0**

votes

**0**answers

58 views

### quantum deformation

The standard quantum groups, say $GL_q(n)$ or $U_q(gl(n))$, depend on the parameter $q$, which in the classical limit tends toward 1. Let $t:=q-1$ be considered as a generic parameter then we can ...