# Tagged Questions

**5**

votes

**1**answer

127 views

### $q$-Deformed Quillenâ€“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 ...

**4**

votes

**1**answer

73 views

### Comodule analogue for statement that a faithful representation of an affine group scheme generates all

If $V$ is a faithful finite dimensional representation of an affine group scheme $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n ...

**18**

votes

**3**answers

520 views

### What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?

In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of ...

**3**

votes

**2**answers

243 views

### Algebraic Groups, Modules, and Comodules

Background:
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H} := \text{Alg}_k\{H; k\},
$$
we recall (see Abe Chapter 4 ...

**2**

votes

**1**answer

151 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 ...

**10**

votes

**1**answer

277 views

### $\text{Rep}(D(G))$ as representation category of a vertex operator algebra

The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as ...

**1**

vote

**0**answers

64 views

### What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?

What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum ...

**16**

votes

**1**answer

242 views

### Is the representation category of quantum groups at root of unity visibly unitary?

Let $\mathfrak g$ be a simple Lie algebra.
By taking the specialization at $q^\ell=1$ of a certain integral versionÂą of the quantum group $U_q(\mathfrak g)$,
and by considering a certain quotient ...

**2**

votes

**0**answers

140 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 ...

**4**

votes

**1**answer

191 views

### Towards a quantum version of Schur's orthogonality relations

This is taken from Timmermann's Invitation to Quantum Groups and Duality.
Hi folks I am struggling a little with a small calculation in the above text.
I will just get right into it.
Lemma 3.2.5
...

**15**

votes

**0**answers

368 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 ...

**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 ...

**6**

votes

**1**answer

300 views

### The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

I asked this question on Math.Stack but have not had any answers.
Question
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
The trivial ...

**0**

votes

**0**answers

176 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\\
...

**3**

votes

**0**answers

216 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 ...

**4**

votes

**1**answer

138 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
$$KK^{-1}=K^{-1}K=1$$
...

**4**

votes

**1**answer

201 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, ...

**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

**1**answer

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

**1**answer

411 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 ...

**3**

votes

**1**answer

175 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

**1**answer

381 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 ...

**8**

votes

**1**answer

515 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 ...

**9**

votes

**0**answers

330 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 $sl(n)$?

**0**

votes

**1**answer

317 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 ...

**6**

votes

**2**answers

761 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 ...

**1**

vote

**0**answers

197 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 ...

**22**

votes

**2**answers

2k views

### When does Lusztig's canonical basis have non-positive structure coefficients?

I've heard asserted in talks quite a few times that Lusztig's canonical basis for irreducible representations is known to not always have positive structure coefficents for the action of $E_i$ and ...

**6**

votes

**2**answers

863 views

### Is there a quantum Hermite reciprocity?

It is well known that there is an isomorphism of $SL_2=SL(V)$ representations
$$
Sym^n(Sym^m(V))\simeq Sym^m(Sym^n(V))
$$
called Hermite reciprocity (discovered in 1854).
My question is: Is there ...

**6**

votes

**1**answer

457 views

### Is there a good reference for the relationship between the Yangian and formal based loop group?

For every finite dimensional semi-simple Lie group $\mathfrak{g}$, we have a loop algebra $\mathfrak{g}[t,t^{-1}]$. This loop algebra has a natural invariant inner product by taking the residue at ...

**5**

votes

**2**answers

193 views

### What is the explanation for the special form of representations of three string braid group constructed using quantum groups information supplied

It is well-known that representations of quantised enveloping algebras give representations of braid groups. For the examples that I know explicitly the representations of the three string braid group ...

**7**

votes

**2**answers

846 views

### What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?

In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...

**5**

votes

**1**answer

671 views

### Does the canonical basis of a tensor product of quantum group representations span the isotypic components of tilting modules?

It's a theorem of Lusztig that if one takes two (or more) representations of a quantum group at a non-root-of-unity choice of $q$, and look at the canonical basis of their tensor product, then each ...

**5**

votes

**1**answer

248 views

### Are there elements of fixed weight in a crystal not killed by too many Kashiwara operators?

I've come across an annoying lemma trying to finish up an argument, and I was hoping one of you guys knew about it.
Question: Given
a weight $\lambda$ of a simple Lie algebra $\mathfrak g$, and
...

**3**

votes

**1**answer

306 views

### Categorifying the group representations

I've heard about this construction on the lecture about higher representation theory:
Given a Lie algebra $g$, one constructs $\mathcal A$, a category whose $K_0$ is the universal enveloping ...

**6**

votes

**5**answers

491 views

### Is there a version of Temperley-Lieb using sl(3) rather than sl(2)?

This question is a spin-off from Sammy Black's question on super Temperley-Lieb. Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the ...

**15**

votes

**2**answers

902 views

### Does the super Temperley-Lieb algebra have a Z-form?

Background Let V denote the standard (2-dimensional) module for the Lie algebra sl2(C), or equivalently for the universal envelope U = U(sl2(C)). The Temperley-Lieb algebra TLd is the algebra of ...

**3**

votes

**3**answers

422 views

### What is a formula for the “group-like Drinfeld element”?

Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...

**4**

votes

**2**answers

406 views

### Are there interesting monoidal structures on representations of quantum affine algebras?

Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...

**3**

votes

**1**answer

288 views

### What is the “right” hermitian structure on tensor products of quantum group representations?

This is pretty specific, but there are some experts around.
So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a ...