**1**

vote

**0**answers

87 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 ...

**5**

votes

**1**answer

293 views

### Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...

**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 ...

**6**

votes

**1**answer

329 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 ...

**4**

votes

**2**answers

325 views

### lie algebras, Kac Moody, and quantum mechanics book

Hi all, I've just finished a graduated course on Kac-Moody algebras, and I'm really looking for some reading in regard to their applications to Quantum Mechanics. Can you help?

**6**

votes

**1**answer

323 views

### Are Turaev--Viro invariants secretly a discretized path integral?

Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with ...

**0**

votes

**0**answers

180 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\\
...

**11**

votes

**2**answers

587 views

### Quantum-Jimbo Algebras: Why Such Fuss About Roots of Unity?

Coming from a Lie algebraic background, I'm trying to branch onto quantum group theory. The divide I see all the time, is $q$ a root of unity or $q$ not a root of unity. I am wondering why is this? If ...

**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 ...

**2**

votes

**1**answer

189 views

### Hopf Duals and Matrix Coefficients

One defines the finite dual of a Hopf algebra $A$ as
$$
H^o := \{f \in H^* ~|~ f(I) = 0, \text{ for some ideal $I$ of $H$ with } \dim_C(H/I) < \infty \}.
$$
As is well-known, $H^o$ has a ...

**1**

vote

**1**answer

122 views

### Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations

The representations of the quantized enveloping algebra $U_q(\frak{sl}_n)$ are labeled by the positive weight lattice $P^+$ of the classical Lie algebra $\frak{sl}_n$. Moreover, the dual Hopf algebra ...

**4**

votes

**2**answers

320 views

### Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation
$$
[E,F] = \frac{K-K^{-1}}{q-q^{-1}}.
$$
To address this problem, one has ...

**3**

votes

**0**answers

123 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 ...

**0**

votes

**0**answers

137 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 ...

**4**

votes

**4**answers

319 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

**1**answer

178 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

**2**answers

517 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

**1**answer

148 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

**0**answers

240 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

**1**answer

111 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

**0**answers

163 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

**1**answer

115 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

**7**answers

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

**1**answer

149 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

**1**answer

192 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 ...

**18**

votes

**9**answers

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

**1**answer

287 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

**1**answer

228 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

**1**answer

90 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

**1**answer

187 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

**3**answers

761 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 ...

**22**

votes

**2**answers

733 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

**0**answers

143 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 ...

**1**

vote

**1**answer

141 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

**1**answer

223 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

**1**answer

105 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

**1**answer

243 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

435 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

**3**answers

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 ...

**4**

votes

**2**answers

482 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

**1**answer

203 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

**1**answer

118 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

**2**answers

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

**3**answers

588 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

**1**answer

188 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 ...

**10**

votes

**1**answer

447 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

**3**answers

496 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

**0**answers

121 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?
...

**10**

votes

**3**answers

885 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 $[X,Y] = H$, $[H,X] = 2X$ ...

**10**

votes

**1**answer

608 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 ...