Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [quantum-groups]

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure.

1
vote
1answer
23 views

Are coefficients of a formal deformation bilinear?

$\newcommand{\planck}\hbar$ I was given the following definition of formal deformation: Let $A$ be an associative $R$-algebra with unit over a commutative ring. A formal deformation of $A$ is an ...
3
votes
0answers
21 views

Reference Request: Jimbo's Proof of Quantum Schur-Weyl Duality

In his seminal 1986 paper "A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke Algebra, and the Yang-Baxter Equation", Jimbo asserted (Proposition 3) that the quantum group associated to $\mathfrak{gl}_n$...
0
votes
0answers
10 views

The intersection of the annihilators of all finite dimensional weight modules is zero

Let's start with some definitions. Let $U_q:= U_q(\mathfrak{gl}_n)$ be the quantum enveloping algebra of $\mathfrak{gl}_n$, generated by the standard generators $\{e_i,f_i, x_j^{\pm}\,|\; i=1,\ldots, ...
1
vote
1answer
105 views

Dual Jacobi identity for Lie bialgebra

I'm studying Lie bialgebras: https://en.wikipedia.org/wiki/Lie_bialgebra. I'm a bit confused about the way of writing the so called "dual Jacobi identity". On Majid's book "a Quantum group Primer" ...
0
votes
1answer
37 views

Hopf Algebra Structure of $U_q(\mathfrak {sl}(2))$.

I am working out of Christian Kassel Quantum Groups. Define $U_q=U_q(\mathfrak {sl}(2))$ as the algebra generated by elements $E,F,K,K^{-1}$ subject to the following relations. $$ KK^{-1}=K^{-1}K=1\\ ...
1
vote
0answers
17 views

The action of tensor product over N terms on a ket.

Equation (6) of the paper titled, Multi-player and Multi-choice quantum game has left me puzzled-after many hours-as to how it is being derived. My working begins from the generic form seen just after ...
0
votes
1answer
28 views

Temperley-Lieb Diagrams and Representations of U_q(sl_2)

A Temperley-Lieb diagram is a crossingless matching of $2n$ points. We think of this matching as living in a rectangle, with $n$ points on top and the other $n$ on the bottom. To $n$ points we can ...
1
vote
0answers
19 views

Indecomposable G-equivariant vector bundles

I am reading the paper "The representation ring of the quantum double of a finite group" by Witherspoon. In Chapter 2 we define a G-equivariant vector bundle on a finite G-set $X$, as a collection of ...
2
votes
0answers
35 views

Simple Lie algebra representations and tensor powers of fundamental representations [duplicate]

Let $\frak{g}$ be a simple Lie algebra over $\mathbb{C}$. We will call a representation of $\frak{g}$ tautological if it is a fundamental representation of smallest dimension. For $V$ a tautological ...
0
votes
2answers
34 views

Multi-deformed numbers

The following deformations of usual numbers are well-known: $$ [n]_q=\frac{q^n-q^{-n}}{q-q^{-1}}, $$ and $$ [n]_{pq}=\frac{p^n-q^{-n}}{p-q^{-1}}. $$ Question. Are there any meaningful further ...
0
votes
0answers
16 views

paper of s.zelenova about skew field of quantum plane

please look at the pic can anyone please help me how to show that that how $F$ becomes a skew field.
2
votes
1answer
75 views

Proof the Lie algebra morphisms induce algebra morphisms on the universal enveloping algebra.

I am struggling to understand Theorem V.2.1 in Christian Kassel's Quantum Groups page 95. The Theorem is stated as follows. Let $L$ be a Lie algebra. Given any associative algebra $A$ and any ...
2
votes
1answer
103 views

Quantum invariants of 2-knots

I'm looking for a status report on analogues of quantum invariants of knots, for the 2-knots (homotopy classes of spheres / other Riemann surfaces embedded into 4-manifolds). Background I'm mostly ...
6
votes
4answers
949 views

Reference for Quantum groups

I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is ...
0
votes
0answers
24 views

Witten-Reshetikhin-Turaev (WRT) Invariant for surgery on the figure 8 at all roots of unity?

I've been trying to find a reference that gives the WRT invariant of a 3-manifold obtained by surgery on the figure 8 knot at an arbitrary root of unity but have only found them at the standard ...
0
votes
1answer
94 views

Sorting out a homological computation concerning Hall algebras

Let me recall some basic definitions concerning Hall algebras given in for example Schiffmann's notes, Hubery's notes or even the excellent master thesis by Sjoerd Beentjes which you can find online. ...
1
vote
1answer
27 views

Quantizing solutions to the reflection algebra

I am trying to find the quantum analogues to classical solutions of Sklyanin's reflection algebra (RE). I have a solution to the classical Poisson bracket for known r-matrix $r(\mu)$ \begin{equation}\...
1
vote
1answer
164 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
2
votes
0answers
58 views

Quantum Groups for Generic q and 3d-TQFT. What breaks?

I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the ...
0
votes
0answers
39 views

How to find normalizer to a subgroup of the Pauli group?

The Pauli operators are given by: $X = \left( \begin{array} { c c } { 0 } & { 1 } \\ { 1 } & { 0 } \end{array} \right) , \quad Y = \left( \begin{array} { c c } { 0 } & { - i } \\ { i } &...
2
votes
1answer
63 views

What is the relationship between universal enveloping algebra $U(g)$ and the QUE algebras $U_{q}(g)$

I just know $U_{q}(sl_2)$ is universal enveloping algebra $U(sl_2)$ when $q$ tends to $1$. Due to Qiaochu Yuan's comment, this is true in general. Conversely, Suppose $sl_2$ has three basises $e, h, f$...
2
votes
1answer
560 views

Graded and Filtered Algebras

This is right out of Kassel's Quantum Groups book which I am self-studying. It is on page 14. The general set-up is this. Let $A$ be a filtered algebra with filtration $F_0(A) \subset F_1(A) \subset \...
4
votes
0answers
56 views

Toy example of deformed diffeomorphism group

Consider a toy example of a diffeomorphism group – the group of diffeomorphisms of a 1-dimensional manifold with a disconnected boundary (2 points). The group is a group of monotonically increasing ...
1
vote
0answers
30 views

Ribbon Element, Equivalence of Definitions

In their book Quantum Groups Chari and Pressley define the ribbon element of a quasitriangular Hopf Algebra algebra $(H,R)$ as a special element $\nu\in H$ such that $\nu$ is central in $H$ $\nu^...
1
vote
0answers
31 views

Computing the ribbon element of $\operatorname{U_q}(\mathfrak{sl}_2)$

I'm trying to understand the computation of the ribbon element of $\operatorname{U_q}(\mathfrak{sl}_2)$. I've laid out an argument by Andre Henriques below. I don't understand how the conclusion (...
1
vote
0answers
57 views

Quantum principal bundles in physics

Recently I was reading in Stephen B. Sontz' "Principal bundles - The quantum case" and in contrast to "the classical case" he offered almost no connections with physical concepts. For quantum groups ...
0
votes
0answers
35 views

Sub-modules translate in sub-co-modules of the dual?

Is it true that if $W$ is a submodule of $V$ then by duality $W^*$ is a sub-co-module of $V^*$? Can anybody confirm that or give an example when this doesn't happen? I think it might be necessary to ...
2
votes
0answers
35 views

Indecomposable modules for the big quantum group

I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
3
votes
1answer
130 views

What is weight lattice modulo coroot lattice?

In Lectures on Tensor Categories and Modular Functors by Bakalov and Kirillov, the $S$ matrix (expression 3.3.7) is expressed in the form $\vert P/kQ^\vee \vert^{-1/2}\times(\cdots)$, where $k\in \...
1
vote
1answer
25 views

Are these 2 definitions of $K$ and $H$ on $U(sl_q(2))$ coherent?

I'm studing $U(sl_q(2))$ and studying how to recover $U(sl(2))$ from $U(sl_q(2))$ I found these two definition for both $H$ and $K$ as formal generators. $$H=\frac{K-K^{-1}}{q-q^{-1}}$$ $$K=q^{H}$$ I'...
2
votes
0answers
42 views

PBW theorem for quantum Lie algebras

Does PBW-theorem exists for quantum Lie algebras? I also appreciate your help in finding papers related to this theorem for the case of Quantum Lie algebras.
0
votes
0answers
43 views

What is wrong with this argument on $U_q(sl(2))$? What is missing to make it precise?

Let us define $U_q(sl(2))$ as the algebra with four generator as usual $$K\,K^{-1}=K^{-1}K=1,$$$$K\,E\,K^{-1}=q^{2}E,\,K\,F\,K^{-1}=q^{-2}F,\\EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.$$ I want to easily show ...
2
votes
1answer
53 views

q-quantization of Lie bialgebras

I am trying to understand the difference between the "Drinfeld" and the "Lusztig" theory of quantum groups, more specifically with respect to the problem of quantization of Lie bialgebras/Poisson Lie ...
6
votes
0answers
1k views

Some concrete examples of $M_q(2)$ points

Given $q \in C$ invertible, Kassel says that a $M_q(2)$ point of an $R$ algebra is a $m=\left(\begin{array}{cc} A & B\\ C & D \end{array}\right)\in R^{4}$ such that $A,\,B,\,C,\,D \in R$ ...
7
votes
3answers
79 views

Concrete cases where $YX=qXY$

I was reading Kassel on the quantum plane and he defines an $R$-point on this plane as a pair of $X$, and $Y$ elements of the non commutative algebra $R$ such that $$YX=qXY,$$ with $q$ invertible. Can ...
3
votes
1answer
139 views

Primitive elements of finite dimensional Hopf algebras

An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that: "If $H$ is a $\...
2
votes
0answers
72 views

Universal enveloping algebra vs algebra of continuous functions

I'm a physicist studying quantum groups, but this question is about the usual classical Lie groups (though it is related to the language that is used to describe the deformation to quantum groups, ...
0
votes
1answer
43 views

Checking antipode on Enveloping algebra of a Lie Algebra

Let $U\left(\mathfrak{g}\right)$ be the enveloping algebra of $\mathfrak{g}$.Let's define comultiplication, counit and antipode as $$\triangle\left(X\right) =X\otimes1+1\otimes X,$$ $$\epsilon\left(X\...
1
vote
2answers
76 views

Checking the antipode for the dual of the Group Hopf Algebra

Consider the vector space $\mathbb{C}G\,=\left\{ f:\,G\longrightarrow\mathbb{C}\right\}$ , and define the following multiplication: $\mu\left(f\otimes g\right)\left(x\right):=f\left(x\right)g\...
3
votes
0answers
40 views

Why are these endomorphisms of $U_q(\mathfrak{g})^-$

I am reading: "On crystal bases of the $q$-analogue of universal enveloping algebras" by Kashiwara. On page 481 - Lemma 3.4.1: For any $P\in U_q(\mathfrak{g})^-$ there exist unique $Q,R\in U_q(...
1
vote
0answers
25 views

Duality between comodules $SL_q(2)$ and $U_q(sl(2))$

There's a welle known Hopf pairing between $SL_q(2)$ and $U_q(sl(2))$, I have no problem in construction such pairing wich can be easly be done by the coefficients of a 2-dimensional irreducible ...
2
votes
1answer
129 views

Reference on correspondence between commutative Hopf Algebras and Groups

Is it true that every commutative Hopf algebra is related to a Group in such a way that the co-multiplication is originated from the multiplication of the group, the antipode from the inverse? Making ...
3
votes
0answers
88 views

What is the relation between crystals and crystal bases?

Kashiwara introduced the concept of crystal bases (Lusztig introduced canonical basis), see for example the article. Kashiwara also introduces the concept of crystals (Section 7). What is the ...
4
votes
1answer
422 views

Definition of the Quantum plane and the Yang Baxter Equation

I was reading this on the quantum plane and the Yang Baxter equation. John Baez says that imposing $$ R(X\otimes X)= X\otimes X $$ $$ R(Y\otimes Y)= Y\otimes Y $$ $$ R(X\otimes Y)=q Y\otimes X $$ $$ R(...
1
vote
0answers
52 views

Bogoliubov Transform

For the operator defined as polynomial is the boson creation and annihilation operators $\hat{a}$, $\hat{a}^\dagger$ such that $[\hat{a},\hat{a}^\dagger] = 1$ $$\hat{L} = A\hat{a}^2 + B\hat{a}^{\...
1
vote
0answers
69 views

Integrable modules of quantum groups.

Let $\widehat{\mathfrak{g}}$ be an affine Lie algebra and $U_q(\widehat{\mathfrak{g}})$ the corresponding quantum affine algebra. Let $V$ be an integrable module of $U_q(\widehat{\mathfrak{g}})$. That ...
0
votes
2answers
56 views

On proposition I.3.2 of 'Quantum groups' by Kassel.

I am reading the book Quantum groups by Kassel. In proposition I.3.2 at the very beginning the reader is asked to show that under the identifications made, the maps $\Delta,\varepsilon$ and $S$ ...
2
votes
1answer
155 views

Intuition behind the relation of commutative Hopf algebra and Groups

I've heard that a commutative Hopf algebra can be thought an algebra construction over the space of functions on a group to the ground field. The product should be the pointwise multiplication, ...
2
votes
0answers
135 views

What are $q$-deformations?

This question has already appeared in a lot of different ways and here is another one. First of all, many people know the typical quantum group $U_q(\mathfrak{sl}_2)$ by generators and relations. ...
2
votes
1answer
75 views

Structure of simple modules of $U_q(\mathfrak{sl}_2)$ at a root of unity.

This is Theorem VI.5.5 in Kassel's Quantum Groups: Let $q$ be a root of unity. Prove: Any simple $U_q$-module of dimension $e$ is isomorphic to a module of the following list: $V(\lambda, a,...