Questions tagged [quantum-groups]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
26 views

Dirac Delta expansion over $SU(2)_{q}$

It is known that the Dirac $\delta(U)$ distribution can be expanded in representations for compact groups. For example concerning the $U(1)$ $$ \delta(\phi)= \frac{1}{2\pi}\sum_{n} e^{in\phi}$$ Or ...
0
votes
0answers
19 views

Definition of natural representation of quantum group.

I have a question about terminology. In the paper, a terminology "natural representation" is used. I don't know the precise definition of natural representation. What is the definition of &...
0
votes
0answers
15 views

A definition of a semiprime ring for Hopf algebras

If $R$ is an associative ring (unital or not) is a well known result that the following conditions are equivalent: (i) For all $a\in R$, $aRa=0$ implies $a=0$. (ii) For all left ideals $I$ of $R$, $I^...
1
vote
0answers
30 views

Quasitriangular structure for infinitely dimensional Drinfeld double?

It is well known that the Drinfeld double $D(H)=H^{*op}\otimes H$ of an Hopf algebra $H$ admits a quasitriangular structure. When $H$ is finitely dimensional, the $R$-matrix can be given by $$R=\sum e^...
1
vote
1answer
39 views

Universal completions of *algebras

I am dealing with two "universal completions" but I am not sure if they are the same thing and would appreciate some guidance. Let $\mathcal{A}$ be a unital *-algebra. A $\mathrm{C}^*$-...
3
votes
1answer
116 views

Good text on quantum groups.

I'm interested in learning about quantum groups about a $C^*$-algebraic perspective. I'm familiar with (the basics) of topology, abstract algebra, measure theory, functional analysis (in particular $...
1
vote
1answer
55 views

The unbounded antipode for Woronowicz's quantum group $\operatorname{SU}_q(2)$

For non-zero $q\in [-1,1]$, Woronowicz's quantum group $\operatorname{SU}_q(2)$ is given as the universal unital $\mathrm{C}^*$-algebra generated by elements $a,c\in C(\operatorname{SU}_q(2))$ subject ...
0
votes
0answers
28 views

$H$-$H$-Bicomodules as $H \otimes H$-comodules

For $R$ and $S$ two rings, an $R$-$S$-bimodule is actually the same thing as a left module over the ring $R \otimes_{\mathbb{Z}} S^{\mathrm{op}}$. It seems the same happens for comodules over two Hopf ...
1
vote
1answer
30 views

Generalization of the concept of homogeneous element

Let $G$ be a finite group. Recall that $kG$ is a Hopf algebra and, since $dim(kG)<\infty$, $(π‘˜πΊ)^βˆ—$ is also a Hopf algebra with its structure dual to that of $π‘˜πΊ$. As it is well known, an ...
3
votes
1answer
115 views

What is the group of group-like elements of a quantum group?

A quantum group is not a group. For example, the Drinfeld-Jimbo "quantum doubles" are Hopf algebras obtained by deforming the universal enveloping algebras of Lie algebras. But in every Hopf algebra,...
1
vote
2answers
80 views

PBW basis for quantum groups and braid group action

I'm trying to check some details from this paper. Let $U_q(\mathfrak{sl}_3)$ be the usual Drinfeld-Jimbo quantum group, with generators $E_1,E_2,F_1,F_2,K_1^{\pm},K_2^{\pm}$ and well known relations, ...
2
votes
0answers
63 views

Why is $q$ sometimes a complex number, but other times a prime power?

In the fields of representation theory and quantum algebra, we often start with some $\mathbf{C}$-algebra and study it's quantization as an algebra over $\mathbf{C}(q)$, using the algebra structure to ...
2
votes
1answer
51 views

Studying quantized algebras, what motivates the choice of base ring?

In the fields of representation theory and quantum algebra, we often start with, for example, some $\mathbf{C}$-algebra $A$ and study a quantization of $A$ by adjoining an indeterminate $q$, or ...
1
vote
1answer
36 views

Studying quantized algebras, why introduce $q^{1/2}$ instead of just $q$?

In the fields of representation theory and quantum algebra, we often study quantized versions of algebraic objects by regarding them as algebras over $\mathbf{C}(q)$, or some subring of $\mathbf{C}(q)$...
1
vote
1answer
147 views

About the quantum plane.

We know taht the quantum plane, denoted by $\mathbb F_q[x,y]$ or $\mathcal{O}_q(\mathbb F^2)$, is the $\mathbb F$-algebra generated by $x$ and $y$ subject to the relation $yx-qxy=0,$ where $q\in \...
2
votes
0answers
61 views

Examples of ribbon Hopf algebras

Let $(H, m, \Delta, u, \epsilon, S, R)$ be a quasi-triangular Hopf algebra, where $H$ is a (finite-dimensional) vector space over a field $\mathbb{K}$ with the structure maps $m: H \otimes H \...
5
votes
1answer
424 views

Dual of a Hopf algebra

Given is a Hopf algebra $(H,m,\eta, \Delta, \epsilon, S)$. We know that there is a dual notion of it, called the dual Hopf algebra on $H^{*}$ as a vector space. It has the natural structure of a Hopf ...
1
vote
1answer
38 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
63 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
1answer
64 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
0answers
20 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
48 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
49 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
36 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
37 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
1answer
68 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\\ ...
2
votes
1answer
97 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 ...
0
votes
1answer
99 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
43 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}\...
3
votes
0answers
65 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 ...
2
votes
1answer
88 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 $e, h, f$...
4
votes
0answers
65 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
40 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
34 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
66 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
39 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
41 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
255 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
27 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
51 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
48 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 ...
7
votes
0answers
2k views

Some concrete examples of $M_q(2)$ points

Given $q \in \mathbb{C}$ invertible, Kassel says that an $M_q(2)$ 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 \...
7
votes
3answers
91 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 ...
2
votes
1answer
78 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 ...
2
votes
0answers
115 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, ...
1
vote
1answer
59 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
116 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 ...
1
vote
0answers
26 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 ...
3
votes
0answers
168 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 ...

1
2 3 4 5