Questions tagged [quantum-groups]

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

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1answer
89 views

Is the multiplier algebra $M(K(H)\otimes C(X))$ isomorphic to $M(K(H))\otimes C(X)$?

In the survey article by Maes and Van Daele in the beginning of section 5, after Def. 5.1, the following claim is made (I will use different notation here, but wanted to give the reference nonetheless)...
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1answer
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Decomposition of VN(G) into direct sum of matrix algebras when G is a compact group. [closed]

Does anyone know any concrete proof of the following statement or any resource that has the proof? When $G$ is a compact group, by using classical Peter-Weyl theory, it follows that $VN(G)$ is a ...
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24 views

Resources for group theory? [duplicate]

I have recently started studying abstract algebra (specifically group theory). So far my learning has consisted of videos giving an intuition (e.g. 3blue1brown) and various lecture notes I've found on ...
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45 views

Anticommutation of Convolution Products on Trace Class Operators of Quantum Groups

Edit: This question has been posted to MathOverflow. Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
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Does quasi-triangularity imply invertibility of the antipode?

I have been studying quantum groups from Christian Kassel's Quantum Groups side-by-side with Majid's Foundations of Quantum Group Theory. I noticed that while Majid proves that the antipode of a quasi-...
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1answer
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Self-duality of $U_q(b_+)$ proof (Majid, A Quantum Groups Primer, Proposition 2.5)

I am reading A Quantum Groups Primer by Shahn Majid, and I'm having trouble filling in the details for the proof of Proposition 2.5, which states that the Hopf algebra $U_q(b_+)$ is self-dual. $ \...
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1answer
51 views

What makes an affine Kac-Moody algebra 'untwisted' or 'twisted'

I'm learning about affine Lie algebras partly from J. Fuchs' Affine Lie Algebras and Quantum Groups. Actually I'm using Kerf and Bauerle but for this question I find Fuchs' labelling of the Dynkin ...
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1answer
22 views

Kac-Moody algebras: why $Z \subset$ span$\Pi^V$

I am working through Volume 1 of Kerf and Bäuerle's book (which I generally find excellent for a humble physicist like myself to learn from), and I'm unfortunately stuck. In Lemma 11.2.1c, they state ...
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21 views

Classification of connected finite affine type A crystal graphs

In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...
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73 views

Invertibility of quasitriangular Hopf algebra element using Sweedler notation

The question concerns part of a theorem in the book Foundations of Quantum Group Theory, Shahn Majid (Cambridge University Press, 1995). More specifically, Theorem 2.3.4 (p.55-57) which I'll rewrite ...
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1answer
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Image of elements of $S_n$ in the Temperley-Lieb algebra

Consider the algebra $A_n$ generated by $u_1,\ldots,u_{n-1}$ subject to relations $u_i^2=-2u_i$, $u_iu_{j}u_i=u_i$ for $|i-j|=1$ and $u_iu_j=u_ju_i$ for $|i-j| \geqslant 2$. The algebra $A_n$ is the ...
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Antiautomorphism of an quantized universal enveloping algebra

I am learning about quantum groups, and I have a question about the construction of certain antiautomorphism of an quantized universal enveloping algebra, $U_q(\mathfrak{g})$. In the book $\ulcorner$...
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Liouville—von Neumann Operator as a member of a Tensor space ( in an Open Quantum System )

$\textbf{Context:}$ I'm posting in Math SE instead of Physics SE because this is really a question about differential geometry and vector spaces. Let $\mathcal{H}$ be a Hilbert Space, then $|\psi\...
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Quantum Fourier Transform

I have a question about QFT. Given two ONB of the Hilbert Space $\mathbb C^G$: $\forall x\in G: |{x}\rangle:=\delta_x$ $\forall \xi\in G: |{\chi_\xi}\rangle:=\frac{1}{\sqrt{|G|}}e^{2\pi i \xi\bullet \...
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1answer
56 views

The Coproduct of the Identity element

For a general Hopf algebra $H$, the coproduct of the identity element is $$\Delta(1) = 1\otimes 1.$$ Now for a finite group $G$ and a field $k$, $k(G)$ forms a Hopf algebra with basis $\{\delta_{g}| g\...
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Commutative Hopf algebra as a smash product

For $A$ a $k$-algebra with a left Hopf action by a Hopf algebra $H$, we can form the smash product $A \# H$ which has underlying space $A \otimes_k H$ and product, $$(a\otimes h)(a'\otimes h') = \sum_{...
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supersymmetry and quantum groups

I have some background in non-commutative geometry (in particular, I am doing research in quantum groups) and I have of course heard several times about the concept of supergeometry and supergroups. ...
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How to understand canonical basis and dual canonical basis are dual to each other in the case of $sl_3$?

Let $\mathfrak{g}=\mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^-$ be a triangular decomposition of a simple Lie algebra. Let $v$ be an indeterminant and let $U_v(\mathfrak{n})$ be the positive ...
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1answer
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Definition of $C^*$-algebraic quantum group

A compact quantum group is a pair $(A, \Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta: A \to A \otimes A$ is a $C^*$-morphism such that (1) $(\Delta \otimes \operatorname{id}_A) \circ \...
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1answer
97 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 ...
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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 &...
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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^...
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1answer
55 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 $\mathrm{C}^*$-...
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1answer
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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 $...
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1answer
71 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 ...
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1answer
39 views

Generalization of the concept of homogeneous element

Let $G$ be a finite group. Recall that $kG$ is a Hopf algebra and, since $dim(kG)<\infty$, $(𝑘𝐺)^∗$ is also a Hopf algebra with its structure dual to that of $𝑘𝐺$. As it is well known, an ...
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1answer
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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,...
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2answers
116 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, ...
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70 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 ...
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1answer
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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 ...
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1answer
44 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 $\mathbf{C}(q)$...
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1answer
379 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 \...
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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 \...
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1answer
960 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 ...
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1answer
41 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 ...
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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 $\mathfrak{gl}_n$...
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1answer
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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, ...
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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 ...
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1answer
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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 ...
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67 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 ...
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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 ...
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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 ...
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1answer
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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\\ ...
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1answer
159 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 ...
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1answer
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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. ...
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1answer
56 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}\...
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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 ...
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1answer
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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 $e, h, f$...
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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 ...
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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^...

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