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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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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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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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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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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.
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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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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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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 ...
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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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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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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 } &...
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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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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 (...
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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 ...
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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 ...
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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, ...
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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 \...
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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'...
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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.
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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 ...
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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 $A,\,B,\,C,\,D \in R$ ...
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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 ...
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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 ...
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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, ...
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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\...
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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\...
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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(...
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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 ...
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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 ...
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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 ...
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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}^{\...
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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 ...
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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 ...
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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(...
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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, ...
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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. ...
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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,...
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Verma module $V(\lambda)$ of $U_q(\mathfrak{sl}_2)$ is not simple $\implies$ $\lambda = \pm q^n$

This is exercise $VI.6.2$ in Kassel's Quantum Groups. My observations are the following: If $V(\lambda)$ is not simple, then there is a surjective module homomorphism $\psi:V(L)\rightarrow V$ to some ...
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Usefulness of representations of quantum groups

For compact quantum groups there exists a rich representation theory. What I still not well understand, how this helps in various calculations. My problem is somehow, that these representations are ...
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Do there exist $2 \times 2$ matrices satisfying $XY = q YX$, $YZ = q ZY$ and $ZX = q XZ$

Let $q = e^{2\pi i /3}$. Do there exist (distinct) $2 \times 2$ matrices with coefficients in $\mathbb{C}$ satisfying the commutator relations: $ XY = qYX $ $ YZ = qZY $ $ ZX = qXZ $ Even thinking ...
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Uniqueness of quantizations

When studying quantum groups, in particular quantized universal enveloping algebras, people will tell you that such a quantization is in some sense unique. More specifically, you might hear that a ...
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group von Neumann algebra and its Plancherel weight

Lat $G$ be a locally compact group and $\mathcal{L}(G)$ its von Neumann algebra equipped with the Plancherel weight $\omega_G$. If $G$ is discrete then $\omega_G$ is a finite trace. Suppose that $\...
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Where does the condition $\pi(X_-)+\pi(X_+)=0$ in dual Lie algebra come from?

I am trying to understand the dual Lie algebra in a Lie bialgebra. In the above article, it is said that: "Let ${\displaystyle {\mathfrak {g}}}$ be any semisimple Lie algebra. To specify a Lie ...
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Generators Relations of the Weyl Group

Suppose we have a Root System and call $s_{\alpha}$ the transformation $$s_{\alpha}\left(\beta\right)=\beta-\frac{2\left\langle \beta,\,\alpha\right\rangle }{\left\langle \alpha,\,\alpha\right\rangle }...
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Question on simplification of terms in a Hopf algebra

Let $H=(H,\mu,\eta,\Delta,\epsilon,S)$ be a Hopf algebra and $S$ is the antipode in $H$. By the definition of antipode $S*id_H=id_H*S=\eta \circ\epsilon $. If I denote the Sweedler's notation by $ \...
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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" ...