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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. (Def: http://en.wikipedia.org/wiki/Quantum_group)

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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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Difference between infinitesimal parameters and group generators

I am getting myself confused regarding the differences between the infinitesimal generators of Lie group and the elements of the Lie algebra, likely due to the fact that I am studying from a physics ...
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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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1answer
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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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Normalization summation of Clebsch Gorden coefficients

I encontered a summation $$\sum_{m_1,m_2}^{m_1+m_2=j} \frac{(j_1+m_1)!(j_2+m_2)!}{(j_1-m_1)!(j_2-m_2)!}\,.$$ in this book p77 when normalizing the Clebsch Gorden coefficients. $j$ here is a constant ...
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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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Algebra proof for quantum mechanics ladder operator

My question contains the word quantum mechanics but is a purely algebraic problem. The so-called ladder operators $a_{-}$ and $a_{+}$ from quantum mechanics are operators that do not commute and the ...
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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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1answer
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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" ...
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Finding an explicit isomorphism between $\mathbb{C}_q[x^{\pm 1},y^{\pm 1}]/\langle x^k-\alpha, y^k-\beta \rangle$ and $M_k(\mathbb{C})$

Consider the quantum torus \begin{align*} \mathbb{C}_q[x^{\pm 1},y^{\pm 1}] = \frac{\mathbb{C}\langle x,y \rangle}{\langle yx-qxy \rangle }[x^{-1}, y^{-1}] \end{align*} where $q \in \mathbb{C}^\times$...
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1answer
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The antipode and the compact quantum group von Neumann algebra

I came across a technical question (at least I believe so) of the following nature. Let $(\mathbb{G},\Delta)$ be a compact quantum group, where $\mathbb{G}$ is a $C^*$-algebra and $\Delta$ is a co-...
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1answer
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Haar state on nonseparable, noncocommutative compact quantum group

I am trying to understand what seems to be a common knowledge, that every compact quantum group has a Haar state. However, each approach that I found on the internet is a bit hard for me to grasp. I ...
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1answer
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States uniquely defined by operators

In this article authors define quantum states $\left|\psi\right\rangle$ (elements of some Hilbert space H - in most simple case a $2\times 1$ complex vector such that $|\psi| = 1$) by a set of ...
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Reconstruction of monoidal categories

Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator). Bruce Westbury ...
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Frobenius’ Method to Solve an ODE (Hydrogen Atom - Radial Equation)

My goal is to find two linearly independent solutions to the ODE $$ r^2\frac{d^2R}{dr^2}+2r\frac{dR}{dr}+[r^2+\lambda r-l(l+1)]R=0 $$ in the interval $[0,\infty)\ni r$, where $R=R(r)$, and $\lambda\...
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Dual Pairing between $U_q(\mathfrak{su}(n))$ and $\mathcal{O}(SU_q(n))$

In page 327 of Klymik and Schmudgen`s book Quantum Groups and their representations, theorem 18 reads: There exists unique dual pairings of the pairs: $U_q(gl_n)$ and $\mathcal{O}(GL_q(n))$, SL_q(n)....
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On root vectors

I don't fully understand the definition of root vectors in the construction of the so called Andruskiewitsch-Schneider Hopf-algebras. See for example section 2 of the paper 'On the Classification of ...