# Questions tagged [quantum-groups]

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

223 questions
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### What is the relation between the construction of Reshetikhin-Turaev and Witten's paper?

I have read that the papers of Reshetikhin and Turaev provide a mathematically rigorous framework for the ideas contained in Witten's paper "Quantum field theory and the Jones polynomial". Can ...
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### 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 $$[a, b]=(q-1)\{a,b\}+o((q-1)^2).$$ ...
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### Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
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### Is there a Relationship between Quantum Groups and Lie Groups?

I know that the Lie Group is all about continuous transformation groups. I know that the quantum group denotes various kinds of noncommutative algebra with additional structure. Transformation group ...
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### The “$\mathbb{Z}_n$-theta function” - what is it? Is it being studied somewhere?

The Jacobi theta function is well known: $$\theta(z, \tau) = \sum_{n=-\infty}^\infty \mathrm{e}^{\pi i n^2 \tau + 2\pi i nz}$$ In Shahn Majid's "Foundations of Quantum Group Theory", you'll find a ...
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### Quantum $SU(2)$ and $3$-sphere

There is a dipheomorphism of $SU(2)$ and $3$-sphere. What happens when you construct quantum $SU(2)$? Do we have an lifting of homeomorphism (or some other morphism) of quantum group to some other ...
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### Towards a Quantum Peter Weyl Theorem

This is taken from Timmermann's Invitation to Quantum Groups and Duality. Let $(A,\Delta)$ be a *-Hopf algebra and let $\chi:V\rightarrow V\otimes A$ be a corepresentation of $(A,\Delta)$ on a vector ...
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### Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
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### A Tensor Product Identification

Hi folks just looking at Timmermann's "Introduction to Quantum Groups and Duality" and looking at the algebra of representative functions on a compact topological group $G$. I take off in Example 3.1....
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### Construction of a Manin triple

Let $\mathfrak{g}$ be a Lie bialgebra and $\mathfrak{g}^*$ be its dual. My question is how to construct a bracket on the direct sum $\mathfrak{g}\oplus\mathfrak{g}^*$ such that we obtain a Manin ...
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### What is an Example of a Co-commutative but not Commutative Quantum Group?

I am looking for 'a' right candidate for an "abelian" quantum group. In a comment to another question it was suggested that the correct candidate was co-commutative. It is straightforward to show ...
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### The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional ...
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### Where do I go from Linear algebra past Calc III to try to learn complex physics (relativity and quantum group theory)?

I'm mainly a programmer, but I have a love for Mathematics that's been, well, insatiable. I've had my eye on learning Quantum Groups and Relativity, but I want to stay in something I can do with ...
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### elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
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### Tensor Product Question in Kassel's Quantum Groups

In Kassel's book on Quantum Groups, I am stuck on the following computation: \begin{eqnarray*} [\Delta (E), \Delta (F)] &=& \Delta (E)\Delta (F)-\Delta (F)\Delta (E)\\ &=& (1\otimes E ...
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### How to show that $SL_q(2)$ and $U_q(\mathfrak{sl}(2))$ are noncommutative and noncocommutative

It is known that the two quantum groups $SL_q(2)$ and $U_q(\mathfrak{sl}(2))$ are both noncommutative and noncocommutative. May I ask how do we show that? I have attempted the following: To prove ...
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### $U_q$ Quantum group and the four variables: $E, F, K, K^{-1}$

In Kassel's book on Quantum groups, it is defined that: "We define $U_q=U_q(\mathfrak{sl}(2) )$ as the algebra generated by the four variables $E$, $F$, $K$, $K^{-1}$ with the relations \begin{...
We know generators and relations for $\mathfrak{sl}_2$: $e, f, h, [e, f]=h, [h,e]=2e, [h,f]=-2f$. Generators and relations for $U_q(\mathfrak{sl}_2)$ are $e, f, k=q^h$, $kek^{-1}=q^2e, kfk^{-1}=q^{-2}... 1answer 106 views ### Prove well-definedness of comultiplication and counit of$GL_q(2)$and$SL_q(2)$. I read a textbook on Quantum Groups by Kassel, and did not understand the following proof of well-definedness of$\Delta$(comult.) and$\epsilon$(counit) on$GL_q(2)$and$SL_q(2)$: (pg 84) The ... 0answers 110 views ### how to calculate the derivative of a plane wave in non commutative geometry Shahn Majid and Eliezer Batista find the derivative of a wave plane in their paper: Non Commutative Geometry of Angular Momentum Space$U \mathfrak{su}(2)$. They obtained the non commutating ... 0answers 137 views ### Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra As is well known, any Lie group$G$has a canonical action on its Lie algebra$\frak{g}$, namely the adjoint action$Ad$. Firstly, let me ask, does this extend to an action of$G$on its enveloping ... 1answer 79 views ### Kernel of a Comodule Map is a Sub-Comodule Let$H$be a Hopf algebra,$(V,\Delta_R)$a right$H$-comodule map, and$f:V \to V$a right$H$-comodule map. Since by definition we must have, for all$v \in V\$, that $$\Delta_R(f(v)) = \sum f(v_{(... 1answer 1k views ### The longest word in Weyl group and positive roots. How to write down a reduced decomposition of the longest word in a Weyl group? For example, how to write down a reduced decomposition of the longest word in type B3 Weyl group? For a decomposition of ... 1answer 100 views ### proving that this coproduct and product get along well enough to make a bialgebra (Quantum Groups, Kassel) I'm working through Kassel's Quantum Groups book and I'm stuck on what looks like a pretty simple problem (ch3, ex 2 in the book): Let k be a field and C= k[t] be a coalgebra with coproduct:$$ \...
It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...