# Questions tagged [quantum-groups]

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

211 questions
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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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### 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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### 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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### 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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### 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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### 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 ...
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 ...