Questions tagged [quantum-groups]
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure.
330 questions
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What is the relation between q-calculus and Lie algebras/quantum groups
On the Wikipedia page for quantum calculus it mentions "In a sense, q-calculus dates back to Leonhard Euler and Carl Gustav Jacobi, but has only recently begun to find usefulness in quantum ...
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Convolution on compact quantum group
Let $\mathbb{G}$ be a compact quantum group in Woronowicz's sense. It is standard to define the convolution by
\begin{align*}
\omega_1*\omega_2&=(\omega_1\otimes\omega_2)\Delta,\\
\omega*a&=(I\...
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A doubt about a proposition in Tuset's book on Quantum Groups [duplicate]
While studying the book "Compact Quantum Groups and their Representation Categories" (by Sergey Neshveyev and Lars Tuset) I am stuck with the proof of Proposition 2.2.8 (Page 43 - AMS 2014 ...
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Any other uses of K-Theory of $\mathrm{C}^{*}$-algebras aside from classification?
I have recently become very interested in K-theory of $\mathrm{C}^{*}$-algebras due to several of the interesting properties one can derive about a $\mathrm{C}^{*}$-algebra $A$ relative to its $K_0(A),...
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Proving the B-Beta Integral for the Faddeev's Quantum Dilogarithm
I am currently reading Lemma 15 in Appendix B of this paper, which has the following proof. The function $G_b$ is the Faddeev's Quantum Dilogarithm and I am struggling to understand the proof of how
$$...
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Basis for generalized eigenspaces with tensor products and a suitable coproduct.
This question comes from when I was learning about quantum groups. I want to know the basis of generalized eigenspaces with tensor products
Assume that $\phi_m(m\ge 0)$ are operators with $\phi_n\...
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How is this antilinear map (charge conjugation) represented as a complex matrix?
I am trying to understand a step in the paper Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock by Michael Stone, Ching-Kai Chiu, Abhishek Roy on page ...
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Is there a version of Ringel's Theorem that works for $O_q^+(SL_2)$ instead of $\mathscr{U}^+_q(\mathfrak{sl}_2)$?
It's a famous theorem of Ringel (later extended by Green and others) that the hall algebra of the $\mathbb{F}_q$-valued representations of an ADE quiver is the (positive part of the) quantum universal ...
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why the quantum determinant commutes with all elements of the Algebra
I do research in the field of integrability where the quantity called the "Quantum determinant" is often used.
This so called Quantum determinant is also used in quantum groups as it is ...
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Intuition about Lie Bialgebra Structure
I am reading Etingof's Lectures on Quantum Groups, Chapter 2. There, the authors define a Lie bialgebra structure on $T_eG$ for $G$ a Lie-Poisson group. Questions:
First, they define a Lie algebra ...
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Group-like projections outside dense Hopf*-algebra of compact quantum group
Let $C(\mathbb{G})$ be an algebra of continuous functions on a compact quantum group.
Let $p\in C(\mathbb{G})$ be a group-like projection, that is a non-zero projection such that
$$\Delta(p)(1_{C(\...
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Explanation of a specific proof that the antipode is an anticoalgebra map
In S. Majid’s Foundations of quantum group theory, the property $(S \otimes S) \circ \Delta h = \tau \circ \Delta \circ Sh$ (Proposition 1.3.1) is proven as follows:
\begin{align*}
Sh_{(2)} \...
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Main idea behind one-parameter groups of $*$-automorphisms
I am reading the basic theory of locally compact quantum groups, and one of the main concepts covered currently is one parameter groups of $*$-automorphisms for a Von Neumann algebra $M \subset B(H)$. ...
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Efficient numerical calculation of Campbell's lemma
Let $A$ and $B$ be square matrices. Consider Campbell's identity:
$e^{A} B e^{-A} = \mathbb{1} + [A, B] + \frac{1}{2!}[A,[A,B]] + \frac{1}{3!}[A,[A,[A,B]]] + \frac{1}{4!}[A,[A,[A,[A,B]]]]+ \dots$
This ...
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Hilbert-Poincare series of a quadratic algebra
Given a graded quadratic algebra, when the associated Hilbert-Poincaré series is a quotient of integer polynomials where the numerator has negative roots and the denominator positive roots? It is ...
