# Questions tagged [quantum-groups]

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

221 questions
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### Relations between Lie algebras and Lie coalgebras.

Let $g^*$ be the dual vector space of a vector space $g$. Suppose that $g^*$ is a Lie algebra and $[,]_{g^*}: \Lambda^2 g^* \to g^*$ satisfies the Jacobi identity. Let $\delta: g \to \Lambda^2 g$ be ...
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### How to compute Casimir elements of $g \otimes g$?

Let $g$ be a Lie algebra. How to compute Casimir elements of $g \otimes g$? I am asking this question because in the book a guide to quantum groups, page 80, there is an equation $r_{12} + r_{21}=t$, ...
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### Defining the dual-comodule of a comodule?

As is well known, every left module has a dual, which is a right module. How does this work for comodules? More explicitly, does there exist a notion of the dual-...
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### arithmetic with quantum integers

Consider the ring $\mathbb{Z}[q^{\pm 1}]$. For $n \in \mathbb{N}$, define the quantum integers: $$[n]_q := \frac{q^n-q^{-n}}{q-q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-(n-3)} + q^{-(n-1)}$$ What ...
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### The 1-Norm on a Quantum Group as a Supremum

To this MO question, Yemon Choi comments that If $\tau$ is a faithful normal trace on a von Neumann algebra $M$, then IIRC $\tau(|x|)$ is equal to the supremum of $|\tau(xy)|$ as $y$ runs over all ...
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Let $N$ be the unipotent subgroup of a Lie group $G$ consisting of all upper triangular matrices and $n$ the Lie algebra of $N$. I found in some paper that there is an action $U_q(n) \times \mathbb{C}... 0answers 118 views ### A “nice” orthogonal basis for translation invariant symmetric polynomials It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take$\Lambda_n $as the graded ring of symmetric polynomials of a field$F$in$n$... 1answer 32 views ### A computation for Manin triple. I am reading the book. I have some questions about the computations in (4.1) on page 40. The computation are in the following. I don't know why $$([[e_r^*, e_k], e_s^*]+[e_r^*, [e_s^*,e_k]],e_l) ... 1answer 151 views ### A reference for the Tannaka-Krein theorem I am looking for a reference for the Tannaka-Krein theorem on compact groups. By the Tannaka-Krein theorem which is also called (classic) Tannaka duality (because of the quantum theory), I mean the ... 2answers 88 views ### Reference request for q-numbers? Let q be an element of a field k (possibly \mathbb{C}), different from -1 and 1. We have$$[n]=\frac{q^n-q^{-n}}{q-q^{-1}}=q^{n-1}+q^{n-2}+\dots+q^{-n+1}$$Where n is a natural number. ... 0answers 36 views ### How to understand the algebra U_A(Lg)? Let g be a complex simple Lie algebra and Lg = g \otimes \mathbb{C}[t, t^{-1}]. Let q be a non-zero complex number and U_q(Lg) the quantum loop algebra corresponding to g. Let A = \mathbb{Z}... 1answer 88 views ### Motivation behind Quasitriangular Hopf algebra I would like to know why it is interesting to define the quasi-triangular structure on a Hop algebra. I understand that the pseudo-co-commutative (the existence of an intertwining operator between the ... 1answer 67 views ### A triangular Hopf algebra and its unitary R-matrix Why is the R-matrix of a Hopf algebra called unitary when it satisfies the relation$$R^{-1}=R_{12},$$I would say invertible, why then call it unitary? Is that a nomenclature that maybe comes from ... 0answers 91 views ### Defining the quantum group U_q(\mathfrak{sl}_2) I've seen two defining relation for U_q(\mathfrak{sl}_2) by the Serre relations$$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$or by taking K=q^H$$KK^{-1}=K^{-1}K=1,\quad [E,... 2answers 112 views ### Action of universal R-matrix of U_q(sl_2) My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let$q \in \mathbb{C}$with$q \neq 0, \pm 1$. ... 1answer 84 views ### What does the notation$U(\frak{g})[[\hbar]]$mean? I'm reading the following motivation for studying quantum group but I'm unfamiliar with the double bracket notation in $$U(\frak{g})[[\hbar]].$$ Is this a special set of polynomials with coefficient ... 1answer 56 views ### Reference request: product in$\mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$. Let$a \otimes b, a'\otimes b' \in \mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$, where$X, Y$are two algebraic varieties. Suppose that algebraic group$T$acts on$X, Y$. Then there are coactions$\delta:...
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 $$[a, b]=(q-1)\{a,b\}+o((q-1)^2).$$ ...