# Tagged Questions

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. (Def: http://en.m.wikipedia.org/wiki/Quantum_group)

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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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### An alternative proof for the units of $U_q(\mathfrak{sl}_2)$ using Ore extensions.

I would like to establish what the set of units are in the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$. First, I recall the definition of the quantized enveloping algebra- throughout the ...
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### Connections of Finite groups and quantum groups

I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...
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### What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
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### How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
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### Fusion rules and comultiplication for a Kac algebra

Let $\mathbb{A}$ be a Kac algebra (i.e. a Hopf C*-algebra) of finite dimension, and $\mathbb{A}^{*}$ its dual : Is there a formula revealing the fusion rules for the irreducible ...
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### $C(G) \otimes C(G) \simeq C(G \times G)$

Let be $G$ a compact group and $C(G)$ the vector space of continue complex functions defined over $G$, $C(G):=\{f:G \to \mathbb{C}\}$, where $f$ is a continue function. How can I prove that  C(G) ...
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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 ...
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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 ...
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 ...