# Questions tagged [quantum-groups]

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

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### Properties of Bicrossed Product multiplcation

I am reading Kassel's Quantum Groups book (the chapter on Drinfeld doubles). In it, there is the following claim: If $H,K\subseteq G$ are groups such that $\forall g\in G$, $\exists!(y,z)\in H\times K$...
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### Do multi-parameter unitary subgroups exist?

I'm working with an $N$-dimensional quantum system that is defined by the following Hamiltonian $$H = H_{\text{drift}} + \sum_{j=1}^n a^j H_{\text{drive}}^j$$ Where $n \ll N$ (in my case $n = 4$ ...
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### q analogue of a number is a polynomial in $_q$

$[m]_q = \frac{q^m−q^{-m}}{q-q^{-1}}$ is the q- analogue of the number $m\in\mathbb{Z}_{\geq0}$. $_q=0$, $_q=1$ and $_q=q+q^{-1}$. I don't know how to prove Every $[m]_q$ can be expressed as ...
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### Why we need an orthonormal basis?

Here is the question I am trying to solve: Prove that $\lambda$ is injective. Here is the definition of the linear map $\lambda$: Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their ...
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### Quantum plane is a bialgebra

I am reading ‘Hopf algebras and their actions on rings’. Susan wrote the quantum plane as an example at 1.3.9 Example. He said $B = k \langle x,y \mid xy = qyx \rangle$, $0 \neq q \in k$ with ...
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### Where does the unitarity structure of $U_q(\mathfrak{sl}_2)$ come from?

It is known that when $q$ is the root of unity, the representation of the quantum group $U_q(\mathfrak{sl}_2)$ is a unitary modular tensor category. However, if we want it to have the dagger structure,...
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