# 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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### What are $q$-deformations?

This question has already appeared in a lot of different ways and here is another one. First of all, many people know the typical quantum group $U_q(\mathfrak{sl}_2)$ by generators and relations. ...
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### Where does the condition $\pi(X_-)+\pi(X_+)=0$ in dual Lie algebra come from?

I am trying to understand the dual Lie algebra in a Lie bialgebra. In the above article, it is said that: "Let ${\mathfrak {g}}$ be any semisimple Lie algebra. To specify a Lie ...
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### induction mathematics in bracket of quantum group

By the induction mathematics . How can prove the following [fm,f1m]=0,where f is generators in quantum group and m greter than 3.
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### States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
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### How to show that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding?

Let $H$ be a bialgebra and ${}_H^H YD$ the category of Yetter-Drinfeld modules over $H$. It is said that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding. ...
### 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:...