# (Mathematical) Applications of Quantum Group

What are some mathematical applications of Quantum Groups?

I have tried researching and found that it is used to find solutions to Yang-Baxter equation.

Any other applications?

Thanks a lot.

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Two amongst many things: 1) you can study representations of algebraic groups/Lie algebras in characteristic $p$ by studying quantized enveloping algebras in characteristic zero at a $p$th root of unity. 2) there is a certain category $S(a)$ depending on a parameter $a$ such that $S(2)$ controls the category of $\mathfrak{sl}_2$-modules. It is natural to ask what $S(a)$ for $a\neq 2$ means, and it turns out $S(q+q^{-1})$ has the same relation to $U_q(\mathfrak{sl}_2)$ that $S(2)$ has to $U(\mathfrak{sl}_2)$. –  mt_ Mar 31 '12 at 16:26