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In Kassel's book on Quantum groups, it is defined that: "We define $U_q=U_q(\mathfrak{sl}(2) )$ as the algebra generated by the four variables $E$, $F$, $K$, $K^{-1}$ with the relations \begin{eqnarray*} &&KK^{-1}=K^{-1}K=1\\ &&KEK^{-1}=q^2 E,\ KFK^{-1}=q^{-2} F\\ and &&[E,F]=\frac{K-K^{-1}}{q-q^{-1}} \end{eqnarray*}" May I ask if there is a way of understanding what are E, F, K and $K^{-1}$? Are there matrix representations of them or anything like that? Sincere thanks for any help. |
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On the one hand, I'd suggest taking that description of $U_q(\mathfrak{sl}_2)$ as a formal definition. That is, think of $U_q(\mathfrak{sl}_2)$ as being the free associative algebra $k\langle E, F, K, K^{-1}\rangle$ modulo the relations you gave. $E$, $F$, $K$ are just formal symbols. You can just as well call them $x$, $y$, and $z$. You can procede in this manner without losing any of the algebraic side of quantum groups. On the other hand, it's helpful to know that $U_q(\mathfrak{sl}_2)$ is meant to be a one-parameter (that parameter is $q$) deformation of the enveloping algebra $U(\mathfrak{sl}_2)$ of the lie algebra $\mathfrak{sl}_2$. Further, that as hopf algebras, $U(\mathfrak{sl}_2)$ and $SL(2)$ are dual to each other. It is in this way that you can get some intuition behind the relations defining the quantum group. |
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