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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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The answer given by mebassett seems to take away the meaning of the elements $E,K,K^{-1},F$ and reduce them to formal symbols $x,y,z$ rather than providing meaning. That is one way to look at them of course; as formal symbols.

To get a better feeling of what these elements mean, you can turn to representation theory. Consider a finite-dimensional representation $V$ of $\mathfrak{U}_q(\mathfrak{sl}_2)$. If the highest weight $\lambda$ of $V$ is dominant integral, which just means $\lambda \in \mathbb{Z}^+$ for representations of $\mathfrak{U}_q(\mathfrak{sl}_2)$, the representation $V$ is the unique finite-dimensional irreducible representation $V_\lambda$ of dimension $\lambda+1$.

The representation $V_\lambda$ is given by the span of a highest weight vector $v_\lambda$ and all of its 'reductions' by the element $F$, i.e. $V_\lambda = \langle v_\lambda, F v_\lambda, F^2 v_\lambda, \dots, F^\lambda v_\lambda \rangle$. The weight of an element $F^k v_\lambda$ is $\lambda - 2k$. The actions of the elements $E,K,K^{-1},F$ on $V_\lambda$ are as follows:

$E v_\lambda = 0$

$E F^k v_\lambda = [k]_q [\lambda - k + 1]_q F^{k-1} v_\lambda$

$K^\pm F^k v_\lambda = q^{\pm(\lambda - 2k)} F^k v_\lambda$

$F F^k v_\lambda = F^{k+1} v_\lambda$

$F^{\lambda+1} v_\lambda = 0$

where $[k]_q = \frac{q^k - q^{-k}}{q - q^{-1}}$ is the $q$-analog of $k$.

As you see, the element $E$ moves an element of the representation up in weight, stopping at the highest weight $\lambda$. Similarly, the element $F$ moves an element of the representation down in weight, stopping at the lowest weight (which is $-\lambda$). The element $K$ acts with weight zero and simply 'registers' the weight of an element of the representation. In fact, $K = q^h$ where $h$ is the single element $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ that generates that Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{sl}_2$. In the more general case of $\mathfrak{U}_q(\mathfrak{sl}_n)$, a generating element $K_i = q^{h_i}$ acts as

$q^{h_i} w = q^{\mu(h_i)} w$

on an element $w$ of weight $\mu$.

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