# $U_q$ Quantum group and the four variables: $E, F, K, K^{-1}$

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.

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.

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 $\mu$.