The Quantum Universal Enveloping Algebra

We start from the universal enveloping algebra of ${\text{sl}}_2(\text{C})$, and then deform it to get the quantum universal enveloping algebra $U_q({\text{sl}}_2(\text{C}))$.
Recall from the example that ${\text{sl}}_2(\text{C})$ is the Lie algebra of $2\times 2$ complex traceless matrices, and it has a standard basis $\{e,f,h\}$ with Lie brackets $[H,E]=2E$, $[H,F]=-2F$ and $[E,F]=H$. Its universal enveloping algebra $U({\text{sl}}_2(\text{C}))$ is a unital associative algebra generated by $E,F,H$.

$q$-Number

Suppose $\mathbb{F}$ is a field, and $q\in \text{F}$ is a nonzero element that is not either $1$ or $-1$. Then the $q$-number of an integer $n\in \mathbb{Z}$ is defined as
$[n{]}_q:=\frac{q^n-q^{-n}}{q-q^{-1}}=q^{n-1}+q^{n-3}+\cdots +q^{-n+1}.$

$U_q({\mathfrak{sl}}_2(\mathbb{C}))$

Consider $q\in {\text{C}}^{\times }$. The quantum universal enveloping algebra $U_q({\text{sl}}_2(\text{C}))$ is the unital associative algebra over $\text{C}$ generated by $E,F,K,K^{-1},L$ subject to the relations:
\begin{gather} KK^{-1}=1,\qquad KEK^{-1}=q^{2}E,\qquad KFK^{-1}=q^{-2}F,\\ [E,F]=L,\qquad (q-q^{-1})L=K-K^{-1}, \\ \qquad [L,E]=q(EK+K^{-1}E), \qquad [L,F]=-q^{-1}(FK+K^{-1}F) . \end{gather}

$U_q({\text{sl}}_2(\text{C}))$ is deformed from $U({\text{sl}}_2(\text{C}))$

When $q=1$, this definition reduces to the usual universal enveloping algebra $U({\text{sl}}_2(\text{C}))$ with $K$ and $L$ identified with $1$ and $H$ (up to a sign), respectively.
We don’t actually need the generator $L$ if $q\ne 1,-1$. In fact, if $q\ne 1,-1$, then $L=\frac{K-K^{-1}}{q-q^{-1}}$. But we keep $L$ in the definition to easily compare with the classical case $q=1$.

Proposition

$U_q:=U_q({\text{sl}}_2(\text{C}))$ has a Hopf algebra structure with comultiplication $\Delta$, counit $\epsilon$ and antipode $s$ defined by
$\begin{array}{rl}\Delta :U_q\to U_q\otimes U_q,& E\mapsto 1\otimes E+E\otimes K,F\mapsto K^{-1}\otimes F+F\otimes 1,\\ & K\mapsto K\otimes K,K^{-1}\mapsto K^{-1}\otimes K^{-1},\\ & L\mapsto K^{-1}\otimes L+L\otimes K,\\ \epsilon :U_q\to \text{C},& E,F,L\mapsto 0,K,K^{-1}\mapsto 1,\\ s:U_q\to U_q,& E\mapsto -E{K}^{-1},F\mapsto -KF,K\mapsto K^{-1},K^{-1}\mapsto K,L\mapsto -L.\end{array}$

Universal $R$-Matrix

Highest Weight Representations