Notation in quantum groups.

Question

The quantum group $U_q(sl_3)$ is generated by $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$ subject to some relations. I read some papers and there is a notation $K_{\lambda}$, where $\lambda$ is a weight. I think that we have $\lambda = a \alpha_1 + b \alpha_2$ for some numbers $a,b$. For example, for fundamental weights $\omega_1, \omega_2$ we have $\omega_1-\omega_2 = \frac{1}{3} \alpha_1 - \frac{1}{3} \alpha_2$. I think that we have $K_1 = K_{\alpha_1}$, $K_2 = K_{\alpha_2}$. Do we have $K_{\omega_1 - \omega_2} = K_1^{\frac{1}{3}}K_2^{-\frac{1}{3}}$? But it seems that $K_1^{1/3}$ and $K_2^{-1/3}$ are not in $U_q(sl_3)$? Any help will be greatly appreciated.

Answer 1

There is a more general definition of quantum groups. Let $P=\bigoplus_{i\in I}\mathbb{Z}\Lambda_i$ denote the weight lattice and $Q=\bigoplus_{i\in I}\mathbb{Z}\alpha_i$ the root lattice of a Lie algebra $\mathfrak{g}$. Then, to each $W$-stable lattice $L$ with $Q\subset L\subset P$ one has a quantum group with generators $E_i$ and $F_i$ ($i\in I$), and $K_\lambda$ ($\lambda\in L$). The key relations are $$ K_\lambda E_i K_\lambda^{-1}=q^{\langle \lambda,\alpha_i^\vee\rangle}E_i\;\;\;\mbox{and}\;\;\; K_\lambda F_i K_\lambda^{-1}=q^{-\langle \lambda,\alpha_i^\vee\rangle}F_i. $$ The point is that the exponents of $q$ in these relations are integers because $L$ is $W$-stable.

There is an analogy with Chevalley groups, where one chooses a $U_{\mathbb{Z}}$-stable $\mathbb{Z}$-subalgebra of $U=U(\mathfrak{g})$. I'm not entirely sure if this is covered in Jantzen's book on Quantum Groups, but is certainly in Lusztig's.