On the center of the quantized enveloping algebra of a simple Lie algebra

Abstract

Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the quantum group $U_q(\frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(\frak{g}))$ is a polynomial algebra if and only if $\frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $\frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ variables with one relation; in case $\frak{g}$ is of type $E_6$, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in fourteen variables with eight relations; in case $\frak{g}$ is of type $A_{n}$, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra described by $n$-sequences.