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

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