COHOMOLOGY OF QUANTIZED FUNCTION ALGEBRAS AT ROOTS OF UNITY

Abstract

Let $G$ be a simply-connected, semisimple algebraic group over $k$, an algebraically closed field of characteristic zero. Let ${\cal O}_{\epsilon}[G]$ be the quantized functionalgebra of $G$ at a primitive $\ell$th root of unity$\epsilon$, and let $\overline{{\cal O}_{\epsilon}[G]}$ be the`restricted' quantized function algebra at $\epsilon$,a finite-dimensional $k$-algebra obtained from${\cal O}_{\epsilon}[G]$ by factoring out a centrally generatedideal. It is known that $\overline{{\cal O}_{\epsilon}[G]}$ isa Hopf algebra. We study the cohomology ring $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$,a graded commutative algebra, and, for any finite-dimensional$\overline{{\cal O}_{\epsilon}[G]}$-module $M$, the$\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module$\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$.We prove that for sufficiently large $\ell$ there is an isomorphism of graded algebras\[\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)\cong k[X_1,\ldots ,X_{2N}],\]where each $X_i$ is homogeneous of degree $2$, and $2N$ equals the number of roots associated to $G$. Moreover we show that in this case $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$is a finitely generated $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module.We also show under much less restrictive conditions on$\ell$ that $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$continues to be a finitely generated graded commutative algebra over which $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$ is afinitely generated module.\vspace{6mm}\noindent1991 Mathematics Subject Classification: 16W30, 17B37, 17B56.