Cohomology of infinitesimal quantum groups, I

. In this paper, we investigate the cohomology of infinitesimal quantum groups (and algebras) associated to classical quantum groups (and algebras) at a root of unity. A main result expresses the Ext-groups between irreducible modules in terms of those for the full quantum group. Under the assumption that certain module categories for the quantum group have a Kazhdan-Lusztig theory (in the sense of Cline, Parshall, and Scott), this permits explicit calculations of cohomology in terms of Kazhdan-Lusztig polynomials. This assumption in turn follows from recently announced results of Kazhdan and Lusztig.

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