| Let k be a field of characteristic p ≠ 2. Let phi be a finite, indecomposable root system, G the simple, simply-connected algebraic group over k having root system phi, and g = Lie(G). Let Uzeta( g ) denote the Lusztig (divided power) quantized enveloping algebra corresponding to g , with parameter q specialized to a primitive ℓ-th root of unity zeta ∈ k. The Frobenius-Lusztig kernels Uzeta(Gr) of U zeta( g ) are certain finite-dimensional Hopf subalgebras of U zeta( g ) that play a role in the (integrable) representation theory of Uzeta( g ) analogous to the role played by the Frobenius kernels G r of G in the rational representation theory of G. If r = 0, then Uzeta (Gr) = uzeta( g ), the "small" quantum algebra discovered by Lusztig [48, 49]. The higher Frobenius-Lusztig kernels of Uzeta( g ) (i.e., those parametrized by values r ≥ 1) exist only if p > 0.;The goal of this dissertation is to study the cohomology of the Frobenius-Lusztig kernels of Uzeta( g ) when p > 0. Our strategy parallels the characteristic zero work of Ginzburg and Kumar [30] and of Bendel, Nakano, Parshall and Pillen [9], as well as the earlier work on Frobenius kernels of algebraic groups by Friedlander and Parshall [27] and Andersen and Jantzen [2]. For r = 0, we show (in most cases) that the cohomology H ·(uzeta( g ), k) of uzeta( g ) is isomorphic as a G-module to the induced module indGPJ S˙u *J (for some subset J of simple roots depending on ℓ). If additionally ℓ ≥ h, h the Coxeter number of g , then H·(uzeta ( g ), k) is isomorphic as an algebra to k[ N ], the coordinate ring of the variety of nilpotent elements in g . For r = 1, we show (under certain restrictions on ℓ, p, and phi) that the cohomology ring H ·(Uzeta(G 1), k) is Noetherian. For arbitrary r ≥ 1, we show that the cohomology rings H·( Uzeta(Ur), k) and H·(Uzeta( Br), k) for the "nilpotent" and Borel subalgebras Uzeta(Ur ) and Uzeta(Br) of Uzeta(Gr) are also Noetherian. |
