Second cohomology groups for Frobenius kernels and related structures

Abstract

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U ) with coefficients in k.

B-cohomology

“…One of the main results in [8] is a complete description of H 2 (B, λ). When p > h we shall recover this result below (see Section 5) so we do not give the statement here.…”

Second cohomology groups for algebraic groups and their Frobenius kernels

“…(2.1) arises from the reduction H 2 (B 1 , k) = H 2 (U 1 , k) T 1 . For more details on how these equations arise see [5]. …”

“…The B-cohomology completes the calculations for the second cohomology groups as shown in [5] and Section 4. The following theorem from Bendel, Nakano, and Pillen [5] states the results for p 3: …”

“…Bendel, Nakano, and Pillen [5] computed the above groups for n = 2 and p 3. There are many theorems that only hold for p 3, which makes the computations for p = 2 harder.…”

“…The above cohomology groups for the case when the field has characteristic 2 are computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen (2007) [5] for p 3. Furthermore, the computations show H 2 (G r , H 0 (λ)) has a good filtration.…”