17 research outputs found
Cohomology of quantum groups: An analog of Kostant's Theorem
We prove the analog of Kostant's Theorem on Lie algebra cohomology in the
context of quantum groups. We prove that Kostant's cohomology formula holds for
quantum groups at a generic parameter , recovering an earlier result of
Malikov in the case where the underlying semisimple Lie algebra . We also show that Kostant's formula holds when is
specialized to an -th root of unity for odd (where is
the Coxeter number of ) when the highest weight of the
coefficient module lies in the lowest alcove. This can be regarded as an
extension of results of Friedlander-Parshall and Polo-Tilouine on the
cohomology of Lie algebras of reductive algebraic groups in prime
characteristic.Comment: 12 page
Second cohomology for finite groups of Lie type
Let be a simple, simply-connected algebraic group defined over
. Given a power of , let
be the subgroup of -rational points. Let be the
simple rational -module of highest weight . In this paper we
establish sufficient criteria for the restriction map in second cohomology
to be an
isomorphism. In particular, the restriction map is an isomorphism under very
mild conditions on and provided is less than or equal to a
fundamental dominant weight. Even when the restriction map is not an
isomorphism, we are often able to describe in
terms of rational cohomology for . We apply our techniques to compute
in a wide range of cases, and obtain new
examples of nonzero second cohomology for finite groups of Lie type.Comment: 29 pages, GAP code included as an ancillary file. Rewritten to
include the adjoint representation in types An, B2, and Cn. Corrections made
to Theorem 3.1.3 and subsequent dependent results in Sections 3-4. Additional
minor corrections and improvements also implemente
First cohomology for finite groups of Lie type: simple modules with small dominant weights
Let be an algebraically closed field of characteristic , and let
be a simple, simply connected algebraic group defined over .
Given , set , and let be the corresponding
finite Chevalley group. In this paper we investigate the structure of the first
cohomology group where is the
simple -module of highest weight . Under certain very mild
conditions on and , we are able to completely describe the first
cohomology group when is less than or equal to a fundamental dominant
weight. In particular, in the cases we consider, we show that the first
cohomology group has dimension at most one. Our calculations significantly
extend, and provide new proofs for, earlier results of Cline, Parshall, Scott,
and Jones, who considered the special case when is a minimal nonzero
dominant weight.Comment: 24 pages, 5 figures, 6 tables. Typos corrected and some proofs
streamlined over previous versio