The VIGRE Algebra Group was led by Brian Boe,
Leonard Chastkofsky and Daniel Nakano.
The other members of the group are
Kenyon J. Platt
The main goal of the VIGRE Algebra Seminar this term was to investigate the cohomology of Lie algebras. In the
first three weeks, Boe and Nakano gave lectures on introductory material on Lie algebras and the definition of Lie algebra
cohomology. Examples were worked out for the unipotent radical of simple Lie algebras of ranks 1 and 2.
The University of Wisconsin, Stout RUI group led by Chris Bendel developed MAGMA code for computing the
first two differentials in Koszul complex. Boe extended this program to compute the dimensions of all Lie algebra cohomology
groups. With this program, the group has begun to experiment with different Lie algebras over various fields of characterstic p.
There were two observations which were evident. First, the Lie algebra cohomology had a palindromic behavior and second,
the weights were always conjugates of the zero weight under the action of the extended affine Weyl group. Wiesner and Nakano
found a proof for the palindromic behavior and Nakano came up with a proof for the weights occuring in the cohomology. These results
were presented to the group.
Several of the students are working on modifications of Boe's computer program. Our hope next semester is to get the students more
involved in the running of the programs. We are also looking at ways to prove comparision results for the Lie algebra cohomology in characterstic
zero (given by Kostant's formula) and the Lie algebra cohomology in characteristic p (which is unknown for low primes).
Spectral sequence techniques will be probably relevant in this analysis.