UNSW Pure Maths Seminar: Drupieski -- Cohomology of infinitesimal algebraic groups and quantized enveloping algebras

           THE UNIVERSITY OF NEW SOUTH WALES

             DEPARTMENT OF PURE MATHEMATICS


_________________DEPARTMENTAL SEMINAR__________________


Speaker: Christopher Drupieski (Virginia)

Title:   Cohomology of infinitesimal algebraic groups
         and quantized enveloping algebras

Date:    Tuesday, 5 August 2008
Time:    12:00 noon
Venue:   RC-3084, The Red Centre, UNSW

Abstract:

In 1984, Andersen and Jantzen computed the structure
of the cohomology ring with trivial coefficients of
the restricted Lie algebra corresponding to a reductive
algebraic group. In 1993, Ginzburg and Kumar adapted
the arguments of Andersen and Jantzen to compute the
cohomology ring with trivial coefficients of the
finite-dimensional quantum enveloping algebra at an
ell-th root of unity defined by Lusztig. In both cases,
lower bounds were assumed on the characteristic p of
the ground field (respectively, the order ell of the
root of unity) in order to achieve key vanishing
results. Beyond some ad hoc calculations by Andersen
and Jantzen, general results for small values of p
and ell remained elusive.

In this talk I will report on some recent results of
Bendel, Nakano, Parshall and Pillen that provide a
uniform method for computing the aforementioned
cohomology groups, as well as some recent results and
some open questions in the quantum mixed case.
Related results on the coordinate rings of nilpotent
varieties may also be touched upon.

Some basic familiarity with group theory and Lie
algebras will be assumed (e.g., root space
decomposition). I will endeavor to present the
remaining content on algebraic groups, Frobenius
kernels, spectral sequences, and quantized enveloping
algebras in a sufficiently elementary fashion that no
previous mastery of these topics will be assumed.


Enquiries to Jonathan Kress, 9385 7078, j.kress@unsw.edu.au