ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: Serganova, Vera, UC Berkeley, USA On representations of Lie superalgebras

We give a brief review and a summary of recent results concerning finitedimensional representations of classical Lie superalgebras. The examples of these superalgebras are general linear Lie superalgebra gl(m, n) and orhosymplectic Lie superalgebra osp(m, n). We present a solution to the problem of calculating characters of irreducible representations. In order to do this one needs to define Kazhdan-Lusztig polynomials for the category of finite-dimensional representations over classical Lie superalgebras and find an algorithm for their calculation. The last can be done using a geometric approach involing cohomology of G-sheaves on a compact homogeneous supermanifold G/P for different parabolic subgroups P. In particular we explain an analogue of Borel-Weil-Bott theorem for flag supemanifold. Finally we discuss an unusual tensor product structure on the category of finite dimensional representations for the family of superalgebras gl(m, n) which is closely related to all above topics.

ICM 1998 Berlin · Plenary and Invited Lectures