|
Abstract:
Heinz Hopf studied the structure of the cohomology of a Lie group treated
as a topological space. The result was a surprisingly simple and elegant
description of this graded algebra in terms of l natural numbers
m1,...,ml called exponents of the Lie group, where l is the
rank of the compact part of the Lie group. Later these exponents were
explicitly determined for all simple groups. Kostant gave an interpretation
of these exponents in terms of a special homomorphism of SL(2) into the
Lie group.
There has been a renewed interest in this circle of ideas, thanks to the
work of Hitchin and the so-called `geometric Langlands programme'.
I will give an elementary and non-technical account of the classical
theory and indicate how the new ideas may throw light on fundamental
questions regarding the classification of Lie groups, etc.
|