Elementary unipotents in simple algebraic groups

Abstract:

The notion of a simple algebraic group is a joint generalization of the notions of a simple Lie group and a finite simple group in the setting of algebraic geometry. A typical example is the group SL_n(R) where R is a commutative ring with 1. It is well-known that if R is a field, SL_n(R) is generated as an abstract group by elementary unipotent matrices, which look like the identity matrix plus just one non-zero element in an off-diagonal position. We will discuss extensions of this basic fact to other simple algebraic groups and to commutative rings R different from fields, most notably, to the rings of polynomials and Laurent polynomials. In the SL_n case, these results were obtained by A. Suslin in 1977, using ideas from D. Quillen's proof of Serre's conjecture on projective modules. In 2012 the speaker completed the case of simple algebraic groups of isotropic rank at least 2.