Hermitian presentations of Chevalley groups II

Abstract

A Chevalley group is called Hermitian if its root system is 3-graded. In this case, the roots of degree 0 are called compact and the remaining ones (those of degree 1 or −1) are called noncompact. Here, we modify the classical Steinberg presentation of a Chevalley group in a way that its generators become the symbols indexed by noncompact roots (the noncompact symbols); the new presentation is referred to as Hermitian. Either a compact symbol (that is, a symbol indexed by a compact root) or the commutator of a compact symbol and a noncompact one can be expressed as a product of noncompact symbols by Chevalley's commutator formula. Combining these expressions, one obtains a formula that displays a pair of concatenated commutators and involves noncompact symbols only. We get a presentation of the same Chevalley group when we replace Chevalley's commutator formula by this new double commutator formula and restrict the other relations to noncompact symbols. The simply-laced case has already been treated in [M.P. De Oliveira, E.W. Ellers, Hermitian presentations of Chevalley groups I, J. Algebra 276 (2004) 371–382. MR2054401 (2005b:20084)]. Here we proceed with an intrinsic investigation of the general case. In the process, we give a detailed analysis of the structure constants as well as higher order constants of the Chevalley algebra associated with our Chevalley group. In particular, we see that there are fewer choices for the signs of the coefficients appearing in the double commutator formula than there are in Chevalley's commutator formula; actually, we show that the former are in one-to-one correspondence with the choice of signs produced by the noncompact vectors in a Chevalley basis of the above Chevalley algebra when seen as a basis for the Lie triple system they span. In the end we give examples of Hermitian presentations for the types B n and C n and a review of basic properties of 3-graded root systems.