Bruhat Presentations for *-Classical Groups

Abstract

We introduce in a general setting, the unitary similitude groups associated to ϵ-hermitian form over a ring A with involution. We concentrate next in the rank 2 case. The classical presentation of GL(2, F), F a field, originating from its Bruhat decomposition can be extended to these unitary similitude groups. Furthermore, the multiplier associated to ϵ-hermitian forms affords a signed non commutative version of the classical determinant of 2 by 2 matrices, which we recover exploiting Grassmann's approach to determinants. Finally, we present various examples of our groups and give some categorical properties of them.