The isomorphisms of unitary groups over noncommutative domains

Abstract

The isomorphisms between projective unitary congruence groups are known when the underlying Witt indices are 33, the underlying spaces are finite dimensional, and the underlying integral domains are commutative [15, 161. Here we extend these results to noncommutative domains possessing a division ring of quotients and to arbitrary dimensions. Our development allows unitary and symplectic groups to be treated simultaneously, applies to collinear groups, and unifies the known theories over commutative domains and noncommutative fields. We consider the class of collinear (unitary or symplectic) groups having “enough projective transvections,” i.e., at least one on each isotropic line (see Sections 2A and 1B). The chief hurdle, as in the commutative case, is to show that in such groups projective transvections are preserved under isomorphism. From this we get a correspondence of isotropic lines to which the Fundamental Theorem of Projective Geometry can be applied. Then it is easy to show that the isomorphism is of the expected form, i.e., induced by an orthogonality-preserving semilinear bijection (reflexive collinear transformation) between the two underlying spaces; in particular, a unitary group is not isomorphic to a symplectic group.