Scalar products of spinors and an extension of Brauer-Wall groups

Abstract

The automorphism groups of scalar products of spinors are determined. Spinors are considered as elements of minimal left ideals of Clifford algebras on quadratic modules, e.g., on orthogonal spaces. Orthogonal spaces of any dimension and arbitrary signature are discussed. For example, the automorphism groups of scalar products of Pauli spinors and Dirac spinors are, respectively, isomorphic to the matrix groups U(2) and U(2, 2). It is found that there are, in general, 32 different types or similarity classes of such automorphism groups, if one considers real orthogonal spaces of arbitrary dimension and arbitrary signature. On a more abstract level this means that the Brauer-Wall group of the real field R, consisting of graded central simple algebras over R, can be extended from the cyclic group of 8 elements to an abelian group with32 elements. This extended Brauer-Wall group is seen to be isomorphic with the group (Z ₈ × Z ₈ )/Z ₂ and it consists of real Clifford algebras with antiinvolutions. Moreover, the subgroup determined by the antiinvolution is isomorphic to the automorphism group of scalar products of spinors.