Structure of spin groups associated with degenerate Clifford algebras

Abstract

Clifford algebras over finite-dimensional vector spaces endowed with degenerate quadratic form contain a nontrivial two-sided nilpotent ideal (the Jacobson radical) generated by the orthogonal complement of such spaces. Thus, they cannot be faithfully represented by matrix algebras. Following the theory of spin representations of classical Clifford algebras, the left regular (spin) representations of these degenerate algebras can be studied in suitably constructed left ideals. First, structure of the group of units of such algebras is examined for a quadratic form of arbitrary rank. It is shown to be a semidirect product of a group generated by the radical and the group of units of a maximal nondegenerate Clifford subalgebra. Next, in the special case of corank 1, Clifford, pin, and spin groups are defined an their structures are described. As an example, a Galilei–Clifford algebra over the Galilei space-time is considered. A covering theorem is then proved analogous to the one well known in the theory of spin and orthogonal groups.