Primitive Idempotents and Indecomposable Left Ideals in Degenerate Clifford Algebras

Abstract

Primitive idempotents of degenerate Clifford algebras are determined. A degenerate Clifford algebra A has a nilpotent Jacobson radical J(A) SO that the factor algebra A = A/J(A) is a non-degenerate Clifford algebra isomorphic to a certain maximal Clifford subalgebra of A. Once primitive mutually annihilating idempotents of A are known, they can be lifted, modulo the radical, to primitive mutually annihilating idempotents of A. Moreover, each decomposition of A into a direct sum of principal indecomposable modules can be lifted to a corresponding decomposition of A. The resulting indecomposable summands of A need not be minimal. As an example, principal indecomposable modules of degenerate Clifford algebras with degeneracy in one dimension are found.