Idempotent structure of Clifford algebras
Spinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras \({\mathbb{R}}_{p,q}\) are shown to be products of mutually nonannihilating commuting idempotent factors \(1/2(1+e_ T)\), where the \(k=q-r_{q-p}\) basis elements \(e_ T\) satisfy \(e^ 2_ T=1\). The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras \(R_{p,q}\) with an anti-involution such that each symmetric element is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when \(p+q\leq 3\) and (p,q)\(\neq (2,1)\).
scientific article; zbMATH DE number 4075302