On factorization properties of algebraic spinors (Q1071092)

Algebraic spinors are considered as elements of at least one minimal left ideal (MLI) of a Clifford algebra. A universal Clifford algebra with its isomorphic representation has been identified as a full ring D(k) (or direct sum D(k)\(\oplus\) D(k)) of square matrices with k rows and with entries from a definite division algebra \(D=R\), C or H (quaternions). The main proposition in this paper is: A matrix M belongs to some MLI of D(k) iff M factorizes as \(M=UV^{\dag}\). The direct sum \(M_ 1\oplus\) \(M_ 2\) of matrices belongs to some MLI of D(k)\(\oplus\) D(k) iff either \(M_ 1=0\) and \(M_ 2=UV^{\dag}\) or \(M_ 1=UV^{\dag}\) and \(M_ 2=0\).