On Invertibility of Clifford Algebras Elements with Disjoint Supports

Abstract

Let Cl(n) be the classical associative Clifford algebra over field ℝ with generators e 1, e 2,..., e n and relations e i e j +e j e i = 0,i ≠ j e i 2 = -1. It’s well known ( see [1]) fact that algebras Cl(n) for different n are isomorphic to some matrix ℝ-algebra or to direct sum of some matrix ℝ-algebras. Therefore, from the formal point of view, the question about invertibility in Cl(n) is equivalent to the question about calculating of determinants of matrices. But,these matrices have sizes approximately equal to 2[n/2] × 2[n/2] and really such calculatings are impossible. But for some classes of elements of algebra Cl(n) a criteria of invertibility may be obtained without above mentioned matrix realizibility of Clifford algebra Cl(n). The trivial example of such a class is the set of all vectors x = ∑x i e i ∊ ℝ n ⊂ cl(n). Indeed we have x 2 = -∑x i 2 and hence vector x is invertible in cl(n) iff x ≠ 0.