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 ² _i = -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 ² _i and hence vector x is invertible in cl(n) iff x ≠ 0.