Bilinear Forms Derived from Lipschitzian Elements in Clifford Algebras

Abstract

In every Clifford algebra ${\mathrm {Cl}}(V,q)$, there is a Lipschitz monoid (or semi-group) ${\mathrm {Lip}}(V,q)$, which is in most cases the monoid generated by the vectors of V. This monoid is useful for many reasons, not only because of the natural homomorphism from the group ${\mathrm {GLip}}(V,q)$ of its invertible elements onto the group ${\mathrm {O}}(V,q)$ of orthogonal transformations. From every non-zero $a\in {\mathrm {Lip}}(V,q)$, we can derive a bilinear form $\phi$ on the support S of a in V; it is q-compatible: $\phi (x,x)=q(x)$ for all $x\in S$. Conversely, every q-compatible bilinear form on a subspace S of V can be derived from an element $a\in {\mathrm {Lip}}(V,q)$ which is unique up to an invertible scalar; and a is invertible if and only if $\phi$ is non-degenerate. This article studies the relations between a, $\phi$ and (when a is invertible) the orthogonal transformation g derived from a; it provides both theoretical knowledge and algorithms. It provides an effective tool for the factorization of lipschitzian elements, based on this theorem: if $(v_1,v_2,\ldots ,v_s)$ is a basis of S, then $a=\kappa \,v_1v_2\ldots v_s$ (for some invertible scalar $\kappa$) if and only if the matrix of $\phi$ in this basis is lower triangular. This theorem is supported by an algorithm of triangularization of bilinear forms.