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.