Bilinear forms derived from Lipschitzian elements in Clifford algebras
Given a quadratic form \(q : V \rightarrow K\) from a certain vector space \(V\) to its base-field \(K\), the Clifford algebra \(\text{Cl}(V,p)\) is defined to be the \(K\)-algebra generated by the elements \(v\) of \(V\) subject to the relations \(v^2=q(v)\). The Lipschitz semi-group (or monoid) \(\text{Lip}(V,q)\) is defined to be the multiplicative sub-monoid of \(\text{Cl}(V,p)\) generated by the scalars from \(K\), the elements of \(V\), and the elements of the form \(1+xy\) with \(x,y \in V\) and \(q(x)=q(y)=b_q(x,y)\) where \(b_q\) is the underlying symmetric bilinear form \(V \times V \rightarrow K\) given by \(b_q(x,y)=q(x+y)-q(x)-q(y)\). The goal of this paper is to study the behavior of the elements in \(\text{Lip}(V,q)\). The support \(S\) of any \(a \in \text{Cl}(V,q)\) is the minimal subspace of \(V\) for which \(a \in \text{Cl}(S,q|_S)\). Theorem 1.1 states that for any nonzero \(a\) in \(\text{Lip}(V,q)\) with support \(S\), there exists a unique bilinear form \(\phi : S \times S \rightarrow K\) such that \[ \forall x \in S, \;x a=x\rfloor_\phi a, \] or equivalently, \[ \forall x \in S, \;a x=a\lfloor_\phi x. \] Theorem 1.2 states that given a basis \(\{v_1,\dots,v_s\}\) of \(S\), \(a\) decomposes as \(\kappa v_1 \dots v_s\) for some \(\kappa \in K\) if and only if the matrix of \(\phi\) with respect to \(\{v_1,\dots,v_s\}\) is lower triangular.
scientific article; zbMATH DE number 6892845