Factorization of Lipschitzian elements
Let \(V\) be a vector space over a field \(K\) and let \(q: V\rightarrow K\) be a quadratic form. Denote as \(\mathrm{Cl}(V,q)\) the corresponding Clifford algebra. The Lipschitz monoid \(\mathrm{Lip}(V,q)\) is the multiplicative monoid generated in \(\mathrm{Cl}(V,q)\) by all scalars in \(K\), all vectors in \(V\) and elements \(1+xy\), where \(x\) and \(y\) are vectors that span a totally isotropic plane. The main results of the paper are the following. It is proved that if \(K\) contains at least three scalars, then every element \(a\in \mathrm{Lip}(V,q)\) such that \(\mathrm{Supp}(a)\) is not totally isotropic can be expressed as follows: \(a=x_1\dots x_s\), where \(x_1\),\dots, \(x_s\) form a base of \(\mathrm{Supp}(a)\). An algorithm of the construction of such a decomposition is proposed.
scientific article; zbMATH DE number 6380694