Universal Algebras $${\mathcal{U}(V,Q)}$$ U ( V , Q ) for Quadratic Mappings $${Q:V \rightarrow V}$$ Q : V → V

Abstract

Let V be a vector space, and Q a quadratic mapping \({V \rightarrow V}\). This article studies the (universal) algebra \({\mathcal{U}(V,Q)}\) generated by the elements \({v \in V}\) with the relations \({v^2=Q(v)}\). The properties of this algebra shall prove to be completely different from those of a Clifford algebra. In particular, for almost all quadratic mappings Q, the canonical mapping \({V \rightarrow \mathcal{U}(V,Q)}\) is the null mapping, and \({{\rm dim}(\mathcal{U}(V,Q))=1}\). Nevertheless, some particular mappings Q may give an interesting algebra \({\mathcal{U}(V,Q)}\). The disappointing properties of the algebras \({\mathcal{U}(V,Q)}\) prove that the advantageous properties of Clifford algebras are not at all self-evident, but must be considered as exceptional.