LAUSR

Let V be a vector space, and Q a quadratic mapping $${V \rightarrow V}$$V→V. This article studies the (universal) algebra $${\mathcal{U}(V,Q)}$$U(V,Q) generated by the elements $${v \in V}$$v∈V with the relations $${v^2=Q(v)}$$v2=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 $${V \rightarrow \mathcal{U}(V,Q)}$$V→U(V,Q) is the null mapping, and $${{\rm dim}(\mathcal{U}(V,Q))=1}$$dim(U(V,Q))=1. Nevertheless, some particular mappings Q may give an interesting algebra $${\mathcal{U}(V,Q)}$$U(V,Q). The disappointing properties of the algebras $${\mathcal{U}(V,Q)}$$U(V,Q) prove that the advantageous properties of Clifford algebras are not at all self-evident, but must be considered as exceptional.