Hodge Operators and Exceptional Isomorphisms between Unitary Groups

We give a generalization of the Hodge operator to spaces (V, h) endowed with a hermitian or symmetric bilinear form h over arbitrary fields, including the characteristic two case.Suitable exterior powers of V become free modules over an algebra K defined using such an operator.This leads to several exceptional homomorphisms from unitary groups (with respect to h) into groups of semi-similitudes with respect to a suitable form over some subfield of K .The algebra K depends on h; it is a composition algebra unless h is symmetric and the characteristic is two.