Hodge Operators and Exceptional Isomorphisms between Unitary Groups

TL;DR

This paper generalizes the Hodge operator for spaces with hermitian or symmetric forms over arbitrary fields, revealing exceptional homomorphisms between unitary groups and other algebraic groups, including in characteristic two.

Contribution

It introduces a generalized Hodge operator applicable over arbitrary fields, leading to new exceptional homomorphisms between unitary groups and semi-similitude groups.

Findings

Established a generalized Hodge operator for arbitrary fields.

Derived exceptional homomorphisms between unitary and semi-similitude groups.

Connected the algebra $K$ to composition algebras and characteristic properties.

Abstract

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.