What question did this study set out to answer?

The aim is to generalize the Hodge operator for spaces with hermitian or symmetric bilinear forms over various fields.

March 29, 2026Open Access

Hodge Operators and Exceptional Isomorphisms between Unitary Groups

Key Points

The aim is to generalize the Hodge operator for spaces with hermitian or symmetric bilinear forms over various fields.
Defined Hodge operator for spaces with given forms
Investigated exterior powers of the space V
Established relationships between unitary groups and semi-similarity groups
Identified exceptional homomorphisms from unitary groups to semi-similarity groups
Showed dependence of algebra K on form h; it's a composition algebra unless h is symmetric in characteristic two

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.

KI fragen

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