Existence of an invariant form under a linear map

TL;DR

This paper investigates conditions under which a vector space over a field admits a non-degenerate hermitian or skew-hermitian form invariant under a given invertible linear transformation, considering involutory automorphisms of the field.

Contribution

It provides necessary and sufficient conditions for the existence of invariant hermitian or skew-hermitian forms under linear maps over fields with involutory automorphisms.

Findings

Characterizes when such invariant forms exist based on the properties of the linear map and field automorphism.

Establishes criteria linking the invertibility of the linear map to the invariance of the forms.

Extends classical results on invariant forms to fields with involutory automorphisms.

Abstract

Let $\F$ be a field of characteristic different from $2$ and $\V$ be a vector space over $\F$. Let $J: \alpha \to \alpha^J$ be a fixed involutory automorphism on $\F$. In this paper we answer the following question: given an invertible linear map $T: \V \to \V$, when does the vector space $\V$ admit a $T$-invariant non-degenerate $J$-hermitian, resp. $J$-skew-hermitian, form?