Canonical matrices of isometric operators on indefinite inner product spaces

TL;DR

This paper provides canonical matrix forms for pairs of operators and forms on vector spaces over various fields, extending classification methods for isometric operators in indefinite inner product spaces.

Contribution

It introduces a unified method to classify isometric operators using quiver representations, applicable to multiple field and form types.

Findings

Canonical matrices are derived for different field and form cases.

The method reduces classification to quiver representation problems.

Applicable to symmetric, skew-symmetric, Hermitian, and skew-Hermitian forms.

Abstract

We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases: (i) F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric; (ii) F is an algebraically closed field or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].