Homogeneous involution on graded division algebras and their polynomial identities

TL;DR

This paper characterizes homogeneous involutions on finite-dimensional graded-division algebras over algebraically closed fields and computes their graded polynomial identities with involution, highlighting their role in graded polynomial identity theory.

Contribution

It introduces the concept of homogeneous involution on graded-division algebras and explicitly computes their graded polynomial identities with involution.

Findings

Homogeneous involutions are characterized on graded-division algebras.

Explicit graded polynomial identities with involution are computed.

Homogeneous involutions naturally arise in graded polynomial identity contexts.

Abstract

In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L. Fonseca and T. de Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution.