Gradings on associative algebras with involution and real forms of classical simple Lie algebras

TL;DR

This paper classifies fine gradings on simple associative algebras with involution over real fields, providing a comprehensive understanding of their structure and implications for real forms of classical simple Lie algebras.

Contribution

It offers a complete classification of fine gradings on simple associative algebras with involution over real fields, linking them to real forms of classical Lie algebras.

Findings

Classified fine gradings on simple associative algebras with involution over real fields.

Connected gradings to real forms of classical simple Lie algebras.

Provided a framework for understanding gradings via group homomorphisms.

Abstract

We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.