Multiplicity free gradings on semisimple Lie and Jordan algebras and skew root

TL;DR

This paper introduces skew root systems to construct multiplicity free gradings on semisimple Lie and Jordan algebras, providing a new approach to understanding their abelian group gradings.

Contribution

It develops skew root systems of Lie and Jordan types and uses them to systematically construct and classify multiplicity free gradings on semisimple Lie and Jordan algebras.

Findings

Constructed three families of skew root systems of Lie type.

Constructed three families of skew root systems of Jordan type.

Identified the corresponding simple Lie and Jordan algebras.

Abstract

A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type respectively, and use them to construct multiplicity free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp. Jordan) algebras are simple. Three families of skew root systems of Lie type (resp. of Jordan type) are constructed and the corresponding Lie (resp. Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.