Fine gradings on simple exceptional Jordan pairs and triple systems

TL;DR

This paper classifies fine abelian group gradings on certain exceptional Jordan pairs and triple systems, computes their Weyl groups, and explores the induced gradings on associated Lie algebras.

Contribution

It provides a complete classification of fine gradings on bi-Cayley and Albert Jordan pairs and systems, including their Weyl groups and induced Lie algebra gradings.

Findings

Classified all fine gradings on bi-Cayley and Albert pairs.

Computed Weyl groups for each grading.

Determined induced gradings on exceptional Lie algebras.

Abstract

We give a classification up to equivalence of the fine group gradings by abelian groups on the Jordan pairs and triple systems of types bi-Cayley and Albert, under the assumption that the base field is algebraically closed of characteristic different from $2$. The associated Weyl groups are computed. We also determine, for each fine grading on the bi-Cayley and Albert pairs, the induced grading on the exceptional simple Lie algebra given by the Tits-Kantor-Koecher construction.