Group gradings on exceptional simple Lie superalgebras

TL;DR

This paper classifies all possible group gradings on certain exceptional simple Lie superalgebras over an algebraically closed field of characteristic zero, extending previous classifications of fine gradings.

Contribution

It provides a comprehensive classification of group gradings on the exceptional Lie superalgebras G(3), F(4), D(2,1;α), and A(1,1), using recent methods and previous results.

Findings

Complete classification of group gradings on G(3), F(4), D(2,1;α)

Classification of gradings on A(1,1) with unique automorphism group

Extension of known fine grading classifications to all group gradings

Abstract

We classify up to isomorphism the gradings by arbitrary groups on the exceptional classical simple Lie superalgebras $G(3)$, $F(4)$ and $D(2,1;\alpha)$ over an algebraically closed field of characteristic $0$. To achieve this, we apply the recent method developed by A. Elduque and M. Kochetov to the known classification of fine gradings up to equivalence on the same superalgebras, which was obtained by C. Draper et al. in 2011. We also classify gradings on the simple Lie superalgebra $A(1,1)$, whose automorphism group is different from the other members of the $A$ series.