Group gradings on the Lie and Jordan superalgebras $Q(n)$

TL;DR

This paper classifies all possible gradings by abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$ over algebraically closed fields, detailing fine gradings and $G$-gradings.

Contribution

It provides a comprehensive classification of gradings on $Q(n)$ superalgebras, including fine gradings and isomorphism classes, expanding understanding of their symmetry structures.

Findings

Complete classification of abelian group gradings on $Q(n)$

Identification of fine gradings up to equivalence

Determination of $G$-gradings up to isomorphism

Abstract

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n \geq 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism.