Group gradings on exceptional simple Lie superalgebras
Sebastiano Argenti, Mikhail Kochetov, Felipe Yasumura
TL;DR
The authors deliver a complete classification of group gradings up to isomorphism on the exceptional simple Lie superalgebras $G(3)$, $F(4)$, and $D(2,1;\alpha)$, and extend the analysis to $A(1,1)$ by leveraging the Elduque–Kochetov framework built on the prior fine-gradings classification. They compute almost fine gradings via the fine gradings, determine Weyl groups, and describe admissible homomorphisms to arbitrary abelian groups $G$, yielding explicit isomorphism-type criteria for each family of gradings. For each algebra, the gradings fall into a finite set of families (two for $G(3)$; five for $F(4)$; several for $D(\alpha)$ depending on the $\alpha$-orbit structure; six for $A(1,1)$), with precise equivalence and Weyl-group data. The results illuminate the graded-structure landscape of these exceptional superalgebras, offering a robust blueprint for understanding symmetry breaking, automorphism action, and representation-theoretic implications in graded contexts. Overall, the work provides a comprehensive, methodically derived map of all $G$-gradings on these key Lie superalgebras and delivers a practical toolkit for further explorations in graded Lie theory and related algebraic structures.
Abstract
We classify up to isomorphism the gradings by arbitrary groups on the exceptional classical simple Lie superalgebras $G(3)$, $F(4)$ and $D(2,1;α)$ 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.