The Dzhumadildaev brackets: a hidden supersymmetry of commutators and the Amitsur-Levitzki--type identities

TL;DR

This paper explores the Dzhumadildaev brackets, revealing a hidden supersymmetry in commutators and generalizing Amitsur-Levitzki identities across various Lie algebra and superalgebra contexts.

Contribution

It uncovers a hidden supersymmetry in commutators through Dzhumadildaev brackets and discusses their potential generalizations beyond existing identities.

Findings

Discovered hidden supersymmetry in commutators

Generalized Amitsur-Levitzki identities to Lie superalgebras

Outlined open problems for future research

Abstract

The Amitsur--Levitzki identity for matrices was generalized in several directions: by Kostant for simple finite-dimensional Lie algebras, by Kirillov (later joined by Kontsevich, Molev, Ovsienko, and Udalova) for simple vectorial Lie algebras with polynomial coefficients, and by Gie, Pinczon, and Ushirobira for the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|n)$. Dzhumadildaev switched the focus of attention in these results by considering the algebra formed by antisymmetrizors and discovered a hidden supersymmetry of commutators. We overview these results and their possible generalizations (open problems).