# 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 algebras and superalgebras.

## Contribution

It uncovers a hidden supersymmetry in commutators via Dzhumadildaev brackets and discusses potential generalizations of Amitsur-Levitzki identities.

## Key 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).