# Infinity-enhancing of Leibniz algebras

## TL;DR

This paper establishes a correspondence between infinity-enhanced Leibniz algebras and differential graded Lie algebras, providing explicit brackets and connecting these structures to tensor hierarchies and $L_$-algebras.

## Contribution

It introduces a new link between infinity-enhanced Leibniz algebras and differential graded Lie algebras, with explicit bracket formulas and applications to tensor hierarchies.

## Key findings

- Explicit brackets for all orders are provided.
- The correspondence with differential graded Lie algebras is established.
- The structure agrees with previous partial results from arXiv:1904.11036.

## Abstract

We establish a correspondence between infinity-enhanced Leibniz algebras, recently introduced in order to encode tensor hierarchies, and differential graded Lie algebras, which have been already used in this context. We explain how any Leibniz algebra gives rise to a differential graded Lie algebra with a corresponding infinity-enhanced Leibniz algebra. Moreover, by a theorem of Getzler, this differential graded Lie algebra canonically induces an $L_\infty$-algebra structure on the suspension of the underlying chain complex. We explicitly give the brackets to all orders and show that they agree with the partial results obtained from the infinity-enhanced Leibniz algebras in arXiv:1904.11036.