Leibniz 2-algebras and twisted Courant algebroids

TL;DR

This paper categorifies Leibniz algebras into 2-term sh Leibniz algebras, revealing their connection to omni-Lie 2-algebras, twisted Courant algebroids, and H-twisted Lie algebroids, and explores their geometric structures.

Contribution

It introduces the categorification of Leibniz algebras and links it to omni-Lie 2-algebras, twisted Courant algebroids, and H-twisted Lie algebroids, providing new insights into their algebraic and geometric structures.

Findings

Categorification of Leibniz algebras as 2-term sh Leibniz algebras

Identification of algebraic structures of omni-Lie 2-algebras and twisted Courant algebroids

Establishment of Dirac structures leading to 2-term L-infinity algebras and H-twisted Lie algebroids

Abstract

In this paper, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac structures of twisted Courant algebroids give rise to 2-term $L_\infty$-algebras and geometric structures behind them are exactly $H$-twisted Lie algebroids introduced in \cite{Grutzmann}.