Rota-Baxter Lie $2$-algebras

TL;DR

This paper introduces Rota-Baxter Lie 2-algebras as a categorification of Rota-Baxter Lie algebras, establishing their equivalence with 2-term Rota-Baxter L-infinity algebras and exploring their crossed modules.

Contribution

It defines Rota-Baxter Lie 2-algebras, proves their categorical equivalence with 2-term Rota-Baxter L-infinity algebras, and connects crossed modules of Rota-Baxter Lie algebras with strict 2-term Rota-Baxter L-infinity algebras.

Findings

Category of Rota-Baxter Lie 2-algebras is equivalent to 2-term Rota-Baxter L-infinity algebras

One-to-one correspondence between strict 2-term Rota-Baxter L-infinity algebras and crossed modules of Rota-Baxter Lie algebras

Construction of crossed modules of Lie algebras from crossed modules of Rota-Baxter Lie algebras

Abstract

In this paper, we introduce the notion of Rota-Baxter Lie $2$-algebras, which is a categorification of Rota-Baxter Lie algebras. We prove that the category of Rota-Baxter Lie $2$-algebras and the category of $2$-term Rota-Baxter $L_\infty$-algebras are equivalent. We introduce the notion of a crossed module of Rota-Baxter Lie algebras and show that there is a one-to-one correspondence between strict $2$-term Rota-Baxter $L_\infty$-algebras and crossed modules of Rota-Baxter Lie algebras. We give the construction of crossed modules of Lie algebras from crossed modules of Rota-Baxter Lie algebras.