Hom-Lie 2-algebras

TL;DR

This paper introduces hom-Lie 2-algebras and related structures, establishing their categorical equivalences, exploring skeletal cases, and constructing hom-analogues of string Lie 2-algebras, advancing the categorification of hom-Lie algebra theory.

Contribution

It defines hom-Lie 2-algebras, proves their categorical equivalence with 2-term $HL_ abla$-algebras, and constructs hom-analogues of string Lie 2-algebras and strict models.

Findings

Category of hom-Lie 2-algebras is equivalent to 2-term $HL_ abla$-algebras

Constructed hom-analogues of string Lie 2-algebras

Established correspondence between strict hom-Lie 2-algebras and crossed modules

Abstract

In this paper, we introduce the notions of hom-Lie 2-algebras, which is the categorification of hom-Lie algebras, $HL_\infty$-algebras, which is the hom-analogue of $L_\infty$-algebras, and crossed modules of hom-Lie algebras. We prove that the category of hom-Lie 2-algebras and the category of 2-term $HL_\infty$-algebras are equivalent. We give a detailed study on skeletal hom-Lie 2-algebras. In particular, we construct the hom-analogues of the string Lie 2-algebras associated to any semisimple involutive hom-Lie algebras. We also proved that there is a one-to-one correspondence between strict hom-Lie 2-algebras and crossed modules of hom-Lie algebras. We give the construction of strict hom-Lie 2-algebras from hom-left-symmetric algebras and symplectic hom-Lie algebras.