Hom-Lie 2-algebras

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.

💡 Research Summary

This paper develops a comprehensive theory of “hom‑Lie 2‑algebras,” which are the categorified analogues of hom‑Lie algebras, together with their hom‑analogue of L∞‑algebras, called HL∞‑algebras. After recalling the necessary background on hom‑Lie algebras (a Lie‑type bracket whose Jacobi identity is twisted by a linear self‑map φ) and on 2‑vector spaces (categories internal to Vect), the authors introduce the main objects.

A hom‑Lie 2‑algebra consists of a 2‑vector space L equipped with (i) a skew‑symmetric bilinear functor