Higher Derived Brackets for Arbitrary Derivations

TL;DR

This paper develops a general construction of higher derived brackets from arbitrary derivations in Lie superalgebras, with applications in geometry and physics, and explores their relation to subalgebra inclusions and generalized $L_{ olinebreak}_{ olinebreak} ext{-} olinebreak}_{ olinebreak} ext{infinity}$-algebras.

Contribution

It introduces a new method to construct higher derived brackets from any derivation, extending previous work and connecting to geometric and physical contexts.

Findings

Higher derived brackets can be generated by arbitrary derivations.

The construction generalizes existing brackets and relates to subalgebra inclusions.

Applications include geometry, physics, and generalized $L_{ ext{infinity}}$-algebras.

Abstract

We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathematical physics. From a totally different viewpoint, we show that higher derived brackets arise when one wants to turn the inclusion map of a subalgebra of a differential Lie superalgebra, with a given complementary subalgebra, into a fibration. (For a non-Abelian complementary subalgebra, this leads to a generalization of $L_{\infty}$-algebras with dropped or weakened (anti)symmetry of the brackets.)