Higher Derived Brackets for Arbitrary Derivations

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.)

24 pages. LaTeX 2e. In the second version a minor style editing was done. Uses a package for commutative diagrams, diagrams.sty, available at ftp://ftp.dante.de/pub/tex/macros/generic/diagrams/taylor/ Extra editing (style, misprints...) in the third version

References in corpus (3)

• Higher derived brackets and homotopy algebras
• Derived brackets
• On odd Laplace operators

Cited by in corpus (24)

• On the category of Lie n-algebroids
• Leibniz algebroids, twistings and exceptional generalized geometry
• A Poincare lemma for sigma models of AKSZ type
• Even symplectic supermanifolds and double field theory
• Equivalences of Higher Derived Brackets
• Schwinger-Dyson equation for non-Lagrangian field theory
• Introduction to graded geometry
• Nonabelian higher derived brackets
• On homotopy Poisson actions and reduction of symplectic Q-manifolds
• "Nonlinear pullbacks" of functions and $L_{\infty }$-morphisms for homotopy Poisson structures
• Star products on graded manifolds and ${\alpha }^{\prime }$-corrections to Courant algebroids from string theory
• Deformation-obstruction theory for diagrams of algebras and applications to geometry
• Higher Derived Brackets and Deformation Theory I
• Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids
• On the infinity category of homotopy Leibniz algebras
• Deformations and homotopy theory of Rota-Baxter algebras of any weight
• Simultaneous Deformations of Lie Algebroids and Lie Subalgebroids
• L-infinity bialgebroids and homotopy Poisson structures on supermanifolds
• On coisotropic deformations of holomorphic submanifolds
• On higher Poisson and Koszul--Schouten brackets
• Equivalences of coisotropic submanifolds
• Deformations of Courant Algebroids and Dirac Structures via Blended Structures
• Commutative $n$-ary superalgebras with an invariant skew-symmetric form
• Cumulants, Koszul brackets and homological perturbation theory for commutative $B{V}_{\infty }$ and $IB{L}_{\infty }$ algebras