Higher Derived Brackets for Arbitrary Derivations

Details Higher Derived Brackets for Arbitrary Derivations Higher Derived Brackets for Arbitrary Derivations

2004-12-09

arxiv.org/abs/math/0412202v3

Mathematics

Quantum Algebra

24 pages. LaTeX 2e. In the second version a minor style editing was done. Uses a package for commutative diagrams, diagrams.st

Scientific paper

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.)

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