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Higher Derived Brackets and Deformation Theory I

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Fusun Akman and Lucian M. Ionescu

The existing constructions of derived Lie and sh-Lie brackets involve
multilinear maps that are used to define higher order differential operators.
In this paper, we prove the equivalence of three different definitions of
higher order operators. We then introduce a unifying theme for building
derived brackets and show that two prevalent derived Lie bracket constructions
are equivalent. Two basic methods of constructing derived strict sh-Lie
brackets are also shown to be essentially the same. So far, each of these
derived brackets is defined on an abelian subalgebra of a Lie algebra. We
describe, as an alternative, a cohomological construction of derived sh-Lie
brackets. Namely, we prove that a unital differential algebra with a graded
homotopy commutative and associative product and an odd, square-zero operator
(that commutes with the differential) gives rise to an sh-Lie structure on the
cohomology via derived brackets. The method is in particular applicable to
differential vertex operator algebras.

Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 385-403