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Homotopy Leibniz algebras and derived brackets. (preperation), available
"... We will give a generalized framework of derived bracket construction. The derived bracket construction provides a method of constructing homotopies. We will prove that a deformation derivation of dg Leibniz algebra (or called dg Loday algebra) induces a strong homotopy Leibniz algebra by the derived ..."
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We will give a generalized framework of derived bracket construction. The derived bracket construction provides a method of constructing homotopies. We will prove that a deformation derivation of dg Leibniz algebra (or called dg Loday algebra) induces a strong homotopy Leibniz algebra by the derived bracket method. 1 Introduction. Given a differential graded (dg) Leibniz algebra (or called dg Loday algebra), (V,[,],d), the modified bracket ±[dx,y] satisfies a graded Leibniz rule on sV, where x,y ∈ V. Such a construction of graded Leibniz algebra is called a derived bracket construction (KosmannSchwarzbach [4, 5]). The derived bracket plays important
THE LIOUVILLE PHENOMENON IN THE DEFORMATION PROBLEM OF COISOTROPICS
, 805
"... Abstract. The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L∞algebra on the shifted foliation complex ..."
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Abstract. The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L∞algebra on the shifted foliation complex (Ω∗[1](F), dF), which allows a concise description of deformations in terms of a MaurerCartan equation. Infinitesimal deformations are given by dFclosed forms, and the relation between infinitesimal deformations and full deformations can be studied in terms of obstruction classes lying in the foliation cohomology H ∗ F. Closely related to the foliation cohomology is Haefliger’s group Ω ∗ c(T/H), an underappreciated model for the leaf space of a foliation. We make integral use of this group in showing solvability and unsolvability of the obstruction equations. We also show the L∞apparatus to be capable of detecting the Liouville/diophantine distinction of KAM theory, and argue for the greater significance of Haefliger’s integrationoverleaves map in passing this fine structure to a geometric model for the leaf space.