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L∞algebras and higher analogues of Dirac structures and Courant algebroids, arXiv:1003.1004
"... Abstract. We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, nonunique) differential form on M, and are equivalent to (and simpler to handle than) the MultiDirac stru ..."
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Cited by 17 (4 self)
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Abstract. We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, nonunique) differential form on M, and are equivalent to (and simpler to handle than) the MultiDirac structures recently introduced in the context of field theory by Vankerschaver, Yoshimura and Marsden. We associate an L∞algebra of observables to every higher Dirac structure, extending work of Baez, Hoffnung and Rogers on multisymplectic forms. Further, applying a recent result of Getzler, we associate an L∞algebra to any manifold endowed with a closed differential form H, via a higher analogue of split Courant algebroid twisted by H. Finally, we study the relations between the L∞algebras appearing above.
Derived brackets and sh leibniz algebras
, 2009
"... We will give a generalized framework of derived bracket construction. It will be shown that a deformation differential provides a strong homotopy (sh) Leibniz algebra structure by derived bracket construction. A relationship between the three concepts, homotopy algebra theory, deformation theory and ..."
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Cited by 8 (1 self)
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We will give a generalized framework of derived bracket construction. It will be shown that a deformation differential provides a strong homotopy (sh) Leibniz algebra structure by derived bracket construction. A relationship between the three concepts, homotopy algebra theory, deformation theory and derived bracket construction, will be discussed. We will prove that the derived bracket construction is a map from the equivalence classes of deformation theory to the one of sh Leibniz algebras. 1 Introduction. Let (V,d,[,]) be a differential graded (dg) vector space, or a complex equipped with a binary bracket product. It is called a dg Leibniz algebra, or sometimes called a dg Loday algebra, if the bracket product satisfies a graded Leibniz identity. When the bracket is skewsymmetric, or graded commutative, the Leibniz identity is equivalent
Deformation theory from the point of view of fibered categories
"... Abstract. We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We ..."
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Abstract. We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a version of Schlessinger’s Theorem in this context, and the Ran–Kawamata vanishing theorem for obstructions. We accompany this with a detailed analysis of three important cases: smooth varieties, local complete intersection subschemes and coherent sheaves.
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.
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