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Title

Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras

Authors 1

Affiliations

  1. Université des Sciences et Technologies de Lille, UFR de Mathématiques, F-59 655 Villeneuve d'Ascq Cedex, France

Abstract

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms of certain associated differential (bi)graded Lie and G(erstenhaber)-algebras; in particular the G-algebra arising from an almost complex structure is a (strict) d(ifferential) G-algebra iff the almost complex structure is integrable. Such G-algebras, endowed with a generator turning them into a B(atalin-)V(ilkovisky)-algebra, occur on the B-side of the mirror conjecture. We generalize a result of Koszul to those dG-algebras which arise from twilled LR-algebras. A special case thereof explains the relationship between holomorphic volume forms and exact generators for the corresponding dG-algebra and thus yields in particular a conceptual proof of the Tian-Todorov lemma. We give a differential homological algebra interpretation for twilled LR-algebras and by means of it we elucidate the notion of a generator in terms of homological duality for differential graded LR-algebras.

Keywords

ENG
differential graded Lie algebra, twilled Lie-Rinehart algebra, Lie-Rinehart algebra, Batalin-Vilkovisky algebra, Gerstenhaber algebra, mirror conjecture, Calabi-Yau manifold, Lie bialgebra

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Banach Center Publications
2000/51/1
Export to PBN, Opens in new tab
Pages:
87-102
Main language of publication
English
Published
2000
Exact and natural sciences