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

Johannes Huebschmann

Banach Center Publications (2000)

  • Volume: 51, Issue: 1, page 87-102
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Huebschmann, Johannes. "Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras." Banach Center Publications 51.1 (2000): 87-102. <http://eudml.org/doc/209047>.

@article{Huebschmann2000,
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.},
author = {Huebschmann, Johannes},
journal = {Banach Center Publications},
keywords = {differential graded Lie algebra; twilled Lie-Rinehart algebra; Lie-Rinehart algebra; Batalin-Vilkovisky algebra; Gerstenhaber algebra; mirror conjecture; Calabi-Yau manifold; Lie bialgebra; almost complex manifold},
language = {eng},
number = {1},
pages = {87-102},
title = {Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras},
url = {http://eudml.org/doc/209047},
volume = {51},
year = {2000},
}

TY - JOUR
AU - Huebschmann, Johannes
TI - Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras
JO - Banach Center Publications
PY - 2000
VL - 51
IS - 1
SP - 87
EP - 102
AB - 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.
LA - eng
KW - differential graded Lie algebra; twilled Lie-Rinehart algebra; Lie-Rinehart algebra; Batalin-Vilkovisky algebra; Gerstenhaber algebra; mirror conjecture; Calabi-Yau manifold; Lie bialgebra; almost complex manifold
UR - http://eudml.org/doc/209047
ER -

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