Higher homotopies and Maurer-Cartan algebras: Quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras

Abstract

Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra-the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra-and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the "space of leaves" and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including as well the Hodge-de Rham spectral sequence.2000 Mathematics Subject Classification. Primary 17B65 17B66 53C12 57R30; secondary 17B55 17B56 17B81 18G40 53C15 55R20 70H45 .

Simultaneous deformations of a Lie algebroid and its Lie subalgebroid

“…Remark 4.23. The L ∞ -algebra in Corollary 4.21 has been first discussed by J. Huebschmann in [34]. Moreover, the L ∞ -algebra in Theorem 4.20 has been extensively described by L. Vitagliano in [35].…”

Representations of Homotopy Lie–Rinehart Algebras

“…LR algebras have analogues up to homotopy, which are known as strong homotopy (SH) LR algebras [21,19,46]. SH LR algebras were introduced by Kjeseth in [21], under the name homotopy Lie-Rinehart pairs, and appear naturally in different geometric contexts, e.g., BRST-BV formalism [22], foliation theory [19,46,47], complex geometry [52], action of L ∞ algebras on graded manifolds (see Section 4.2 of this paper). Kjeseth's definition of a homotopy Lie-Rinehart pair makes use of the coalgebra concepts of subordinate derivation sources, and resting coderivations.…”

“…Nonetheless, SH LR algebras are already enough for many applications. Namely, they appear in the BRST-BV formalism [22], foliation theory [19,46,47], complex geometry [52]. Moreover, as shown in Section 3 (see also [6]), one can associate a SH LR algebra to a homotopy Poisson algebra (in the sense of Cattaneo-Felder [7]).…”

Extensions of Lie-Rinehart algebras and cotangent bundle reduction

“…We mention in passing that Lie-Rinehart algebras yield substantial new insight elsewhere, see [12]- [19] and, in particular, are a crucial tool in a systematic approach to singular Kähler reduction [20]- [22]. In particular Lie-Rinehart algebras provide the appropriate algebraic tool for a satisfactory description of the behavior of the theory at singularities.…”