Representations of Homotopy Lie-Rinehart Algebras

Abstract

I propose a definition of left/right connection along a strong homotopy Lie-Rinehart algebra. This allows me to generalize simultaneously representations up to homotopy of Lie algebroids and actions of strong homotopy Lie algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie-Rinehart connections.

v2: 29 pages, 3 tables, title changed, examples and references added. v3: 32 pages, examples added. Typos and a (minor) mistake corrected. v4: typos corrected, accepted for publication on Math. Proc. Camb. Phil. Soc., Comments still welcome!

References in corpus (11)

• On the category of Lie n-algebroids
• Supergroupoids, double structures, and equivariant cohomology
• Deformations of coisotropic submanifolds and strong homotopy Lie algebroids
• Lie-Rinehart algebras, descent, and quantization
• Characteristic classes associated to Q-bundles
• L-infinity algebra actions
• On the Strong Homotopy Associative Algebra of a Foliation
• Homotopy BV algebras in Poisson geometry
• Multi derivation Maurer-Cartan algebras and sh-Lie-Rinehart algebras
• Poisson cohomology and quantization
• BFV-complex and higher homotopy structures

Cited by in corpus (21)

• Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
• Shifted derived Poisson manifolds associated with Lie pairs
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Vector Bundle Valued Differential Forms on $\mathbb{N}Q$-manifolds
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Multi derivation Maurer-Cartan algebras and sh-Lie-Rinehart algebras
• Jacobi bundles and the BFV-complex
• Deformations and homotopy theory of Rota-Baxter algebras of any weight
• Deformation of Hom-Lie-Rinehart algebras
• Higher Koszul brackets on the cotangent complex
• Polyvector fields and polydifferential operators associated with Lie pairs
• Atiyah classes of strongly homotopy Lie pairs
• Calculus of multilinear differential operators, operator $L_{\infty }$-algebras and ${IBL}_{\infty }$-algebras
• Kirillov structures up to homotopy
• On classification and deformations of Lie-Rinehart superalgebras
• $L_{\infty }$-actions of Lie algebroids
• Lie algebroids as $L_{\infty }$ spaces
• Quantization of (-1)-Shifted Derived Poisson Manifolds
• $A_{\infty }$-Algebras from Lie Pairs
• Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid
• Cumulants, Koszul brackets and homological perturbation theory for commutative $B{V}_{\infty }$ and $IB{L}_{\infty }$ algebras