Shifted derived Poisson manifolds associated with Lie pairs

Abstract

We study the shifted analogue of the "Lie--Poisson" construction for $L_{\infty }$ algebroids and we prove that any $L_{\infty }$ algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair $(L,A)$, the space $\mathrm{tot}{\Omega }_A^{\bullet }({\Lambda }^{\bullet }(L/A))$ admits a degree $(+1)$ derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley--Eilenberg differential $d_A^{\mathrm{Bott}}:{\Omega }_A^{\bullet }({\Lambda }^{\bullet }(L/A))\to {\Omega }_A^{\bullet +1}({\Lambda }^{\bullet }(L/A))$ as unary $L_{\infty }$ bracket. This degree $(+1)$ derived Poisson algebra structure on $\mathrm{tot}{\Omega }_A^{\bullet }({\Lambda }^{\bullet }(L/A))$ is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley--Eilenberg hypercohomology $\mathbb{H}({\Omega }_A^{\bullet }({\Lambda }^{\bullet }(L/A)),d_A^{\mathrm{Bott}})$ admits a canonical Gerstenhaber algebra structure.

37 pages

References in corpus (4)

• Deformations of coisotropic submanifolds and strong homotopy Lie algebroids
• Shifted Poisson and symplectic structures on derived N-stacks
• Formal Deformations of Dirac Structures
• Formality for g-manifolds

Cited by in corpus (10)

• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Hopf algebras arising from dg manifolds
• Shifted coisotropic structures for differentiable stacks
• Quantization of (-1)-Shifted Derived Poisson Manifolds
• Cohomology of hemistrict Lie 2-algebras
• $A_{\infty }$-Algebras from Lie Pairs
• The standard cohomology of regular Courant algebroids
• Hyperkähler structures on leaves of hyper-Lie Poisson manifolds
• Supersymmetric Poisson and Poisson-supersymmetric sigma models