2017/12/31 by Ruggero Bandiera, Zhuo Chen, Mathieu Stiénon +1
Mathematics · #Advanced Topics in Algebra #Algebraic number #Algebraic structure #Algebraic structures and combinatorial models #Degree (music) #Exterior algebra #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Lie algebra #Lie algebroid #Poisson algebra #Poisson distribution #Poisson manifold #math.DG #math.QA
paper · pdf · doi:10.1007/s00220-019-03457-w
published as Comm. Math. Phys. 375.3 (2020), pp. 1717-1760 · 37 pages
openalex created_date 2017/12/22 · arxiv created 2018/12/19 · openalex publication_date 2019/06/10 · arxiv updated 2021/03/10 · openalex updated_date 2026/08/06
We study the shifted analogue of the "Lie--Poisson" construction for L_∞ algebroids and we prove that any L_∞ 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 totΩ\bulletA(Λ^\bullet(L/A)) admits a degree (+1) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley--Eilenberg differential dABott:Ω\bulletA(Λ^\bullet(L/A))→ Ω\bullet +1A(Λ^\bullet(L/A)) as unary L_∞ bracket. This degree (+1) derived Poisson algebra structure on totΩ\bulletA(Λ^\bullet(L/A)) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley--Eilenberg hypercohomology ℍ(Ω\bulletA(Λ^\bullet(L/A)),dABott) admits a canonical Gerstenhaber algebra structure.