Factorization of Isomorphisms of $(H,\theta )$-twisted Lie algebroids

Abstract

We study the isomorphism groupoid $\mathcal{T}(M)$ of $\theta$-almost twisted Poisson ($\theta$-atP) structures on a smooth manifold $M$, focusing on the internal structure of its morphisms. A morphism in $\mathcal{T}(M)$ is a $C^{\infty }(M)$-linear isomorphism $Φ:\gO^1(M)\to\gO^1(M)$ that simultaneously intertwines the anchor maps and the $(H,\theta )$-twisted Koszul brackets associated with two $\theta$-atP structures. Every such morphism induces a canonical isomorphism in $\theta$-atP cohomology. We define a classifying functor $$\Delta :\mathrm{Mor}(\mathcal{T}(M))⟶(Z_{\mathrm{dR}}^1(M),+),\Delta (\Phi )={\theta }^{\prime }-\theta ,$$ which is additive under composition and partitions the morphisms into two complementary families: the sub-groupoid ${\mathcal{T}}_{\mathrm{fix}}=\ker \Delta$ of isomorphisms preserving $\theta$, and the family ${\mathcal{T}}_{\mathrm{mod}}$ of isomorphisms shifting $\theta$. We describe each element in this partition.

15 pages