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 $\Phi:\gO^1(M)\to\gO^1(M)$ 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 $$ \Delta : \mathrm{Mor}(\mathcal{T}(M)) \longrightarrow (Z^1_{\mathrm{dR}}(M) ,+), \qquad \Delta(\Phi)=\theta' - \theta, $$ 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.