Basic sections of LA-groupoids
Antonio Maglio, Fabricio Valencia
TL;DR
This work addresses modeling the space of sections for stacky Lie algebroids presented by LA-groupoids with injective core-anchor maps. It introduces the notion of basic sections on $E/\partial(C)$ and proves that they form a Morita-invariant Lie algebra, Morita-equivalent to the traditional Lie 2-algebra of multiplicative sections via a canonical isomorphism $\Psi: \Gamma_{\operatorname{mult}}(V)/\operatorname{im}(\delta) \to \Gamma_{\operatorname{bas}}(V)$. This yields a simpler, Morita-invariant model for stacky sections that remains equivalent to the multiplicative-sections model, extending the idea of basic vector fields from foliation groupoids to general LA-groupoids. The paper develops the theory through Bott representations and Atiyah algebroids, and illustrates it with examples in basic vector fields, basic derivations, and basic forms/jets on Poisson and Jacobi groupoids, with connections to $0$-shifted geometry and potential reduction procedures for structured groupoids.
Abstract
We define the notion of basic section of an LA-groupoid whose core-anchor map is injective. Such a notion turns out to be Morita invariant, so that it provides a simpler model for the sections of the stacky Lie algebroids presented by such LA-groupoids, yet equivalent to the well-known model provided by their multiplicative sections.