On the reduced space of multiplicative multivectors

TL;DR

This paper investigates the structure of multiplicative multivectors on Lie groupoids, introduces a reduced space invariant under Morita equivalence, and explores its relation to Lie algebroid cohomology and jet groupoids.

Contribution

It provides a canonical decomposition of multiplicative multivectors, establishes a relation between the reduced space and cohomology, and links the multiplicative and differential reduced spaces via infinitesimal analysis.

Findings

Decomposition formula for multiplicative multivectors.

Relation between reduced space and jet groupoid cohomology.

Connection between multiplicative and differential reduced spaces through the Van Est map.

Abstract

A strict Lie $2$-algebra $\Gamma(\wedge^\bullet A) \stackrel{T}{\rightarrow} \mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ is associated with any Lie groupoid $\mathcal{G}$. Here, $\Gamma(\wedge^\bullet A)$ is the Schouten algebra of the tangent Lie algebroid $A$ of $\mathcal{G}$ and $\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ is the space of multiplicative multivectors on $\mathcal{G}$. The quotient ${R}_{\mathrm{mult}}^\bullet:=\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})/\mathrm{Img} T$, a Morita invariant of $\mathcal{G}$, is called the reduced space of multiplicative multivectors. We prove a canonical decomposition formula of elements in $\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ and establish a key relation between ${R}_{\mathrm{mult}}^k$ and the cohomology $\mathrm{H} ^1(\mathfrak{J} \mathcal{G},\wedge^k A)$ where $\mathfrak{J} \mathcal{G}$ is the jet groupoid of $\mathcal{G}$ and $1\leqslant k\leqslant \mathrm{rank} A$. We also study ${R}_{\mathrm{diff}}^\bullet $, the reduced space of Lie algebroid differentials on $A$. By taking infinitesimals, $\bar{\delta}: $ ${R}_{\mathrm{mult}}^\bullet $ $\to $ ${R}_{\mathrm{diff}}^\bullet $, the two reduced spaces are related. We find that the kernel of $\bar{\delta}$ is isomorphic to the kernel of the Van Est map $\mathrm{H}^1(\mathcal{G},\wedge^k \ker\rho)\to \mathrm{H}^1(A,\wedge^k \ker\rho)$, where $\rho$ is the anchor of $A$.