On the reduced space of multiplicative multivectors

Abstract

A strict Lie $2$-algebra $\Gamma ({\wedge }^{\bullet }A)\stackrel{T}{\to }{\mathfrak{X}}_{\mathrm{mult}}^{\bullet }(\mathcal{G})$ 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 $1⩽k⩽\mathrm{rank}A$. 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 $A$.

Structure optimized, references added and typos removed

References in corpus (4)

• Linear and multiplicative 2-forms
• Lie theory of multiplicative tensors
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Coadjoint orbits of Lie groupoids