Glanon groupoids

TL;DR

This paper introduces Glanon groupoids, a unified framework for various complex and symplectic structures on Lie groupoids, and establishes their infinitesimal counterparts and integration theorems.

Contribution

It defines Glanon groupoids, studies their Lie algebroids, and proves a bijection with their infinitesimal structures, extending integration results to holomorphic Poisson groupoids.

Findings

Established a bijection between Glanon Lie algebroids and source-connected Glanon groupoids.

Unified symplectic, holomorphic, and Poisson structures within a single framework.

Proved the integration theorem for holomorphic Poisson groupoids.

Abstract

We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that there is a bijection between Glanon Lie algebroids and source-simply connected and source-connected Glanon groupoids. As a consequence, we recover various integration theorem and obtain the integration theorem for holomorphic Poisson groupoids.