The weak Lie 2-algebra of multiplicative forms on a quasi-Poisson groupoid

TL;DR

This paper constructs a new weak Lie 2-algebra structure from multiplicative forms on quasi-Poisson groupoids, linking it to multivector fields and extending Lie 2-algebra theory in quasi-Poisson geometry.

Contribution

It introduces a graded weak Lie 2-algebra from multiplicative forms on quasi-Poisson groupoids and establishes a morphism to compare with multivector field-based Lie 2-algebras.

Findings

Constructed a weak Lie 2-algebra from multiplicative forms

Established a morphism linking multiplicative forms and multivector fields

Explicitly described the structure for IM 1-forms in quasi-Lie bialgebroids

Abstract

Berwick-Evens and Lerman recently showed that the category of vector fields on a geometric stack has the structure of a Lie $2$-algebra. Motivated by this work, we present a construction of graded weak Lie $2$-algebras associated with quasi-Poisson groupoids based on the space of multiplicative forms on the groupoid and differential forms on the base manifold. We also establish a morphism between the Lie $2$-algebra of multiplicative multivector fields and the weak Lie $2$-algebra of multiplicative forms, allowing us to compare and relate different aspects of Lie $2$-algebra theory within the context of quasi-Poisson geometry. As an infinitesimal analogy, we explicitly determine the associated weak Lie $2$-algebra structure of IM $1$-forms along with differential $1$-forms on the base manifold for any quasi-Lie bialgebroid.