Actions of Lie 2-algebras and comomentum maps

TL;DR

This paper introduces the concept of Lie 2-algebra actions on manifolds, generalizes existing algebraic structures for 2-plectic manifolds, and defines comomentum maps as lifts of these actions, with detailed examples.

Contribution

It defines 2-actions of Lie 2-algebras on manifolds, introduces a generalized Lie 2-algebra D^2 (M, ω), and formalizes comomentum maps as morphisms, extending prior frameworks.

Findings

Defined 2-actions of Lie 2-algebras on manifolds.

Introduced the generalized Lie 2-algebra D^2 (M, ω).

Provided explicit examples classified by algebraic properties.

Abstract

In this paper we introduce the notion of a 2-action of a Lie 2-algebra on an arbitrary manifold M. Furthermore, in [Rog12], given a n-plectic manifold (M, $\omega$), the authors consider a Lie Infinity-algebra L$\infty$ (M, $\omega$), which is a higher analogue of the Poisson algebra of observables associated to a symplectic manifold. This Lie Infinity-algebra reduces to a Lie 2-algebra L^2 (M, $\omega$) when (M, $\omega$) is 2-plectic. Following ideas of N.L. Delgado [Del18], we introduce the Lie 2-algebra D^2 (M, $\omega$), which generalises the Lie 2-algebra L^2 (M, $\omega$) and its extension containing Hamiltonian pairs. Given a two-plectic manifold (M, $\omega$) and a Lie 2-algebra g_1 $\oplus$ g_0 acting on M we define a comomentum map as a lift of the action, i.e., as a Lie 2-algebra morphism from g_1 $\oplus$ g_0 to the extension of the Lie 2-algebra D^2 (M, $\omega$). In an appendix, we discuss very explicitly numerous examples, classified according to their algebraic properties.