Actions of Lie 2-algebras and comomentum maps

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.