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Actions of Lie 2-algebras and comomentum maps

Philippe Bonneau, Véronique Chloup-Arnould, Angela Gammella, Tilmann Wurzbacher

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, $ω$), the authors consider a Lie Infinity-algebra L$\infty$ (M, $ω$), 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, $ω$) when (M, $ω$) is 2-plectic. Following ideas of N.L. Delgado [Del18], we introduce the Lie 2-algebra D^2 (M, $ω$), which generalises the Lie 2-algebra L^2 (M, $ω$) and its extension containing Hamiltonian pairs. Given a two-plectic manifold (M, $ω$) 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, $ω$). In an appendix, we discuss very explicitly numerous examples, classified according to their algebraic properties.

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, ), the authors consider a Lie Infinity-algebra L (M, ), 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, ) when (M, ) is 2-plectic. Following ideas of N.L. Delgado [Del18], we introduce the Lie 2-algebra D^2 (M, ), which generalises the Lie 2-algebra L^2 (M, ) and its extension containing Hamiltonian pairs. Given a two-plectic manifold (M, ) and a Lie 2-algebra g_1 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 g_0 to the extension of the Lie 2-algebra D^2 (M, ). In an appendix, we discuss very explicitly numerous examples, classified according to their algebraic properties.
Paper Structure (29 sections, 29 theorems, 78 equations)

This paper contains 29 sections, 29 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Basic results on Lie 2-algebras
  3. -algebras
  4. Lie 2-algebras and their morphisms
  5. Skeletal Lie 2-algebras
  6. Strict Lie 2-algebras
  7. Lie 2-algebras and 3-cocycles
  8. Multi Cartan calculus
  9. Multicontraction operator and multi Lie derivative
  10. Multi Cartan commutation rules
  11. Action of a Lie 2-algebra (2-action)
  12. Lie 2-algebras of observables
  13. Hamiltonian vector fields and forms
  14. Rogers's Lie 2-algebra of observables
  15. Delgado's Lie 2-algebra of observables
  16. ...and 14 more sections

Key Result

Proposition 2.4

A Lie 2-algebra is a graded vector space $L=L_{-1}\oplus L_0$, with a collection of three multilinear maps $(l_1,l_2,l_3)$, where $l_1:L_{-1}\mapsto L_0$, $l_2$ can be decomposed in its "pure" part $l_2^p:L_0\times L_0\mapsto L_0$, antisymmetric and its "mixed" part $l_2^m :L_{-1}\times L_0\mapsto L

Theorems & Definitions (121)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 111 more