Multisymplectic Structures and Higher Momentum Maps

TL;DR

This paper reorganizes results from a master's thesis on multisymplectic structures, constructing high L-infinity algebras from Gerstenhaber algebras, transferring L-infinity structures via cochain homotopies, and establishing their equivalences, motivated by multi-symplectic geometry.

Contribution

It introduces new methods to construct and relate L-infinity algebras from Gerstenhaber algebras and cochain complexes, advancing the understanding of multisymplectic structures.

Findings

Constructed high L-infinity algebra from any Gerstenhaber algebra

Transferred L-infinity structure via cochain null-homotopy

Established equivalence between cochain-homotopies and L-infinity algebra lifts

Abstract

This document is a reorganization of the results on the Master Thesis of the same title written by the author under the supervision of Dr. Christian Blohmann at the University of Bonn in 2014. There are three main results in this document. The first one is a construction of a high L-infinity algebra out of any Gerstenhaber algebra. The second one uses a cochain null-homotopy from the Chevalley-Eilenberg complex associated to a graded Lie algebra to any cochain complex D to induce (transfer) the L-infinity algebra structure from the Lie alegebra to Q(D), a resolution of D. Lastly, the third one establishes the equivalence between cochain-homotopies to Q(D) and L-infinity algebra lifts satisfying certain properties. All the three results are motivated from the study of multi-symplectic structures. In particular, the definition of comomentum maps in multi-symplectic manifolds.