$L_{\infty }$-morphisms between twisted Courant $r$-Lie algebras and untwisted Courant $(r+1)$-Lie algebroids

Abstract

In "Lie infinity algebras and higher analogues of Dirac structures and Courant algebroids" [arXiv:1003.1004], Marco Zambon constructs an $L_{\infty }$-algebra associated with any higher standard or twisted Courant algebroid (also known as a Vinogradov algebroid), and exhibits an explicit $L_{\infty }$-morphism from the Lie algebra associated with a standard Lie algebroid twisted by a closed 2-form to the Lie-2 algebra of the standard Courant algebroid. He poses the question of whether analogous canonical $L_{\infty }$-morphisms exist in higher degrees -- namely, for any standard higher Courant algebroid twisted by a closed $(r+1)$-form. We adfirmatively answer this question, presenting a general framework that naturally yields such canonical $L_{\infty }$-morphisms for arbitrary $r$, while at the same time clarifying the geometrical and homotopical structures underlying the construction. We also show how this framework accommodates the canonical morphism between Roger's observable $L_{\infty }$-algebra of a pre-$r$-plectic manifold and the higher Courant algebra described by Zambon and one of the authors in "Observables on multisymplectic manifolds and higher Courant algebroids" [arXiv:2209.05836].

30 pages