Narrow Results

Let $\omega$ be a closed, non-degenerate differential form of arbitrary degree. Associated to it there are an $L_{\infty }$-algebra of observables, and an $L_{\infty }$-algebra of sections of the higher Courant algebroid twisted by $\omega$. Our main result is the existence of an $L_{\infty }$-embedding of the former into the latter. We display explicit formulae for the embedding, involving the Bernoulli numbers. When $\omega$ is an integral symplectic form, the embedding can be realized geometrically via the prequantization construction, and when $\omega$ is a 3-form the embedding was found by Rogers in 2010. Further, in the presence of homotopy moment maps, we show that the embedding is compatible with gauge transformations.

In this paper, first we give the notion of a crossed homomorphism on a $3$-Lie algebra with respect to an action on another $3$-Lie algebra, and characterize it using a homomorphism from a 3-Lie algebra to the semidirect product 3-Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota–Baxter operators of weight $1$ on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on $3$-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an $L_{\infty }$-algebra whose Maurer–Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted $L_{\infty }$-algebra that controls deformations of a given crossed homomorphism on $3$-Lie algebras.

We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the $L_{\infty }$-algebras governing these deformation problems.

First we show that, associated to any Poisson vector field $E$ on a Poisson manifold $(M,\pi )$, there is a canonical Lie algebroid structure on the first jet bundle $J^{1}M$ which, depends only on the cohomology class of $E$. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and $L\infty$-algebras.

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L_∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, with such an SH Lie pair (L, A) and any A-module E, there is associated a canonical cohomology class, the Atiyah class [α^E], which generalizes the earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [α^L/A] induces a graded Lie algebra structure on ${\text{H}}_{\text{CE}}^{•}(A,L/A[-2])$, and the Atiyah class [α^E] of any A-module E induces a Lie algebra module structure on ${\text{H}}_{\text{CE}}^{•}(A,E)$. Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.