Formality of Kapranov's brackets in K\"ahler geometry via pre-Lie deformation theory

We recover some recent results by Dotsenko, Shadrin and Vallette on the Deligne groupoid of a pre-Lie algebra, showing that they follow naturally by a pre-Lie variant of the PBW Theorem. As an application, we show that Kapranov's $L_\infty$ algebra structure on the Dolbeault complex of a K\"ahler manifold is homotopy abelian and independent on the choice of K\"ahler metric up to an $L_\infty$ isomorphism, by making the trivializing homotopy and the $L_\infty$ isomorphism explicit.Comment: This is a new version of the old "Formality of Kapranov's brackets on pre-Lie algebras". To appear in Int. Math. Res. No

How to discretize the differential forms on the interval

We provide explicit quasi-isomorphisms between the following three algebraic structures associated to the unit interval: i) the commutative dg algebra of differential forms, ii) the non-commutative dg algebra of simplicial cochains and iii) the Whitney forms, equipped with a homotopy commutative and homotopy associative, i.e. $C_\infty$, algebra structure. Our main interest lies in a natural `discretization' $C_\infty$ quasi-isomorphism $\varphi$ from differential forms to Whitney forms. We establish a uniqueness result that implies that $\varphi$ coincides with the morphism from homotopy transfer, and obtain several explicit formulas for $\varphi$, all of which are related to the Magnus expansion. In particular, we recover combinatorial formulas for the Magnus expansion due to Mielnik and Pleba\'nski.Comment: 29 pages, extended abstract, typos fixe

Algebraic models of local period maps and Yukawa algebras

We describe some L-infinity model for the local period map of a compact Kaehler manifold. Applications include the study of deformations with associated variation of Hodge structure constrained by certain closed strata of the Grassmannian of the de Rham cohomology. As a byproduct we obtain an interpretation in the framework of deformation theory of the Yukawa coupling.Comment: to appear in Letters in Mathematical Physic

Eulerian idempotent, pre-Lie logarithm and combinatorics of trees

The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a canonical projection from the enveloping algebra $U(\mathfrak{g})$ to $\mathfrak{g}$. The Baker-Campbell-Hausdorff product and the Magnus expansion can both be expressed in terms of the Eulerian idempotent, which makes it interesting to establish explicit formulas for the latter. We show how to reduce the computation of the Eulerian idempotent to the computation of a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The problem of finding formulas for the pre-Lie logarithm, which is interesting in its own right -- being related to operad theory, numerical analysis and renormalization -- is addressed using techniques inspired by umbral calculus. As a consequence of our analysis, we find formulas both for the Eulerian idempotent and the pre-Lie logarithm in terms of the combinatorics of trees.Comment: Preliminary version. Comments are welcome

Nonabelian higher derived brackets

Let M be a graded Lie algebra, together with graded Lie subalgebras L and A such that as a graded space M is the direct sum of L and A, and A is abelian. Let D be a degree one derivation of M squaring to zero and sending L into itself, then Voronov's construction of higher derived brackets associates to D a L-infinity structure on A[-1]. It is known, and it follows from the results of this paper, that the resulting L-infinity algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras i : (L,D,[, ]) -> (M,D,[, ]). We prove this fact using homotopical transfer of L-infinity structures, in this way we also extend Voronov's construction when the assumption A abelian is dropped: the resulting formulas involve Bernoulli numbers. In the last section we consider some example and some further application.Comment: v3: several changes in the expositio

Shifted derived Poisson manifolds associated with Lie pairs

We study the shifted analogue of the "Lie--Poisson" construction for $L_\infty$ algebroids and we prove that any $L_\infty$ algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair $(L,A)$, the space $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ admits a degree $(+1)$ derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley--Eilenberg differential $d_A^{\operatorname{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\to \Omega^{\bullet +1}_A(\Lambda^\bullet(L/A))$ as unary $L_\infty$ bracket. This degree $(+1)$ derived Poisson algebra structure on $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley--Eilenberg hypercohomology $\mathbb{H}(\Omega^{\bullet}_A(\Lambda^\bullet(L/A)),d_A^{\operatorname{Bott}})$ admits a canonical Gerstenhaber algebra structure.Comment: 37 page