Eulerian idempotent, pre-Lie logarithm and combinatorics of trees

E-print/Working paper (2017)

The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra g, the Eulerian idempotent is a canonical projection from the enveloping algebra U(g) to g. The ... [more ▼]

The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra g, the Eulerian idempotent is a canonical projection from the enveloping algebra U(g) to 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. [less ▲]

Equivalences of coisotropic submanifolds

in Journal of Symplectic Geometry (2017), 15(1), 107-149

We study the role that Hamiltonian and symplectic diffeomorphisms play in the deformation problem of coisotropic submanifolds. We prove that the action by Hamiltonian diffeomorphisms corresponds to the ... [more ▼]

We study the role that Hamiltonian and symplectic diffeomorphisms play in the deformation problem of coisotropic submanifolds. We prove that the action by Hamiltonian diffeomorphisms corresponds to the gauge-action of the L-infinity-algebra of Oh and Park. Moreover we introduce the notion of extended gauge-equivalence and show that in the case of Oh and Park's L-infinity-algebra one recovers the action of symplectic isotopies on coisotropic submanifolds. Finally, we consider the transversally integrable case in detail. [less ▲]

How to discretize the differential forms on the interval

E-print/Working paper (2016)

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 ... [more ▼]

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-∞, algebra structure. Our main interest lies in a natural `discretization' C-∞ quasi-isomorphism φ from differential forms to Whitney forms. We establish a uniqueness result that implies that φ coincides with the morphism from homotopy transfer, and obtain several explicit formulas for φ, all of which are related to the Magnus expansion. In particular, we recover combinatorial formulas for the Magnus expansion due to Mielnik and Plebanski. [less ▲]

Flat Z-graded connections and loop spaces

E-print/Working paper (2015)

The pull back of a flat bundle E→X along the evaluation map π:LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally ... [more ▼]

The pull back of a flat bundle E→X along the evaluation map π:LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally generalizes to flat Z-graded connections on X. Our main result is that the restriction of this holonomy automorphism to the based loop space ΩX of X provides an A-infinity quasi-equivalence between the dg category of flat Z-graded connections on X and the dg category of representations of C(ΩX), the dg algebra of singular chains on ΩX. [less ▲]

Holonomies for connections with values in L_infty algebras

in Homology, Homotopy and Applications (2014), 16(1), 89-118

Given a flat connection on a manifold M with values in a filtered L-infinity-algebra g, we construct a morphism, generalizing the holonomies of flat connections with values in Lie algebras. The ... [more ▼]

Given a flat connection on a manifold M with values in a filtered L-infinity-algebra g, we construct a morphism, generalizing the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheim's A-infinity version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space Rd, and to generalizations of the holonomy representations of braid groups. [less ▲]

A Survey on Stability and Rigidity Results for Lie algebras

in Indagationes Mathematicae (2014), 25(5), 957-976

We give simple and unified proofs of the known stability and rigidity results for Lie algebras, Lie subalgebras and Lie algebra homomorphisms. Moreover, we investigate when a Lie algebra homomorphism is ... [more ▼]

We give simple and unified proofs of the known stability and rigidity results for Lie algebras, Lie subalgebras and Lie algebra homomorphisms. Moreover, we investigate when a Lie algebra homomorphism is stable under all automorphisms of the codomain (including outer automorphisms). [less ▲]

The A_infty de Rham theorem and integration of representations up to homotopy

in International Mathematics Research Notices (2013), 2013(16), 3790-3855

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A-infinity functor from the representations up to homotopy of a Lie algebroid A to those of its ... [more ▼]

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A-infinity functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an A-infinity version of de Rham's theorem due to Gugenheim. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa and Block-Smith, to the case of certain differential graded manifolds. [less ▲]

Lie theory for representations up to homotopy

Presentation (2011, June 13)

Talk on my joint work with Camilo Arias Abad on the Lie theory of representation up to homotopy.

Moduli of coisotropic Sections and the BFV-complex

in Asian Journal of Mathematics (2011), 15(1), 71-100

We consider the local deformation problem of coisotropic submanifolds inside symplectic or Poisson manifolds. To this end the groupoid of coisotropic sections (with respect to some tubular neighbourhood ... [more ▼]

We consider the local deformation problem of coisotropic submanifolds inside symplectic or Poisson manifolds. To this end the groupoid of coisotropic sections (with respect to some tubular neighbourhood) is introduced. Although the geometric content of this groupoid is evident, it is usually a very intricate object. We provide a description of the groupoid of coisotropic sections in terms of a differential graded Poisson algebra, called the BFV-complex. This description is achieved by constructing a groupoid from the BFVcomplex and a surjective morphism from this groupoid to the groupoid of coisotropic sections. The kernel of this morphism can be easily chracterized. As a corollary we obtain an isomorphism between the moduli space of coisotropic sections and the moduli space of geometric Maurer–Cartan elements of the BFV-complex. In turn, this also sheds new light on the geometric content of the BFV-complex. [less ▲]

BFV-complex and higher homotopy structures

in Communications in Mathematical Physics (2009), 286(2), 399443

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold ... [more ▼]

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L-infinity quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead. [less ▲]