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Eulerian idempotent, pre-Lie logarithm and combinatorics of trees ; Schätz, Florian 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 ▲] Detailed reference viewed: 59 (0 UL)Deformations of pre-symplectic structures and the Koszul L-infty-algebra Schätz, Florian ; E-print/Working paper (2017) We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms ... [more ▼] We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_\infty$-algebra, which we call the Koszul $L_\infty$-algebra. This $L_\infty$-algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold, and its proper geometric understanding relies on Dirac geometry. In addition, we show that a quotient of the Koszul $L_{\infty}$-algebra is isomorphic to the $L_\infty$-algebra which controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed. [less ▲] Detailed reference viewed: 80 (0 UL)Equivalences of coisotropic submanifolds Schatz, Florian ; 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 ▲] Detailed reference viewed: 113 (2 UL)Formal connections for families of star products ; ; Schatz, Florian in Communications in Mathematical Physics (2016), 342(2), 739-768 We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points ... [more ▼] We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection defined in. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled by the derivations of the star products. Moreover we show that if the parameter space has trivial first cohomology group any two flat formal connections are related by an automorphism of the family of star products. [less ▲] Detailed reference viewed: 126 (4 UL)How to discretize the differential forms on the interval ; Schätz, Florian 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 ▲] Detailed reference viewed: 80 (2 UL)Higher holonomies: comparing two constructions ; Schatz, Florian in Differential Geometry and its Applications (2015), 40 We compare two different constructions of higher dimensional parallel transport. On the one hand, there is the two dimensional parallel transport associated to 2-connections on 2-bundles studied by Baez ... [more ▼] We compare two different constructions of higher dimensional parallel transport. On the one hand, there is the two dimensional parallel transport associated to 2-connections on 2-bundles studied by Baez-Schreiber, Faria Martins-Picken and Schreiber-Waldorf. On the other hand, there are the higher holonomies associated to flat superconnections as studied by Igusa, Block-Smith and Arias Abad-Schätz. We first explain how by truncating the latter construction one obtains examples of the former. Then we prove that the 2-dimensional holonomies provided by the two approaches coincide. [less ▲] Detailed reference viewed: 131 (5 UL)Flat Z-graded connections and loop spaces ; Schatz, Florian 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 ▲] Detailed reference viewed: 136 (8 UL)Reidemeister torsion for flat superconnections ; Schatz, Florian in Journal of Homotopy and Related Structures (2014), 9(2), 579-606 We use higher parallel transport -- more precisely, the integration A-infinity-functor constructed in -to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger ... [more ▼] We use higher parallel transport -- more precisely, the integration A-infinity-functor constructed in -to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger-Müller theorem, namely that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu. [less ▲] Detailed reference viewed: 103 (2 UL)Holonomies for connections with values in L_infty algebras ; Schatz, Florian 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 ▲] Detailed reference viewed: 125 (2 UL)Highlights 2013 Schatz, Florian Article for general public (2014) General overview of my research interests concerning generalized notions of holonomy. Detailed reference viewed: 57 (3 UL)A Survey on Stability and Rigidity Results for Lie algebras ; ; Schatz, Florian 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 ▲] Detailed reference viewed: 132 (4 UL)Deformations of coisotropic submanifolds for fibrewise entire Poisson structures Schatz, Florian ; in Letters in Mathematical Physics (2013), 103(7), 777-791 We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the L-infinity-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo ... [more ▼] We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the L-infinity-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case, we recover results previously obtained by Oh-Park. Moreover we consider the extended deformation problem and prove its obstructedness. [less ▲] Detailed reference viewed: 99 (1 UL)The A_infty de Rham theorem and integration of representations up to homotopy ; Schatz, Florian 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 ▲] Detailed reference viewed: 71 (2 UL)Introduction to supergeometry ; Schatz, Florian in Reviews in Mathematical Physics (2012), 23(6), 669-690 These notes are based on a series of lectures given by the first author at the school of `Poisson 2010', held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super- and graded ... [more ▼] These notes are based on a series of lectures given by the first author at the school of `Poisson 2010', held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super- and graded manifolds, cohomological vector fields, graded symplectic structures, reduction and the AKSZ-formalism. [less ▲] Detailed reference viewed: 115 (1 UL)Lie theory for representations up to homotopy Schatz, Florian Presentation (2011, June 13) Talk on my joint work with Camilo Arias Abad on the Lie theory of representation up to homotopy. Detailed reference viewed: 57 (1 UL)Deformations of Lie brackets and representations up to homotopy ; Schatz, Florian in Indagationes Mathematicae (2011), 22 We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on ... [more ▼] We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets. [less ▲] Detailed reference viewed: 107 (1 UL)Moduli of coisotropic Sections and the BFV-complex Schatz, Florian 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 ▲] Detailed reference viewed: 136 (1 UL)Invariance of the BFV complex Schatz, Florian in Pacific Journal of Mathematics (2010), 248(2), 453-474 The BFV-formalism was introduced to handle classical systems, equipped with symmetries. It associates a differential graded Poisson algebra to any coisotropic submanifold S of a Poisson manifold (M, \Pi ... [more ▼] The BFV-formalism was introduced to handle classical systems, equipped with symmetries. It associates a differential graded Poisson algebra to any coisotropic submanifold S of a Poisson manifold (M, \Pi). However the assignment (coisotropic submanifold) -> (differential graded Poisson algebra) is not canonical, since in the construction several choices have to be made. One has to fix: 1. an embedding of the normal bundle NS of S into M as a tubular neighbourhood, 2. a connection on NS and 3. a special element Omega. We show that different choices of a connection and an element Omega -- but with the tubular neighbourhood fixed -- lead to isomorphic differential graded Poisson algebras. If the tubular neighbourhood is changed too, invariance can be restored at the level of germs. [less ▲] Detailed reference viewed: 41 (0 UL)BFV-complex and higher homotopy structures Schatz, Florian 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 ▲] Detailed reference viewed: 110 (0 UL)Equivalences of Higher Derived Brackets ; Schatz, Florian in Journal of Pure and Applied Algebra (2008), 212(11), 2450-2460 This note elaborates on Th. Voronov’s construction of L-infinity-structures via higher derived brackets with a Maurer–Cartan element. It is shown that gauge equivalent Maurer–Cartan elements induce L ... [more ▼] This note elaborates on Th. Voronov’s construction of L-infinity-structures via higher derived brackets with a Maurer–Cartan element. It is shown that gauge equivalent Maurer–Cartan elements induce L-infinity-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed. [less ▲] Detailed reference viewed: 115 (0 UL) |
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