Logarithmic Algebroids and Meromorphic Line Bundles and Gerbes
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Mechanics, Control and Lie Algebroids, Geometry Meeting 2009
Mechanics, Control and Lie algebroids Geometry Meeting 2009 Eduardo Martínez University of Zaragoza [email protected] Escola Universitaria Politécnica, Ferrol, October 29–30, 2009 Abstract The most relevant ideas and results about mechanical systems defined on Lie algebroids are presented. This was a program originally proposed by Alan Weinstein (1996) and developed by many authors. 1 Several reasons for formulating Mechanics on Lie algebroids The inclusive nature of the Lie algebroid framework: under the same formalism one can consider standard mechanical systems, systems on Lie algebras, systems on semidirect products, systems with symme- tries. The reduction of a mechanical system on a Lie algebroid is a mechan- ical system on a Lie algebroid, and this reduction procedure is done via morphisms of Lie algebroids. Well adapted: the geometry of the underlying Lie algebroid deter- mines some dynamical properties as well as the geometric structures associated to it (e.g. Symplectic structure). Provides a natural way to use quasi-velocities in Mechanics. 2 Introduction Lagrangian systems Given a Lagrangian L 2 C1(TQ), the Euler-Lagrange equations define a dynamical system q_i = vi d @L @L = : dt @vi @qi Variational Calculus. Symplectic formalism 4 Geodesics of left invariant metrics For instance, if k is a Riemannian metric on a manifold Q , the Euler- 1 Lagrange equations for L(v) = 2 k(v; v) are the equation for the geodesics rvv = 0 If Q = G is a Lie group and k is left invariant, then L defines a function 1 l on the Lie algebra g, by restriction l(ξ) = 2 k(ξ; ξ). -
Regular Jacobi Structures and Generalized Contact Bundles
Regular Jacobi Structures and Generalized Contact Bundles Jonas Schnitzer∗ Dipartimento di Matematica Università degli Studi di Salerno Via Giovanni Paolo II, 132 84084 Fisciano (SA) Italy June 14, 2021 Abstract A Jacobi structure J on a line bundle L → M is weakly regular if the sharp map J ♯ : J 1L → DL has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5- dimensional nilmanifold is equipped with an invariant generalized contact structure, 2) we show that, unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that do not possess a compatible generalized contact structure. Contents 1 Introduction 2 2 Preliminaries and Notation 2 3 Tranversally Complex Jacobi Structures and Generalized Contact Bundles 8 arXiv:1806.10489v1 [math.DG] 27 Jun 2018 4 The Splitting of the Spectral Sequence of a Transversally Complex Subbundle 10 5 Examples I: Five dimensional nilmanifolds 15 6 Examples II: Contact Fiber Bundles 19 7 A Counterexample 21 References 23 ∗[email protected] 1 1 Introduction Generalized geometry was set up in the early 2000’s by Hitchin [9] and further developed by Gualtieri [8], and the literature about them is now rather wide. -
Holomorphic Equivariant Cohomology of Atiyah Algebroids and Localization
arXiv:0910.2019 [math.CV] SISSA Preprint 65/2009/fm Holomorphic equivariant cohomology of Atiyah algebroids and localization U. Bruzzo Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, 34151Trieste, Italy; Istituto Nazionale di Fisica Nucleare, Sezione di Trieste E-Mail: [email protected] V. Rubtsov Universit´ed'Angers, D´epartement de Math´ematiques,UFR Sciences, LAREMA, UMR 6093 du CNRS, 2 bd. Lavoisier, 49045 Angers Cedex 01, France; ITEP Theoretical Division, 25 Bol. Tcheremushkinskaya, 117259, Moscow, Russia E-mail: [email protected] Abstract. In a previous paper we developed an equivariant cohomology the- ory associated to a set of data formed by a differentiable manifold M carrying the action of a Lie group G, and a Lie algebroid A on M equipped with a compatible infinitesimal action of G. We also obtained a localization formula for a twisted version of this cohomology, and from it, a Bott-type formula. In this paper we slightly modify that theory to obtain a localization formula for an equivariant cohomology associated to an equivariant holomorphic vector bun- dle. This formula encompasses many residues formulas in complex geometry, in particular we shall show that it admits as a special case Carrell-Lieberman's residue formula [7, 6]. Date: 10 October 2009 2000 Mathematics Subject Classification: 32L10, 53C12, 53C15, 53D17, 55N25, 55N91 The authors gratefully acknowledge financial support and hospitality during the respective vis- its to Universit´ed'Angers and SISSA. Support for this work was provided by misgam, by P.R.I.N \Geometria sulle variet`aalgebriche," the I.N.F.N. -
ON a NON-ABELIAN POINCARÉ LEMMA 1. Introduction It Is Well
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 8, August 2012, Pages 2855–2872 S 0002-9939(2011)11116-X Article electronically published on December 15, 2011 ON A NON-ABELIAN POINCARE´ LEMMA THEODORE TH. VORONOV (Communicated by Varghese Mathai) Abstract. We show that a well-known result on solutions of the Maurer– Cartan equation extends to arbitrary (inhomogeneous) odd forms: any such form with values in a Lie superalgebra satisfying dω + ω2 = 0 is gauge- equivalent to a constant, ω = gCg−1 − dg g−1. This follows from a non-Abelian version of a chain homotopy formula making use of multiplicative integrals. An application to Lie algebroids and their non- linear analogs is given. Constructions presented here generalize to an abstract setting of differen- tial Lie superalgebras where we arrive at the statement that odd elements (not necessarily satisfying the Maurer–Cartan equation) are homotopic — in a certain particular sense — if and only if they are gauge-equivalent. 1. Introduction It is well known that a 1-form with values in a Lie algebra g satisfying the Maurer–Cartan equation 1 (1.1) dω + [ω, ω]=0 2 possesses a ‘logarithmic primitive’ locally or for a simply-connected domain (1.2) ω = −dg g−1 for some G-valued function, where G is a Lie group with the Lie algebra g.Here and in the sequel we write all formulas as if our algebras and groups were matrix; it makes the equations more transparent, and certainly nothing prevents us from rephrasing them in an abstract form. Therefore the main equation (1.1) will also be written as (1.3) dω + ω2 =0. -
LIE GROUPOIDS and LIE ALGEBROIDS LECTURE NOTES, FALL 2017 Contents 1. Lie Groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Ex
LIE GROUPOIDS AND LIE ALGEBROIDS LECTURE NOTES, FALL 2017 ECKHARD MEINRENKEN Abstract. These notes are under construction. They contain errors and omissions, and the references are very incomplete. Apologies! Contents 1. Lie groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Exercises 9 2. Foliation groupoids 10 2.1. Definition, examples 10 2.2. Monodromy and holonomy 12 2.3. The monodromy and holonomy groupoids 12 2.4. Appendix: Haefliger’s approach 14 3. Properties of Lie groupoids 14 3.1. Orbits and isotropy groups 14 3.2. Bisections 15 3.3. Local bisections 17 3.4. Transitive Lie groupoids 18 4. More constructions with groupoids 19 4.1. Vector bundles in terms of scalar multiplication 20 4.2. Relations 20 4.3. Groupoid structures as relations 22 4.4. Tangent groupoid, cotangent groupoid 22 4.5. Prolongations of groupoids 24 4.6. Pull-backs and restrictions of groupoids 24 4.7. A result on subgroupoids 25 4.8. Clean intersection of submanifolds and maps 26 4.9. Intersections of Lie subgroupoids, fiber products 28 4.10. The universal covering groupoid 29 5. Groupoid actions, groupoid representations 30 5.1. Actions of Lie groupoids 30 5.2. Principal actions 31 5.3. Representations of Lie groupoids 33 6. Lie algebroids 33 1 2 ECKHARD MEINRENKEN 6.1. Definitions 33 6.2. Examples 34 6.3. Lie subalgebroids 36 6.4. Intersections of Lie subalgebroids 37 6.5. Direct products of Lie algebroids 38 7. Morphisms of Lie algebroids 38 7.1. Definition of morphisms 38 7.2. -
Lie Algebroids, Gauge Theories, and Compatible Geometrical Structures
LIE ALGEBROIDS, GAUGE THEORIES, AND COMPATIBLE GEOMETRICAL STRUCTURES ALEXEI KOTOV AND THOMAS STROBL Abstract. The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper is supposed to analyze these compatibil- ities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove furthermore that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized Riemannian structure. Key words and phrases: Lie algebroids, Riemannian foliations, symplectic realization, generalized geometry, symmetries and reduction, gauge theories. Contents 1. Introduction 1 2. Motivation via Mathematical Physics 3 arXiv:1603.04490v2 [math.DG] 12 Apr 2019 3. Connections on Lie algebroids 8 4. Lie algebroids over Riemannian manifolds 14 5. Lie algebroids over manifolds with other geometric structures 21 Appendix A: Flat Killing Lie algebroids, simple examples and facts 24 References 25 Date: March 7, 2018. 0 LIE ALGEBROIDS WITH COMPATIBLE GEOMETRICAL STRUCTURES 1 1. Introduction Lie algebroids were introduced in the 1960s by Pradines [23] as a formalization of ideas going back to works of Lie and Cartan. -
THE ATIYAH ALGEBROID of the PATH FIBRATION OVER a LIE GROUP Contents 1. Introduction 2 2. Review of Transitive Lie Algebroids 3
THE ATIYAH ALGEBROID OF THE PATH FIBRATION OVER A LIE GROUP A. ALEKSEEV AND E. MEINRENKEN Abstract. Let G be a connected Lie group, LG its loop group, and π : P G → G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A = T (P G)/LG of this bundle. Given a symmetric bilinear form on g and the corresponding central extension of Lg, we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form η ∈ Ω3(G) arises as an obstruction. This involves the construction of a ∗ ∗ 2-form ̟ ∈ Ω2(P G)LG = Γ(∧2A ) with d̟ = π η. In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the form η, and for their G-equivariant extensions. Contents 1. Introduction 2 2. Review of transitive Lie algebroids 3 2.1. Lie algebroids 3 2.2. Transitive Lie algebroids 4 2.3. Pull-backs 5 2.4. Equivariant transitive Lie algebroids 6 3. A lifting problem for transitive Lie algebroids 6 3.1. The lifting problem 6 3.2. Splittings 7 3.3. The form ̟ 8 3.4. The cohomology class [η] as an obstruction class 9 3.5. The equivariant lifting problem 10 3.6. Relation with Courant algebroids 12 4. The Atiyah algebroid A → G 13 4.1. The bundle of twisted loop algebras 13 4.2. The Lie algebroid A → G 13 4.3. The bundle P G → G 15 4.4. -
Rota–Baxter Algebras, Singular Hypersurfaces, and Renormalization on Kausz Compactifications
ROTA{BAXTER ALGEBRAS, SINGULAR HYPERSURFACES, AND RENORMALIZATION ON KAUSZ COMPACTIFICATIONS MATILDE MARCOLLI AND XIANG NI Abstract. We consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole sub- traction. We apply this formalism to the unrenormalized momentum space Feynman amplitudes, viewed as (divergent) integrals in the complement of the determinant hypersurface. We lift the in- tegral to the Kausz compactification of the general linear group, whose boundary divisor is normal crossings. We show that the Kausz compactification is a Tate motive and the boundary divisor is a mixed Tate configuration. The regularization of the integrals that we obtain differs from the usual renormalization of physical Feynman amplitudes, and in particular it always gives mixed Tate periods. 1. Introduction In this paper, we consider the problem of extracting periods of algebraic varieties from a class of divergent integrals arising in quantum field theory. The method we present here provides a regular- ization and extraction of finite values that differs from the usual (renormalized) physical Feynman amplitudes, but whose mathematical interest lies in the fact that it always gives a period of a mixed Tate motive. The motive considered is provided by the Kausz compactification of the general lin- ear group. The regularization procedure we propose is modeled on the algebraic renormalization method, based on Hopf algebras of graphs and Rota{Baxter algebras, as originally developed by Connes and Kreimer [13] and by Ebrahmi-Fard, Guo, and Kreimer [19]. -
Courant Algebroids in Parabolic Geometry
COURANT ALGEBROIDS IN PARABOLIC GEOMETRY STUART ARMSTRONG AND RONGMIN LU Abstract. Let p be a Lie subalgebra of a semisimple Lie algebra g and (G,P ) be the corresponding pair of connected Lie groups. A Cartan geometry of type (G,P ) associates to a smooth manifold M a principal P -bundle and a Cartan connection, and a parabolic geometry is a Cartan geometry where P is parabolic. We show that if P is parabolic, the adjoint tractor bundle of a Cartan geometry, which is isomorphic to the Atiyah algebroid of the principal P -bundle, admits the structure of a (pre-)Courant algebroid, and we identify the topological obstruction to the bracket being a Courant bracket. For semisimple G, the Atiyah algebroid of the principal P -bundle associated to the Cartan geometry of (G,P ) admits a pre-Courant algebroid structure if and only if P is parabolic. Contents 1. Introduction 1 1.1. Motivation 2 1.2. Non-exact Courant algebroids in parabolic geometry 2 1.3. Outline of the paper 3 2. The Atiyah sequence and principal connections 4 2.1. The Atiyah sequence 4 2.2. Principal and Lie algebroid connections 5 3. The adjoint tractor bundle 6 3.1. Cartan geometry 6 3.2. Parabolic geometry 8 3.3. Weyl structures in parabolic geometry 11 4. Transitive Courant algebroids 12 4.1. Basic properties and structure 12 4.2. TheCourantbracketandthefirstPontrjaginclass 15 4.3. (Pre-)CourantstructuresonAtiyahalgebroids 20 5. Concluding remarks 21 Acknowledgements 22 arXiv:1112.6425v4 [math-ph] 23 Sep 2012 References 22 1. Introduction In this paper, we show that non-exact transitive Courant algebroids, possibly twisted by a closed 4-form, appear naturally in parabolic geometries. -
Cartan Connections and Lie Algebroids*
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 061, 13 pages Cartan Connections and Lie Algebroids? Michael CRAMPIN Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B–9000 Gent, Belgium Address for correspondence: 65 Mount Pleasant, Aspley Guise, Beds MK17 8JX, UK E-mail: [email protected] Received March 23, 2009, in final form June 07, 2009; Published online June 14, 2009 doi:10.3842/SIGMA.2009.061 Abstract. This paper is a study of the relationship between two constructions associated with Cartan geometries, both of which involve Lie algebroids: the Cartan algebroid, due to [Blaom A.D., Trans. Amer. Math. Soc. 358 (2006), 3651–3671], and tractor calculus [Capˇ A., Gover A.R., Trans. Amer. Math. Soc. 354 (2001), 1511–1548]. Key words: adjoint tractor bundle; algebroid connection; algebroid representation; Cartan connection; Cartan geometry; Lie algebroid; tractor calculus 2000 Mathematics Subject Classification: 53B05; 53C05; 53C15 1 Introduction In 2006 Anthony D. Blaom published a very interesting paper [1] in which he proposed a wide- ranging generalization of Cartan’s idea of a geometry as a Klein geometry deformed by curvature. The essential ingredient of Blaom’s conception is a Lie algebroid with a linear connection (co- variant derivative) that is compatible with the algebroid bracket in a certain sense. Blaom calls such an algebroid a Cartan algebroid, and – rather unfortunately in the circumstances – he calls a connection with the requisite property a Cartan connection. A Cartan geometry in the origi- nal sense (perhaps one should add ‘as reformulated by Sharpe [4]’) of a principal bundle with a Cartan connection form – what Blaom would call a classical Cartan connection – gives rise to a Cartan algebroid, as one would expect. -
Coherent Sheaves on Calabi-Yau Manifolds, Picard-Fuchs Equations and Potential Functions Dissertation Zur Erlangung Des Doktorgr
Coherent Sheaves on Calabi-Yau manifolds, Picard-Fuchs equations and potential functions Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg im Breisgau vorgelegt von Kathrin Natalie Peternell Februar 2018 Dekan der Fakult¨atf¨urMathematik und Physik Prof. Dr. Gregor Herten Erstgutachter Priv.-Doz. Dr. Emanuel Scheidegger Zweitgutachter Prof. Sheldon Katz, PhD Datum der Promotionspr¨ufung 21. Juni 2018 Contents 1 Introduction 1 2 Preliminaries 15 2.1 Notation . 15 2.2 Singularities . 15 2.3 Projective fibre spaces . 16 3 Deformation Theory for geometric objects 19 3.1 Functors of Artin rings . 19 3.2 Deformation theory of a compact complex manifold . 21 3.3 Deformation theory of a submanifold in a fixed complex man- ifold . 22 3.4 Deformation theory of a coherent sheaf on a fixed complex manifold . 23 3.5 Deformation theory of a pair consisting of a smooth divisor in a complex manifold . 23 3.6 Simultaneous deformations of a coherent sheaf and its com- plex base manifold . 25 3.7 Deformations of projective bundles . 26 3.8 Simultaneous deformations of a holomorphic vector bundle and a section . 29 4 Variation of Hodge structures on the cohomology of special complex manifolds 31 4.1 Variation of pure Hodge structures . 31 4.2 Logarithmic differentials and residues . 32 4.2.1 The complex of logarithmic differential forms . 32 4.2.2 The residue map . 33 4.2.3 The notion of mixed Hodge structures and variations of mixed Hodge structures . 41 4.2.4 The mixed Hodge structure on the hypercohomology of the complex of logarithmic differential forms . -
Geometric Structures Encoded in the Lie Structure of an Atiyah Algebroid
CORE Metadata, citation and similar papers at core.ac.uk Provided by Open Repository and Bibliography - Luxembourg Geometric structures encoded in the Lie structure of an Atiyah algebroid Janusz Grabowski,∗ Alexei Kotov, Norbert Poncin † Abstract We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are nec- essarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of sections for Atiyah algebroids associated to principal bundles with semisimple struc- ture groups. For instance we prove that in the semisimple case the Lie algebras of sections are isomorphic if and only if the corresponding Lie algebroids are, or, as well, if and only if the integrating principal bundles are locally isomorphic. Finally, we apply these results to describe the isomorphisms of sections in the case of reductive structure groups – surprisingly enough they are no longer determined by vector bundle isomorphisms and involve divergences on the base manifolds. MSC 2000: 17B40, 55R10, 57R50, 58H05 (primary), 16S60, 17B20, 53D17 (secondary) Keywords: Algebraic characterization of space, Lie algebroid, Atiyah sequence, principal bundle, Lie algebra of sections, isomorphism, semisimple Lie group, reductive Lie group, divergence. 1 Introduction The concept of groupoid was introduced in 1926 by the German mathematician Heinrich Brandt. A groupoid is a small category in which any morphism is invertible. Topological and differential groupoids go back to Charles Ehresmann [Ehr59] and are transitive Lie groupoids in the sense of A.