Higher Contact Geometry and L Algebras

Higher Contact Geometry and L Algebras

Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Higher Contact Geometry and L¥ Algebras Luca Vitagliano University of Salerno, Italy IMPAN, Warsaw, May 14, 2014 Luca Vitagliano Higher Contact Geometry and L¥ Algebras 1 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Introduction: Symplectic Geometry A symplectic manifold is a manifold M equipped with a closed, non- degenerate differential2-form w. The original motivation for symplectic geometry comes from analytical mechanics: the phase space of many classical systems is a symplectic manifold! Actually, symplectic geometry pervades both differential geometry and mathematical physics: Hamiltonian systems, Poisson ge- ometry, Lie algebroids, Courant algebroids, K¨ahlergeometry, etc.. Remark One can attach an algebraic structure to any symplectic manifold (M, w), namely a Poisson bracket f−, −g on the algebra C¥(M). The Poisson bracket f−, −g plays a key role in numerous contexts: integra- bility of Hamiltonian systems, action of Lie groups on symplectic manifolds / moment maps, geometric quantization, etc. Luca Vitagliano Higher Contact Geometry and L¥ Algebras 2 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Introduction: Contact Geometry A contact manifold is a manifold M equipped with a maximally non- integrable, hyperplane distribution C. The original motivation for contact geometry comes from first order scalar PDEs: the first jet space of hypersurfaces is a contact manifold! Ac- cordingly, contact geometry has numerous applications both in dif- ferential geometry and mathematical physics: Jet spaces, control theory, geometric quantization, geometric optics, thermodynamics, etc.. Remark One can attach an algebraic structure to any contact manifold (M, C), namely a Jacobi bracket f−, −g on the module G(TM/C). The Jacobi bracket f−, −g plays a key role in various contexts: symmetries of PDEs, integration by characteristics, etc. Contact geometry can be seen as a part of symplectic geometry and the Jacobi bracket can be derived from a suitable Poisson bracket! Luca Vitagliano Higher Contact Geometry and L¥ Algebras 3 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Introduction: Higher Symplectic Geometry A multisymplectic manifold is a manifold M equipped with a closed, non-degenerate, higher degree differential form w. The original motivation for multisymplectic geometry comes from clas- sical field theory: the phase space of many field theories is a multisym- plectic manifold! Remark C. Rogers and M. Zambon showed that, similarly as for symplectic manifolds, one can attach an algebraic structure to any multisymplec- tic manifold (M, w), namely an L¥ algebra g(M, w), which plays a similar role as the Poisson algebra of a symplectic manifold: action of Lie groups on multisymplectic manifolds / homotopy moment maps, geomet- ric (pre-)quantization of field theories. Luca Vitagliano Higher Contact Geometry and L¥ Algebras 4 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Introduction: Higher Contact Geometry A multicontact manifold is a manifold M equipped with a maximally non-integrable distribution C of higher codimension. The motivation for multicontact geometry comes from the geometry of PDEs: finite jet spaces are multicontact manifolds! Remark Similarly as for contact manifolds, one can attach an algebraic struc- ture to any multicontact manifold (M, C), namely an L¥ algebra g(M, C), which plays a similar role as the Jacobi line bundle of a con- tact manifold: concrete applications are still to be explored!. Multicontact geometry can be seen as a part of multisymplectic geom- etry and the “multicontact” L¥ algebra can be derived from a suitable “multisymplectic” L¥ algebra! Luca Vitagliano Higher Contact Geometry and L¥ Algebras 5 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Introduction: Homotopy Algebras Let A be a type of algebra (associative, Lie. etc.). A homotopy A alge- bra structure on a chain complex is a set of operations that satisfy the axioms of A only up to homotopy (in fact, a coherent system of higher homotopies). Remark Homotopy algebras appear as a consequence of the interaction be- tween algebraic structures and homology/homotopy. For instance, homotopy Lie algebras often govern formal deformation problems of al- gebraic/geometric structures. Remark Homotopy algebras do also appear in geometry as higher versions of standard algebras. For instance symplectic : multisymplectic = Lie algebra : homotopy Lie algebra Luca Vitagliano Higher Contact Geometry and L¥ Algebras 6 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Outline 1 Symplectic and Contact Manifolds 2 Multisymplectic and Multicontact Manifolds 3 Strong Homotopy Lie Algebras 4 L¥ Algebras from Multisymplectic and Multicontact Geometry Luca Vitagliano Higher Contact Geometry and L¥ Algebras 7 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Outline 1 Symplectic and Contact Manifolds 2 Multisymplectic and Multicontact Manifolds 3 Strong Homotopy Lie Algebras 4 L¥ Algebras from Multisymplectic and Multicontact Geometry Luca Vitagliano Higher Contact Geometry and L¥ Algebras 8 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Symplectic Manifolds Let M be a smooth manifold. Definition A symplectic structure on M is a non-degenerate2-form w such that dw = 0. The pair (M, w) is a symplectic manifold. Example Let N be a manifold. M := T∗ N possesses a tautological 1-form q: qp(x) := p(p∗(x)), x 2 Tp M (p : M −! N). w := dq is a canonical symplectic structure on M. This example plays a distinguished role in classical mechanics! Luca Vitagliano Higher Contact Geometry and L¥ Algebras 9 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry The Poisson Algebra of a Symplectic Manifold Let (M, w) be a symplectic manifold. There is a binary bracket f−, −g in C¥(M) defined as follows: X(M) −! W1(M), X 7−! w(X, −) possesses an inverse 1 W (M) −! X(M), d f 7−! X f . Then f f , gg := w(X f , Xg). Remark (M, f−, −g) is a Poisson manifold, i.e. f−, −g is a Lie bracket, f−, −g is a derivation in each argument. Luca Vitagliano Higher Contact Geometry and L¥ Algebras 10 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Distributions on Manifolds Let M be a smooth manifold. Definition A distribution on M is a vector subbundle D ⊂ TM. The curvature form of D is the following vector bundle valued2-form on D: wD : G(D) × G(D) −! G(TM/D), (X, Y) 7−! [X, Y] mod D Remark D is integrable iff wD = 0. More generally, ker wD consists of vector fields in D whose flows preserve D, i.e. characteristic symmetries. Proposition The characteristic distribution KD := ker wD is integrable. Luca Vitagliano Higher Contact Geometry and L¥ Algebras 11 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry Contact Manifolds Let M be a smooth manifold. Definition A contact structure on M is an hyperplane distribution C, such that wC is non-degenerate. The pair (M, C) is a contact manifold. The line- bundle L := TM/C is the Jacobi bundle of (M, C). Example Let N be a manifold, dim N = n, and M := Gr(TN, n − 1). There is a canonical line bundle L ! M, whose fiber at y 2 Gr(Tx N, n − 1) is Ly := Tx N/y. Moreover, M possesses a tautologicalL -valued1-form J: Jy(x) := p∗(x) mod y, x 2 Ty M (p : M −! N). ker J is a canonical contact structure on M. This example plays a distinguished role for first order scalar PDEs! Luca Vitagliano Higher Contact Geometry and L¥ Algebras 12 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry The Kirillov Algebroid of a Contact Manifold Let (M, C) be a contact manifold. There is a binary bracket f−, −g in G(L) defined as follows: 0 −! G(C) −! X(M) −! G(L) −! 0 possesses a canonical splitting G(L) −! X(M), l 7−! Xl. Then fl, mg := [Xl, Xm] mod C. Remark (L, f−, −g) is a Kirillov algebroid, i.e. f−, −g is a Lie bracket, f−, −g is a first order differential operator in each argument. Luca Vitagliano Higher Contact Geometry and L¥ Algebras 13 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L¥ Algebras from Multisymplectic and Multicontact Geometry

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    36 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us