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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
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