Nijenhuis tensors in generalized geometry Yvette Kosmann-Schwarzbach ematiques Laurent Schwartz, ´ Centre de Math´ Ecole Polytechnique, France Bi-Hamiltonian Systems and All That International Conference in honour of Franco Magri University of Milan Bicocca, 27 September-1 October, 2011 Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
It all started in 1977 ◮ Franco Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162 [18 April 1977]. ◮ A geometrical approach to the nonlinear solvable equations. in Nonlinear evolution equations and dynamical systems (Lecce, 1979), Lecture Notes in Phys., 120, Springer, 1980, 233–263. N ′ u ( X , N u Y ) − N ′ u ( Y , N u X ) = N u ( N ′ u ( X , Y ) − N ′ u ( Y , X )) . 1980, I. M. Gel’fand and Irene Dorfman, Tudor Ratiu. 1981, B. Fuchssteiner and A. S. Fokas (recursion operators are ‘hereditary operators’). ◮ with Carlo Morosi, A geometrical characterization of integrable hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S 19, Milan, 1984. (Re-issued: Universit` a di Milano Bicocca, Quaderno 3, 2008. http://home.matapp.unimib.it) [ NX , NY ] − N ([ NX , Y ] + [ X , NY ]) + N 2 [ X , Y ] = 0 . Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
Prehistory In fact, there was a prehistory for this story: KdV and its (first) Hamiltonian structure. ◮ Clifford Gardner, John Greene, Martin Kruskal, Robert Miura, Norman Zabusky, 1965, 1968, 1970, 1974. ◮ Ludwig Faddeev and Vladimir Zakharov, 1971. ◮ Israel Gel’fand and Leonid Dikii, 1975. ◮ Peter Lax, 1976, “recursion formula of Lenart”. ◮ Peter Olver, 1977, “recursion operator”. (See the historical notes in Olver’s book, and “Andrew Lenard: A Mystery Unraveled” by Jeffery Praught and Roman G. Smirnov, SIGMA 1 (2005).) In the early 1980’s, Benno Fucchsteiner, Dan Gutkin, Giuseppe Marmo, Boris Konopelchenko, Orlando Ragnisco, ... Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
1983, Pseudocociclo di Poisson ◮ Pseudocociclo di Poisson e strutture PN gruppale, applicazione al reticolo di Toda, Magri’s unpublished manuscript, Milan 1983. r -matrices, the modified Yang-Baxter equation and Poisson-Lie groups avant la lettre = “Hamiltonian Lie groups” =“Poisson-Drinfeld groups” = “Poisson-Lie groups” Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
From Quaderno S 19 (1984) to generalized geometry ◮ with C. Morosi, Su possibili applicazioni della riduzione di strutture geometriche nella teoria dei sistemi dinamici, AIMETA Trieste 1984, 135–144. ◮ with C. Morosi and Orlando Ragnisco, Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications, Comm. Math. Phys. 99 (1985), 115–140. ◮ with C. Morosi, Old and new results on recursion operators: an algebraic approach to KP equation, in Topics in soliton theory and exactly solvable nonlinear equations (Oberwolfach, 1986), World Sci., 1987, 78–96. ◮ with C. Morosi and G. Tondo, Nijenhuis G -manifolds and Lenard bicomplexes: a new approach to KP systems, Comm. Math. Phys. 115 (1988), 457–475. Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
From Quaderno S 19 (1984) to generalized geometry (cont’d) ◮ yks, The modified Yang-Baxter equation and bi-Hamiltonian structures, in Differential geometric methods in theoretical physics (Chester, 1988), World Sci., 1989, 12–25. “The results presented here are joint work with Franco Magri.” ◮ with yks, Poisson–Nijenhuis structures, Ann. Inst. H. Poincar´ e Phys. Th´ eor. 53 (1990), 35–81. PN-structures on “differential Lie algebras” =Lie d-rings = pseudo-Lie algebras = (K,R)-Lie algebras = ´ Elie Cartan spaces = Lie modules = Lie–Cartan pairs = Lie–Rinehart algebras ≃ Lie algebroids ◮ with yks, Dualization and deformation of Lie brackets on Poisson manifolds, in Differential geometry and its applications (Brno, 1989), World Sci., 1990, 79–84. Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
From Quaderno S 19 (1984) to generalized geometry (cont’d) ◮ with Pablo Casati and Marco Pedroni, Bi-Hamiltonian manifolds and Sato’s equations, in Integrable systems, The Verdier Memorial Conference (Luminy, 1991), Progr. Math., 115, Birkh¨ auser, 1993, 251–272. ◮ 1993, Peter Olver (Canonical forms for bi-Hamiltonian systems) ◮ 1993, Rober Brouzet, Pierre Molino and Javier Turiel (G´ eom´ etrie des syst` emes bihamiltoniens) ◮ 1993, Gel’fand and Ilya Zakharevich (On the local geometry of a bi-Hamiltonian structure) , 1998 Panasyuk (Veronese webs for bi-Hamiltonian structures) ◮ 1994, 1997, Izu Vaisman (A lecture on Poisson–Nijenhuis structures) ◮ .................................... ◮ .................................... ◮ 316 items for “bi-hamiltonian” or “bihamiltonian” in MathSciNet, including 13 by Franco Magri and co-authors, and ??? by other participants in this conference (3 by yks). Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
From Quaderno S 19 (1984) to generalized geometry (cont’d) ◮ yks, The Lie bialgebroid of a Poisson-Nijenhuis manifold, Lett. Math. Phys. 38 (1996), 421–428. “This result was first conjectured by Magri during a conversation that we held at the time of the Semestre Symplectique at the Centre ´ Emile Borel (1994).” (The program on Symplectic Geometry was the first organized in the new Centre ´ Emile Borel in the renovated Institut Henri Poincar´ e in Paris.) Meanwhile the theory of Lie algebroids, Lie bialgebroids, generalized tangent bundles and Courant algebroids developped. ◮ yks and Vladimir Rubtsov, Compatible structures on Lie algebroids and Monge-Amp` ere operators, Acta Appl. Math. , 109 (2010), 101-135. ◮ yks, Nijenhuis structures on Courant algebroids, Bull. Brazilian Math. Soc. , to appear (arXiv1102.1410). Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
What is new? Our aim is to • point out the new features in the theory of Nijenhuis operators on generalized tangent bundles of manifolds, • study the (infinitesimal) deformations of generalized tangent bundles. • show that PN-structures and ΩN-structures on a manifold define (infinitesimal) deformations of its generalized tangent bundle. Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
Generalized tangent bundles The generalized tangent bundle of a smooth manifold, M , is T M = TM ⊕ T ∗ M equipped with • the canonical fibrewise non-degenerate, symmetric, bilinear form � X + ξ, Y + η � = � X , η � + � Y , ξ � , • the Dorfman bracket [ X + ξ, Y + η ] = [ X , Y ] + L X η − i Y ( d ξ ) , X , Y vector fields, sections of TM , ξ , η differential 1-forms, sections of T ∗ M . The Dorfman bracket is a derived bracket, i [ X ,η ] = [[ i X , d ] , e η ]. For derived brackets, see yks, Ann. Fourier 1996, LMP 2004. Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
Properties of the Dorfman and Courant brackets The Dorfman bracket is not skew-symmetric, but since it is a derived bracket, it is a Leibniz (Loday) bracket, i.e. , it satisfies the Jacobi identity in the form [ u , [ v , w ]] = [[ u , v ] , w ] + [ v , [ u , w ]] , u , v sections of T M = TM ⊕ T ∗ M . The Courant bracket is the skew-symmetrized Dorfman bracket, [ X + ξ, Y + η ] == [ X , Y ] + L X η − L Y ξ + 1 2 � X + ξ, Y + η � . The Courant bracket is skew-symmetric but it does not satisfy the Jacobi identity. T M is called the double of TM . It is a Courant algebroid. More generally, the double of a Lie bialgebroid is a Courant algebroid. Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
Two relations Define ∂ : C ∞ ( M ) → Γ( T ∗ M ) by � Z , ∂ f � = Z · f . We shall make use of the relations, [ u , v ] + [ v , u ] = ∂ � u , v � , and � [ u , v ] , w � + � v , [ u , w ] � = � u , ∂ � v , w �� . Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
Questions Define the Nijenhuis torsion of an endomorphism N of T M ? Define the Nijenhuis operators on T M ? in particular the generalized complex structures? Application to the deformation of structures? Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
Nijenhuis torsion Let N be an endomorphism of T M , i.e. , a (1 , 1)-tensor on the vector bundle T M . We define the Nijenhuis torsion, or simply the torsion of N by ( T N )( u , v ) == [ N u , N v ] − N ([ N u , v ] + [ u , N v ]) + N 2 [ u , v ] . for all sections u , v of T M . (Here [ , ] is the Dorfman bracket.) An endomorphism of T M whose torsion vanishes is called a Nijenhuis operator on T M . Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry
Recommend
More recommend
