Nijenhuis tensors in generalized geometry

Transcription

1 Yvette Kosmann-Schwarzbach Centre de Mathématiques Laurent Schwartz, École Polytechnique, France Bi-Hamiltonian Systems and All That International Conference in honour of Franco Magri University of Milan Bicocca, 27 September-1 October, 2011

2 It all started in 1977 Franco Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), [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, 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, (Re-issued: Università di Milano Bicocca, Quaderno 3, [NX, NY ] N([NX, Y ] + [X, NY ]) + N 2 [X, Y ] = 0.

3 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, Ludwig Faddeev and Vladimir Zakharov, Israel Gel fand and Leonid Dikii, 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,...

4 1983, Pseudocociclo di Poisson Pseudocociclo di Poisson e strutture PN gruppale, applicazione al reticolo di Toda, Magri s unpublished manuscript, Milan r-matrices, the modified Yang-Baxter equation and Poisson-Lie groups avant la lettre = Hamiltonian Lie groups = Poisson-Drinfeld groups = Poisson-Lie groups

5 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, with C. Morosi and Orlando Ragnisco, Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications, Comm. Math. Phys. 99 (1985), 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, with C. Morosi and G. Tondo, Nijenhuis G-manifolds and Lenard bicomplexes: a new approach to KP systems, Comm. Math. Phys. 115 (1988),

6 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, The results presented here are joint work with Franco Magri. with yks, Poisson Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), PN-structures on differential Lie algebras =Lie d-rings = pseudo-lie algebras = (K,R)-Lie algebras = Élie 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,

7 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, Birkhäuser, 1993, , Peter Olver (Canonical forms for bi-hamiltonian systems) 1993, Rober Brouzet, Pierre Molino and Javier Turiel (Géométrie des systèmes 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) 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).

8 From Quaderno S 19 (1984) to generalized geometry (cont d) yks, The Lie bialgebroid of a Poisson-Nijenhuis manifold, Lett. Math. Phys. 38 (1996), This result was first conjectured by Magri during a conversation that we held at the time of the Semestre Symplectique at the Centre Émile Borel (1994). (The program on Symplectic Geometry was the first organized in the new Centre Émile Borel in the renovated Institut Henri Poincaré 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-Ampère operators, Acta Appl. Math., 109 (2010), yks, Nijenhuis structures on Courant algebroids, Bull. Brazilian Math. Soc., to appear (arxiv ).

9 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.

10 Generalized tangent bundles The generalized tangent bundle of a smooth manifold, M, is TM = TM T M equipped with the canonical fibrewise non-degenerate, symmetric, bilinear form the Dorfman bracket X + ξ, Y + η = X, η + Y, ξ, [X + ξ, Y + η] = [X, Y ] + L X η i Y (dξ), 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.

11 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 TM = 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 X + ξ, Y + η. 2 The Courant bracket is skew-symmetric but it does not satisfy the Jacobi identity. TM is called the double of TM. It is a Courant algebroid. More generally, the double of a Lie bialgebroid is a Courant algebroid.

12 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.

13 Questions Define the Nijenhuis torsion of an endomorphism N of TM? Define the Nijenhuis operators on TM? in particular the generalized complex structures? Application to the deformation of structures?

14 Nijenhuis torsion Let N be an endomorphism of TM, i.e., a (1, 1)-tensor on the vector bundle TM. 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 TM. (Here [, ] is the Dorfman bracket.) An endomorphism of TM whose torsion vanishes is called a Nijenhuis operator on TM.

15 Deformed bracket We define, for all u, v Γ(TM), [u, v] N = [N u, v] + [u, N v] N [u, v]. Then (T N )(u, v) = [N u, N v] N [u, v] N. Question. When is the deformed bracket a (new) Leibniz bracket? Answer. Necessary and sufficient condition for [u, v] N to be a Leibniz bracket: the torsion of N is a Leibniz cocycle. Sufficient condition for [u, v] N to be a Leibniz bracket: the torsion of N vanishes. If the torsion of N vanishes, then N [u, v] N = [N u, N v], i.e., N is a morphism from (Γ(TM), [, ] N ) to (Γ(TM), [, ]).

16 Warning The torsion of N : TM TM is a map, T N : Γ(TM) Γ(TM) Γ(TM). Unlike the usual case of tangent bundles (and Lie algebroids), T N is not in general C (M)-linear in both arguments, and in general it is not skew-symmetric. Consequence. We shall have to consideer the case of endomorphisms of TM which are skew-symmetric and whose square is proportional to the identity.

17 Skew-symmetric endomorphisms of TM Let N be an endomorphism of TM, i.e., a (1, 1)-tensor on the vector bundle TM. It is skew-symmeric if N + t N = 0, i.e., if it is of the form ( ) N π N = ω t, N where N : TM TM and t N : T M T M is the transpose of N, π : T M TM is a bivector on M, and ω : TM T M is a 2-form on M. The skew-symmetric endomorphisms of TM are those that leave the bilinear form, infinitesimally invariant. More generally, one can consider paired endomorphisms, satisfying N + t N = 2κ Id TM, where κ is a scalar. See J. Cariñena, J. Grabowski and G. Marmo, Courant algebroid and Lie bialgebroid contractions, J. Phys. A 37(2004).

18 Generalized almost complex structures A skew-symmetric endomorphism N of TM such that N 2 = λ Id TM, where λ = 1, 0 or 1, is called a generalized almost cps structure on TM (or on M ). A generalized almost cps structure is called a generalized cps structure if its torsion vanishes. These definitions are due to Izu Vaisman. Here cps stands for complex, product or subtangent. In particular, an endomorphism N of TM is called a generalized almost complex structure if it is skew-symmetric and N 2 = Id TM. Clearly, if N is an almost complex structure on M (i.e., N : TM «TM and N π N 2 = Id M ), and if either π or ω vanishes, then N = ω t is a N generalized almost complex structure on TM.

19 Use of the Courant bracket One can also define the torsion TN C of an endomorphism N with respect to the Courant bracket, replacing the Dorfman bracket by its skew-symmetrization in the preceding formulas. The relation between the two torsions is (T C N )(u, v) = 1 2 ((T N )(u, v) (T N )(v, u)). Proposition (i) For a skew-symmetric endomorphism N, ) (T N C T N (u, v) = 1 ( u, N 2 v N 2 u, v ), 2 for all sections u and v of TM. (ii) If N is proportional to a generalized almost cps structure, both torsions, TN C and T N, coincide.

20 Questions Consider a smooth manifold M and N a skew-symmetric endomorphism of TM. When is T N a C (M)-linear section of TM 2 (TM)? Answer When N is proportional to a generalized almost cps structure. When is T N a section of 3 (TM)? Answer When N is proportional to a generalized almost cps structure. Compare T N with the Nijenhuis torsion τ N of N? Answer When N is proportional to a generalized almost cps structure, they are equal in a suitable sense.

21 Properties of the torsion. Lack of C -linearity and of skew-symmetry Let N be a skew-symmetric endomorphism of TM. Lack of C -linearity. It is clear that (T N )(u, fv) = f (T N )(u, v), but (T N )(fu, v) = f (T N )(u, v) + u, v N 2 ( f ) u, N 2 v f. Lack of skew-symmetry. Using the fact that N is skew-symmetric, we obtain (T N )(u, v) + (T N )(v, u) = N 2 u, v u, N 2 v.

22 Associated 3-tensor In order to determine whether T N determines a skew-symmetric covariant 3-tensor, we use the skew-symmetry of N and relations stated above to obtain (T N )(u, v), w + (T N )(u, w), v = N 2 [u, w] [u, N 2 w], v. Theorem If N is proportional to a generalized almost cps structure (i.e., N is skew-symmetric and N 2 = λ Id TM ), the torsion of N is C (M)-linear in both arguments and skew-symmetric, and defines a skew-symmetric covariant 3-tensor on TM, T N, by T N (u, v, w) = (T N )(u, v), w.

23 Tensors on M and tensors on TM To a tensor t TM 2 (T M) we associate t in 3 (TM T M) defined by t(x + ξ, Y + η, Z + ζ) = t(x, Y ), ζ + t(y, Z), ξ + t(z, X ), η, for all X, Y, Z TM and ξ, η, ζ T M.

24 Torsion of N and torsion of N If the torsion T N of N = ( ) N 0 0 t defines a skew-symmetric N 3-tensor T N on TM, we can compare it with the (1, 2)-tensor on M which is the torsion, τ N, of N. Theorem Let M be a smooth manifold. Let N : TM TM be a (1, 1)-tensor, and let N be ( the skew-symmetric ) endomorphism of N 0 TM T M with matrix 0 t. N If N is proportional to an almost cps structure on M, then T N = τ N.

25 Explicit form of equation T N = τ N The explicit form of equation T N = τ N is (T N )(X + ξ, Y + η), Z + ζ = (τ N )(X, Y, ζ) + (τ N )(Y, Z, ξ) + (τ N )(Z, X, η), for all sections X + ξ, Y + η, Z + ζ of TM T M.

26 Deformations of Dorfman brackets The preceding theorem implies that, ( if N 2 = ) λid TM and N is a N 0 Nijenhuis tensor on M, then N = 0 t, is a Nijenhuis N operator on TM. The preceding theorem admits a converse. Theorem( ) N 0 If N = 0 t, is a Nijenhuis operator on TM, then N necessarily N 2 is a scalar mulitple of the identity of TM and the torsion of N vanishes.

27 The double of a deformed bracket ( ) N 0 Under the hypothesis that N = 0 t is a Nijenhuis N operator, the following constructions give the same result: Constructing the double of the deformed bracket [, ] N, [X + ξ, Y + η] (N) = [X, Y ] N + L N X η i Y (d N ξ), where d N and L N X are defined by Y, d Nξ = NY, dξ and L N X = i X d N + d N i X. Deforming the Dorfman bracket by N, [u, v] N = [N u, v] + [u, N v] N [u, v], where u = X + ξ, v = Y + η are sections of TM, [X + ξ, Y + η] = [X, Y ] + L X η i Y (dx), and N (X + ξ) = NX t Nξ.

28 Weakly deforming tensors The preceding result is valid under a much weaker hypothesis. We do not assume that N 2 is proportional to the identity of TM, only that the torsion of N vanishes. In fact, in general, the torsion of N does not vanish, but N is a weakly deforming tensor, i.e., a quadratic expression in N defined in terms of the big bracket of sections of (TM T M) is a cocycle in the cohomology associated with the Lie bracket of sections of TM, and it remains true that [, ] N is a Leibniz bracket on TM, and [, ] N is the double of the deformed bracket [, ] N on TM.

29 Example: PN-structures and deforming tensors Various types of composite structures on TM give rise to deformations of the Dorfman bracket of the double of TM. Proposition Let N be a (1, 1)-tensor, and π a bivector on M such that Nπ = π t N. If (π, N) is a PN-structure on M, then the skew-symmetric endomorphism of TM T M, N = is a weakly deforming tensor. ( N π 0 t N Consequence. When (π, N) is a PN-structure on M, then [, ] N is a Courant algebroid structure on TM T M, the double of the Lie bialgebroid ((TM, [, ] N ), (T M, [, ] π )), where [ξ, η] π = L π(ξ) η L π(η) ξ d(π(ξ, η)) is the Fuchssteiner-Magri-... bracket of differential 1-forms on the Poisson manifold (M, π). For the more general case of Lie algebroids, see yks-rubtsov [2010]. ),

30 PN-structures where N 2 is proportional to the identity If N 2 is proportional to the identity of TM, and if π is a bivector such that Nπ = π t N, then N 2 is proportional to the identity of TM T M, and T N is identified with τ N 1 2 [π, π] + 1 2C(π, N), where C(π, N) is the concomitant whose vanishing expresses the compatibility of π and N. Proposition If N 2 is proportional to the identity of TM, and π is a bivector such that Nπ = π t N, then T N = 0 if and only if (π, N) is a PN-structure.

31 Example: ΩN-structures and deforming tensors We can also relate ΩN-structures to deforming tensors, although there is no obvious analogue to the previous proposition. Proposition Let N be a (1, 1)-tensor, and ω a bivector on M such that ωn = t Nω. If (ω, N) is an ΩN-structure on A, then the ( N 0 skew-symmetric endomorphism of TM T M, N = ω is a weakly deforming tensor. t N ),

32 More generally We have in fact presented the particular case of the trivial Lie bialgebroid (TM, T M) of a more general theory (see yks, 2011) that deals with Nijenhuis operators on Courant algebroids. Our description of Nijenhuis structures and related concepts relies on the use of Roytenberg s graded Poisson bracket on the minimal symplectic realization of a Courant algebroid [2002], and on its interplay with the big bracket (Roytenberg [2002], yks [1992, 2004, 2005, 2011]). We have argued that, in the deformation theory of a Courant structure by a skew-symmetric tensor, the decisive property is not the vanishing of the Nijenhuis torsion of the tensor but the property of operators on Courant algebroids which we call weakly deforming. Related work in progress : Paulo Antunes, Camille Laurent-Gengoux and Joana Nunes da Costa on compatible structures on Courant algebroids.

33 Further problems Investigate PN -structures in generalized geometry, Investigate bi-hamiltomian structures and bi-hamiltonian systems in generalized geometry, Investigate ΩN -structures in generalized geometry, Investigate the role of the Nijenhuis tensors and define Nijenhuis relations in the theory of Dirac pairs on general Courant algebroids (Dirac pairs generalize the bi-hamiltonian structures.) Relate Nijenhuis operators on Courant algebroids to recursion operators acting on conservation laws...? Integrable systems in generalized geometry...?

34 For Franco Magri on his 65th birthday, auguri meilleurs voeux de bon anniversaire best wishes ***

35 Advances in the theory of Nijenhuis operators Fuchssteiner [1997] for general algebraic structures, Bedjaoui-Tebbal [2000], on the study of contractions of Lie algebras (in the sense of Inonü-Wigner), Cariñena, Grabowski and Marmo [2001], on the study of contractions and deformations of both Lie algebras and Leibniz (Loday) algebras, Cariñena, Grabowski and Marmo [2004], on the study of Leibniz algebroids, in particular Courant algebroids, Clemente-Gallardo and Nunes da Costa [2004] on the case of Courant algebroids, Grabowski [2006] on the supermanifold approach to Courant algebroids.

36 Some references on Lie bialgebras, Lie bialgebroids, Courant algebroids B. Kostant and S. Sternberg, Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Physics 176, P. Lecomte et C. Roger, Modules et cohomologies des bigèbres de Lie, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990). yks, Jacobian quasi-bialgebras and quasi-poisson Lie groups, Contemp. Math. 132, yks, From Poisson algebras to Gerstenhaber algebras, Ann. Fourier 46 (1996). yks, Derived brackets, Lett. Math. Phys. 69 (2004). T. Voronov, Contemp. Math. 315, D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002). D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Contemp. Math. 315, Zhang-Ju Liu, A. Weinstein, Ping Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997).