nijenhuis tensors in generalized geometryfalqui/magri2011/slides/kosmann.pdf · from quaderno s 19...

Nijenhuis tensors in generalized geometry

Yvette Kosmann-SchwarzbachCentre de Mathematiques Laurent Schwartz, Ecole Polytechnique, France

Bi-Hamiltonian Systems and All ThatInternational 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

I Franco Magri, A simple model of the integrable Hamiltonianequation, J. Math. Phys. 19 (1978), 1156–1162 [18 April 1977].

I A geometrical approach to the nonlinear solvable equations. inNonlinear evolution equations and dynamical systems (Lecce,1979), Lecture Notes in Phys., 120, Springer, 1980, 233–263.

N ′u(X ,NuY )− N ′u(Y ,NuX ) = Nu(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’).

I with Carlo Morosi, A geometrical characterization ofintegrable hamiltonian systems through the theory ofPoisson-Nijenhuis manifolds, Quaderno S 19, Milan, 1984.(Re-issued: Universita di Milano Bicocca, Quaderno 3, 2008.http://home.matapp.unimib.it)

[NX ,NY ]− N([NX ,Y ] + [X ,NY ]) + N2[X ,Y ] = 0.


Prehistory

In fact, there was a prehistory for this story:KdV and its (first) Hamiltonian structure.

I Clifford Gardner, John Greene, Martin Kruskal, Robert Miura,Norman Zabusky, 1965, 1968, 1970, 1974.

I Ludwig Faddeev and Vladimir Zakharov, 1971.

I Israel Gel’fand and Leonid Dikii, 1975.

I Peter Lax, 1976, “recursion formula of Lenart”.

I Peter Olver, 1977, “recursion operator”.

(See the historical notes in Olver’s book, and “Andrew Lenard: A MysteryUnraveled” by Jeffery Praught and Roman G. Smirnov, SIGMA 1 (2005).)

In the early 1980’s, Benno Fucchsteiner, Dan Gutkin, GiuseppeMarmo, Boris Konopelchenko, Orlando Ragnisco, ...


1983, Pseudocociclo di Poisson

I Pseudocociclo di Poisson e strutture PN gruppale,applicazione al reticolo di Toda,Magri’s unpublished manuscript, Milan 1983.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 andPoisson-Lie groups avant la lettre

= “Hamiltonian Lie groups”=“Poisson-Drinfeld groups” = “Poisson-Lie groups”


From Quaderno S 19 (1984) to generalized geometry

I with C. Morosi, Su possibili applicazioni della riduzione distrutture geometriche nella teoria dei sistemi dinamici,AIMETA Trieste 1984, 135–144.

I with C. Morosi and Orlando Ragnisco, Reduction techniquesfor infinite-dimensional Hamiltonian systems: some ideas andapplications, Comm. Math. Phys. 99 (1985), 115–140.

I with C. Morosi, Old and new results on recursion operators:an algebraic approach to KP equation, in Topics in solitontheory and exactly solvable nonlinear equations(Oberwolfach, 1986), World Sci., 1987, 78–96.

I with C. Morosi and G. Tondo, Nijenhuis G -manifolds andLenard bicomplexes: a new approach to KP systems, Comm.Math. Phys. 115 (1988), 457–475.


From Quaderno S 19 (1984) to generalized geometry (cont’d)

I yks, The modified Yang-Baxter equation and bi-Hamiltonianstructures, in Differential geometric methods in theoreticalphysics (Chester, 1988), World Sci., 1989, 12–25.“The results presented here are joint work with Franco Magri.”

I with yks, Poisson–Nijenhuis structures, Ann. Inst. H. PoincarePhys. Theor. 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

I with yks, Dualization and deformation of Lie brackets onPoisson manifolds, in Differential geometry and itsapplications (Brno, 1989), World Sci., 1990, 79–84.



I with Pablo Casati and Marco Pedroni, Bi-Hamiltonianmanifolds and Sato’s equations, in Integrable systems, TheVerdier Memorial Conference (Luminy, 1991), Progr. Math.,115, Birkhauser, 1993, 251–272.

I 1993, Peter Olver (Canonical forms for bi-Hamiltonian systems)

I 1993, Rober Brouzet, Pierre Molino and Javier Turiel(Geometrie des systemes bihamiltoniens)

I 1993, Gel’fand and Ilya Zakharevich (On the local geometry of a

bi-Hamiltonian structure), 1998 Panasyuk (Veronese webs for

bi-Hamiltonian structures)

I 1994, 1997, Izu Vaisman (A lecture on Poisson–Nijenhuis structures)

I ....................................I ....................................I 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).



I yks, The Lie bialgebroid of a Poisson-Nijenhuis manifold, Lett.Math. Phys. 38 (1996), 421–428.“This result was first conjectured by Magri during aconversation that we held at the time of the SemestreSymplectique at the Centre Emile Borel (1994).”

(The program on Symplectic Geometry was the first organized in the newCentre Emile Borel in the renovated Institut Henri Poincare in Paris.)

Meanwhile the theory of Lie algebroids, Lie bialgebroids,generalized tangent bundles and Courant algebroids developped.

I yks and Vladimir Rubtsov, Compatible structureson Lie algebroids and Monge-Ampere operators,Acta Appl. Math., 109 (2010), 101-135.

I yks, Nijenhuis structures on Courant algebroids,Bull. Brazilian Math. Soc., to appear (arXiv1102.1410).


What is new?

Our aim is to

• point out the new features in the theory of Nijenhuis operatorson generalized tangent bundles of manifolds,

• study the (infinitesimal) deformations of generalized tangentbundles.

• show that PN-structures and ΩN-structures on a manifold define(infinitesimal) deformations of its generalized tangent bundle.


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

〈X + ξ,Y + η〉 = 〈X , η〉+ 〈Y , ξ〉,

• the Dorfman bracket

[X + ξ,Y + η] = [X ,Y ] + LXη − iY (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] = [[iX , d ], eη].For derived brackets, see yks, Ann. Fourier 1996, LMP 2004.


Properties of the Dorfman and Courant brackets

The Dorfman bracket is not skew-symmetric, but since it is aderived bracket, it is a Leibniz (Loday) bracket, i.e., it satisfies theJacobi 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 ] + LXη − LY ξ +1

2〈X + ξ,Y + η〉.

The Courant bracket is skew-symmetric but it does not satisfy theJacobi 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.


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〉〉.


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?


Nijenhuis torsion

Let N be an endomorphism of TM, i.e., a (1, 1)-tensor on thevector bundle TM.

We define the Nijenhuis torsion, or simply the torsion of N by

(TN )(u, v) == [Nu,N v ]−N ([Nu, 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 aNijenhuis operator on TM.


Deformed bracket

We define, for all u, v ∈ Γ(TM),

[u, v ]N = [Nu, v ] + [u,N v ]−N [u, v ] .

Then(TN )(u, v) = [Nu,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 Leibnizbracket: 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 = [Nu,N v ],i.e., N is a morphism from (Γ(TM), [ , ]N ) to (Γ(TM), [ , ]).


Warning

The torsion of N : TM → TM is a map,

TN : Γ(TM)× Γ(TM)→ Γ(TM).

Unlike the usual case of tangent bundles (and Lie algebroids), TNis not in general C∞(M)-linear in both arguments, and in generalit is not skew-symmetric.

Consequence. We shall have to consideer the case ofendomorphisms of TM which are skew-symmetric and whosesquare is proportional to the identity.


Skew-symmetric endomorphisms of TM

Let N be an endomorphism of TM, i.e., a (1, 1)-tensor on thevector bundle TM. It is skew-symmeric if N + tN = 0, i.e., if it isof the form

N =

(N πω − tN

),

where N : TM → TM and tN : T ∗M → T ∗M is the transpose ofN,π : T ∗M → TM is a bivector on M, andω : TM → T ∗M is a 2-form on M.

The skew-symmetric endomorphisms of TM are those that leavethe bilinear form 〈 , 〉 infinitesimally invariant.

More generally, one can consider paired endomorphisms, satisfyingN + tN = 2κ IdTM , where κ is a scalar. See J. Carinena, J. Grabowski andG. Marmo, Courant algebroid and Lie bialgebroid contractions,J. Phys. A 37(2004).


Generalized almost complex structures

A skew-symmetric endomorphism N of TM such thatN 2 = λ IdTM , where λ = −1, 0 or 1, is called ageneralized almost cps structure on TM (or “on M”).A generalized almost cps structure is called a generalized cpsstructure 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 almostcomplex structure if it is skew-symmetric and

N 2 = −IdTM .

Clearly, if N is an almost complex structure on M (i.e., N : TM → TM and

N2 = −IdM), and if either π or ω vanishes, then N =

„N πω − tN

«is a

generalized almost complex structure on TM.


Use of the Courant bracket

One can also define the torsion T CN of an endomorphism N with

respect to the Courant bracket, replacing the Dorfman bracket byits skew-symmetrization in the preceding formulas. The relationbetween the two torsions is

(T CN )(u, v) =

1

2((TN )(u, v)− (TN )(v , u)) .

Proposition

(i) For a skew-symmetric endomorphism N ,(T CN − TN

)(u, v) =

1

2

(∂〈u,N 2v〉 − N 2∂〈u, v〉

),

for all sections u and v of TM.(ii) If N is proportional to a generalized almost cps structure, bothtorsions, T C

N and TN , coincide.


Questions

Consider a smooth manifold M and N a skew-symmetricendomorphism of TM.

• When is TN a C∞(M)-linear section of TM ⊗ ∧2(TM)∗?Answer When N is proportional to a generalized almost cpsstructure.

• When is TN a section of ∧3(TM)?Answer When N is proportional to a generalized almost cpsstructure.

• Compare TN with the Nijenhuis torsion τN of N?Answer When N is proportional to a generalized almost cpsstructure, they are equal in a suitable sense.


Properties of the torsion. Lack of C∞-linearityand of skew-symmetry

Let N be a skew-symmetric endomorphism of TM.

Lack of C∞-linearity.It is clear that

(TN )(u, fv) = f (TN )(u, v),

but

(TN )(fu, v) = f (TN )(u, v) + 〈u, v〉N 2(∂f )− 〈u,N 2v〉∂f .

Lack of skew-symmetry.Using the fact that N is skew-symmetric, we obtain

(TN )(u, v) + (TN )(v , u) = N 2∂〈u, v〉 − ∂〈u,N 2v〉.


Associated 3-tensor

In order to determine whether TN determines a skew-symmetriccovariant 3-tensor, we use the skew-symmetry of N and relationsstated above to obtain

〈(TN )(u, v),w〉+ 〈(TN )(u,w), v〉 = 〈N 2[u,w ]− [u,N 2w ], v〉.

TheoremIf N is proportional to a generalized almost cps structure (i.e., Nis skew-symmetric and N 2 = λ IdTM), the torsion of N isC∞(M)-linear in both arguments and skew-symmetric, and defines

a skew-symmetric covariant 3-tensor on TM, TN , by

TN (u, v ,w) = 〈(TN )(u, v),w〉 .


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.


Torsion of N and torsion of N

If the torsion TN of N =

(N 00 −tN

)defines a skew-symmetric

3-tensor TN on TM, we can compare it with the (1, 2)-tensor onM which is the torsion, τN , of N.

TheoremLet M be a smooth manifold. Let N : TM → TM be a(1, 1)-tensor, and let N be the skew-symmetric endomorphism of

TM ⊕ T ∗M with matrix

(N 00 − tN

).

If N is proportional to an almost cps structure on M, then

TN = τN .


Explicit form of equation TN = τN

The explicit form of equation TN = τN is

〈(TN )(X + ξ,Y + η),Z + ζ〉= (τN)(X ,Y , ζ) + (τN)(Y ,Z , ξ) + (τN)(Z ,X , η),

for all sections X + ξ,Y + η,Z + ζ of TM ⊕ T ∗M.


Deformations of Dorfman brackets

The preceding theorem implies that, if N2 = λIdTM and N is a

Nijenhuis tensor on M, then N =

(N 00 − tN

), is a Nijenhuis

operator on TM.

The preceding theorem admits a converse.

Theorem

If N =

(N 00 − tN

), is a Nijenhuis operator on TM, then

necessarily N2 is a scalar mulitple of the identity of TM and thetorsion of N vanishes.


The double of a deformed bracket

Under the hypothesis that N =

(N 00 − tN

)is a Nijenhuis

operator, the following constructions give the same result:

• Constructing the double of the deformed bracket [ , ]N ,

[X + ξ,Y + η](N) = [X ,Y ]N + LNXη − iY (dNξ),

where dN and LNX are defined by 〈Y ,dNξ〉 = 〈NY , dξ〉 and

LNX = iXdN + dN iX .

• Deforming the Dorfman bracket by N ,

[u, v ]N = [Nu, v ] + [u,N v ]−N [u, v ],

where u = X + ξ, v = Y + η are sections of TM,[X + ξ,Y + η] = [X ,Y ] + LXη − iY (dx), andN (X + ξ) = NX − tNξ.


Weakly deforming tensors

The preceding result is valid under a much weaker hypothesis. Wedo not assume that N2 is proportional to the identity of TM, onlythat the torsion of N vanishes. In fact, in general, the torsion of Ndoes not vanish, but

• N is a weakly deforming tensor, i.e., a quadratic expression in Ndefined in terms of the “big bracket” of sections of∧•(TM ⊕ T ∗M) is a cocycle in the cohomology associated withthe 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.


Example: PN-structures and deforming tensors

Various types of composite structures on TM give rise todeformations of the Dorfman bracket of the double of TM.

Proposition

Let N be a (1, 1)-tensor, and π a bivector on M such thatNπ = π tN. If (π,N) is a PN-structure on M, then the

skew-symmetric endomorphism of TM ⊕ T ∗M, N =

(N π0 − tN

),

is a weakly deforming tensor.

Consequence. When (π,N) is a PN-structure on M, then [ , ]N isa Courant algebroid structure on TM ⊕ T ∗M, the double of theLie bialgebroid ((TM, [ , ]N), (T ∗M, [ , ]π)), where[ξ, η]π = Lπ(ξ)η − Lπ(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].


PN-structures where N2 is proportional to the identity

If N2 is proportional to the identity of TM, and if π is a bivectorsuch that Nπ = π tN, then N 2 is proportional to the identity ofTM ⊕ T ∗M, and TN is identified with τN − 1

2 [π, π] + 12 C (π,N),

where C (π,N) is the concomitant whose vanishing expresses thecompatibility of π and N.

Proposition

If N2 is proportional to the identity of TM, and π is a bivectorsuch that Nπ = π tN, then TN = 0 if and only if (π,N) is aPN-structure.


Example: ΩN-structures and deforming tensors

We can also relate ΩN-structures to deforming tensors, althoughthere is no obvious analogue to the previous proposition.

Proposition

Let N be a (1, 1)-tensor, and ω a bivector on M such thatωN = tNω. If (ω,N) is an ΩN-structure on A, then the

skew-symmetric endomorphism of TM ⊕ T ∗M, N =

(N 0ω − tN

),

is a weakly deforming tensor.


More generally

We have in fact presented the particular case of the trivial Liebialgebroid (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 relieson the use of Roytenberg’s graded Poisson bracket on the minimalsymplectic realization of a Courant algebroid [2002], and on itsinterplay with the big bracket (Roytenberg [2002], yks [1992, 2004,2005, 2011]).

We have argued that, in the deformation theory of a Courantstructure by a skew-symmetric tensor, the decisive property is notthe vanishing of the Nijenhuis torsion of the tensor but the propertyof operators on Courant algebroids which we call weakly deforming.•Related work in progress :

I Paulo Antunes, Camille Laurent-Gengoux and Joana Nunes daCosta on compatible structures on Courant algebroids.


Further problems

I Investigate PN -structures in generalized geometry,

I Investigate bi-Hamiltomian structures and bi-Hamiltoniansystems in generalized geometry,

I Investigate ΩN -structures in generalized geometry,

I Investigate the role of the Nijenhuis tensors and defineNijenhuis relations in the theory of Dirac pairs on generalCourant algebroids (Dirac pairs generalize the bi-Hamiltonianstructures.)

I Relate Nijenhuis operators on Courant algebroids to recursionoperators acting on conservation laws..........?

I Integrable systems in generalized geometry.....................?


For Franco Magri on his 65th birthday,

augurimeilleurs voeux de bon anniversaire

best wishes

***


Advances in the theory of Nijenhuis operators

• Fuchssteiner [1997] for general algebraic structures,• Bedjaoui-Tebbal [2000], on the study of contractions of Liealgebras (in the sense of Inonu-Wigner),• Carinena, Grabowski and Marmo [2001], on the study ofcontractions and deformations of both Lie algebras and Leibniz(Loday) algebras,• Carinena, Grabowski and Marmo [2004], on the study of Leibnizalgebroids, in particular Courant algebroids,• Clemente-Gallardo and Nunes da Costa [2004] on the case ofCourant algebroids,• Grabowski [2006] on the supermanifold approach to Courantalgebroids.


Some references onLie bialgebras, Lie bialgebroids, Courant algebroids

B. Kostant and S. Sternberg, Symplectic reduction, BRS cohomology,and infinite-dimensional Clifford algebras, Ann. Physics 176, 1987.

P. Lecomte et C. Roger, Modules et cohomologies des bigebres de Lie, C.R. Acad. Sci. Paris Ser. I Math. 310 (1990).

yks, Jacobian quasi-bialgebras and quasi-Poisson Lie groups, Contemp.Math. 132, 1992.

yks, From Poisson algebras to Gerstenhaber algebras, Ann. Fourier 46(1996).

yks, Derived brackets, Lett. Math. Phys. 69 (2004).

T. Voronov, Contemp. Math. 315, 2002.

D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds,Lett. Math. Phys. 61 (2002).

D. Roytenberg, On the structure of graded symplectic supermanifoldsand Courant algebroids, in Contemp. Math. 315, 2002.

Zhang-Ju Liu, A. Weinstein, Ping Xu, Manin triples for Lie bialgebroids,J. Differential Geom. 45 (1997).


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