Nijenhuis tensors in generalized geometry Yvette Kosmann-Schwarzbach - - PowerPoint PPT Presentation

Nijenhuis tensors in generalized geometry

Yvette Kosmann-Schwarzbach

Centre de Math´ ematiques Laurent Schwartz, ´ 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, 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’).

◮ 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 ]) + N2[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applicazione al reticolo di Toda, Magri’s unpublished manuscript, Milan 1983.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

• n 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
• n 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 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.

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 TM = TM ⊕ T ∗M. The Courant bracket is the skew-symmetrized Dorfman bracket, [X + ξ, Y + η] == [X, Y ] + LXη − LY ξ + 1 2X + ξ, Y + η. 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.

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 TM? Define the Nijenhuis operators on TM? 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 TM, i.e., a (1, 1)-tensor on the vector bundle TM. We define the Nijenhuis torsion, or simply the torsion of N by (TN )(u, v) == [Nu, Nv] − N([Nu, v] + [u, Nv]) + 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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Deformed bracket

We define, for all u, v ∈ Γ(TM), [u, v]N = [Nu, v] + [u, Nv] − N[u, v] . Then (TN )(u, v) = [Nu, Nv] − 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 = [Nu, Nv],

i.e., N is a morphism from (Γ(TM), [ , ]N ) to (Γ(TM), [ , ]).

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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), TN 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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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 + tN = 0, i.e., if it is

• f the form

N = N π ω − tN

• ,

where N : TM → TM and tN : 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 + tN = 2κ IdTM, where κ is a scalar. See J. Cari˜ nena, J. Grabowski and

• G. Marmo, Courant algebroid and Lie bialgebroid contractions,
• J. Phys. A 37(2004).

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Generalized almost complex structures

A skew-symmetric endomorphism N of TM such that N 2 = λ IdTM, 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 = −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 π ω − tN « is a generalized almost complex structure on TM.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Use of the Courant bracket

One can also define the torsion T C

N 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 ((TN )(u, v) − (TN )(v, u)) .

Proposition

(i) For a skew-symmetric endomorphism N,

• T C

N − 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, both torsions, T C

N and TN , coincide.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Questions

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

• When is TN a C ∞(M)-linear section of TM ⊗ ∧2(TM)∗?

Answer When N is proportional to a generalized almost cps structure.

• When is TN a section of ∧3(TM)?

Answer When N is proportional to a generalized almost cps structure.

• Compare TN with the Nijenhuis torsion τN of N?

Answer When N is proportional to a generalized almost cps structure, they are equal in a suitable sense.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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 (TN )(u, fv) = f (TN )(u, v), but (TN )(fu, v) = f (TN )(u, v) + u, vN 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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Associated 3-tensor

In order to determine whether TN determines a skew-symmetric covariant 3-tensor, we use the skew-symmetry of N and relations stated above to obtain (TN )(u, v), w + (TN )(u, w), v = N 2[u, w] − [u, N 2w], v.

Theorem

If N is proportional to a generalized almost cps structure (i.e., N is skew-symmetric and N 2 = λ IdTM), the torsion of N is C ∞(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 .

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Torsion of N and torsion of N

If the torsion TN of N = N −tN

• defines a skew-symmetric

3-tensor TN on TM, we can compare it with the (1, 2)-tensor on M which is the torsion, τN, 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 TM ⊕ T ∗M with matrix N − tN

• .

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

• TN =

τN .

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Deformations of Dorfman brackets

The preceding theorem implies that, if N2 = λIdTM and N is a Nijenhuis tensor on M, then N = N − tN

• , is a Nijenhuis
• perator on TM.

The preceding theorem admits a converse.

Theorem

If N = N − tN

• , is a Nijenhuis operator on TM, then

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

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

The double of a deformed bracket

Under the hypothesis that N = N − tN

• is a Nijenhuis
• perator, the following constructions give the same result:
• Constructing the double of the deformed bracket [ , ]N,

[X + ξ, Y + η](N) = [X, Y ]N + LN

Xη − iY (dNξ),

where dN and LN

X are defined by Y , dNξ = NY , dξ and

LN

X = iXdN + dNiX.

• Deforming the Dorfman bracket by N,

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

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

Weakly deforming tensors

The preceding result is valid under a much weaker hypothesis. We do not assume that N2 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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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 tN. If (π, N) is a PN-structure on M, then the skew-symmetric endomorphism of TM ⊕ T ∗M, N = N π − tN

• ,

is a weakly deforming tensor.

• 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 = 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].

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

PN-structures where N2 is proportional to the identity

If N2 is proportional to the identity of TM, and if π is a bivector such that Nπ = π tN, then N 2 is proportional to the identity of TM ⊕ T ∗M, and TN is identified with τN − 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 N2 is proportional to the identity of TM, and π is a bivector such that Nπ = π tN, then TN = 0 if and only if (π, N) is a PN-structure.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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 = tNω. If (ω, N) is an ΩN-structure on A, then the skew-symmetric endomorphism of TM ⊕ T ∗M, N = N ω − tN

• ,

is a weakly deforming tensor.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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

• n 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

• f 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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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

• perators acting on conservation laws..........?

◮ Integrable systems in generalized geometry.....................?

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

For Franco Magri on his 65th birthday,

auguri meilleurs voeux de bon anniversaire best wishes ***

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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 Inon¨ u-Wigner),

• Cari˜

nena, Grabowski and Marmo [2001], on the study of contractions and deformations of both Lie algebras and Leibniz (Loday) algebras,

• Cari˜

nena, 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.

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry

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, 1987.

• P. Lecomte et C. Roger, Modules et cohomologies des big`

ebres de Lie, C.

• R. Acad. Sci. Paris S´
• er. 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 supermanifolds

and Courant algebroids, in Contemp. Math. 315, 2002. Zhang-Ju Liu, A. Weinstein, Ping Xu, Manin triples for Lie bialgebroids,

• J. Differential Geom. 45 (1997).

Bi-Hamiltonian Systems ... In honour of Franco Magri Nijenhuis tensors in generalized geometry