FLAT PENCILS OF SYMPLECTIC CONNECTIONS AND
HAMILTONIAN OPERATORS OF DEGREE 2
JAMES T. FERGUSON
Abstract. Bi-Hamiltonian structures involving Hamiltonian operators of de-
gree 2 are studied. Firstly, pairs of degree 2 operators are considered in terms
of an algebra structure on the space of 1-forms, related to so-called Fermionic
Novikov algebras. Then, degree 2 operators are considered as deformations of
hydrodynamic type Poisson brackets.
1. Introduction
Hamilton’s equations for a finite-dimensional system with position coordinates
qi and associated momenta pi,
are understood geometrically as describing the flow of a vector field XH which is
associated with the Hamiltonian function H(q1, . . . , qn, p1 . . . , pn) by the formula
XH(f) = {f,H}, where {·, ·} is the Poisson bracket:
{f, g} =
. (1)
More generally, one defines a Poisson bracket on an n-dimensional manifoldM as
a map C∞(M)× C∞(M) → C∞(M), (f, g) 7→ {f, g}, satisfying, for any functions
f, g, h on M :
(1) antisymmetry: {f, g} = −{g, f} ,
(2) linearity: {af + bg, h} = a{f, h}+ b{g, h} for any constants a, b ,
(3) product rule: {fg, h} = f{g, h}+ g{f, h} ,
(4) Jacobi identity: {{f, g}, h}+ {{g, h}, f}+ {{h, f}, g} = 0 .
The conditions 1-3 identify {·, ·} as a bivector: a rank two, antisymmetric, con-
travariant tensor field ω on M . It can therefore be represented, by introducing
coordinates {ui} on M , as a matrix of coefficients ωij , giving
ω = ωij
{f, g} = ωij
. (2)
The Jacobi identity places the following constraint on the components of ω:
+ ωjr
+ ωkr
= 0 . (3)
If the matrix ωij is non-degenerate, we may introduce its inverse ωij , satisfying
rj = δ
i . The Jacobi identity for ω
ij is equivalent to the closedness of ωij .
Date: April 11, 2007.
http://arxiv.org/abs/0704.1401v1
2 JAMES T. FERGUSON
We refer to a closed non-degenerate two-form as a symplectic form, and a mani-
fold equipped with one as a symplectic manifold. Darboux’s theorem asserts that
on any 2n-dimensional symplectic manifold there exists a set of local coordinates
{q1, . . . , qn, p1 . . . , pn} in which the Poisson bracket takes the form (1); i.e. the
components of ωij , and so those of ωij , are constant.
One may also introduce Poisson brackets on infinite-dimensional manifolds. The
loop space of a finite-dimensional manifold M , L(M), is the space of smooth maps
u : S1 →M . Poisson brackets relating Hamiltonians to flows in L(M) will therefore
act on functionals mapping L(M) → R. In [5],[6] Dubrovin and Novikov studied
the so-called Poisson brackets of differential-geometric type, which are of the form
{f, g} =
dx (4)
where ui are coordinates on the target space M , and x is the coordinate on S1.
P ij is a matrix of differential operators (in d
), with no explicit dependence on x,
which is assumed to be polynomial in the derivatives uix, u
xx, . . . . If {·, ·} defines a
Poisson bracket on the loop space then P is referred to as a Hamiltonian operator.
There is a grading on such operators, preserved by diffeomorphisms of M , given
by assigning degree 1 to d
, and degree n to the nth x-derivative of each field ui. An
important class is the hydrodynamic type Poisson brackets, which are homogeneous
of degree 1:
P ij = gij(u)
(u)ukx .
According to the programme set out by Novikov [15], differential-geometric type
Poisson brackets on L(M) should be studied in terms of finite-dimensional differ-
ential geometry on the target space M . When expanded as a polynomial in d
the field derivatives, the coefficients, which are functions of the fields ui alone, can
often be naturally related to known objects of differential geometry, or else used to
define new ones. In the hydrodynamic case, for instance, with gij non-degenerate,
P is Hamiltonian if and only if gij is a flat metric on M and Γkij = −girΓ
j are the
Christoffel symbols of its Levi-Civita connection.
In [7] Dubrovin considered the geometry of bi-Hamiltonian structures of Hy-
drodynamic operators, that is pairs of such operators compatible in the sense of
[13], that every linear combination of them also determines a Poisson bracket. In
particular, he introduced a multiplication of covectors on M and expressed the
compatibility of the operators in terms of a quadratic relations on this algebra.
This paper is principally concerned with Hamiltonian operators which are ho-
mogeneous of degree 2. Section 2 presents the differential geometry of such op-
erators, and in particular relates the subclass which can be put into a constant
form by a change of coordinates on M to symplectic connections. Section 3 then
considers pairs of operators from this subclass, and the algebraic constraints their
compatibility places upon the associated multiplication. In section 4 inhomoge-
neous bi-Hamiltonian structures consisting of a degree 1 and a degree 2 operator
are studied.
2. Hamiltonian Operators of Degree 2
We begin with a review of known results on Hamiltonian operators of degree 2:
ij = aij
x + c
xx, (5)
in which the matrix aij is assumed to be non-degenerate. Such operators have been
considered already in, for example, [17], [14], [4], [15], in which the (conditional)
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 3
Darboux theorem has been discussed. In preparation for the bi-Hamiltonian theory
we present these results without the use of special coordinates.
Under the change of coordinates ũi = ũi(up) the coefficients in P ij transform as
bpqr − 2
∂ũk∂ũs
apq ,
cpqr −
∂ũk∂ũs
apq ,
cpqrs +
∂ũk∂ũl
∂2ũj
∂uq∂us
∂ũ(k
∂ũl)
bpqr +
∂3ũj
∂uq∂ur∂us
∂2ũj
∂uq∂ur
∂ũk∂ũl
, (6)
where the brackets denote symmetrisation. So in particular aij transforms as a
rank 2 contravariant tensor on the target space and b
and c
are related to
Christoffel symbols of connections by b
= −2airΓ̄
and c
= −airΓ
. Call
these connections ∇̄ and ∇ respectively.
The transformation rules for c
are not determined uniquely by those for P ,
since (5) sees only the part symmetric in k and l. To fix c
, we always assume the
antisymmetric part is zero. Denote by aij the inverse of a
ij defined by aira
rj = δ
The condition that the operation defined in (4) is skew-symmetric and satisfies
the Jacobi identity places constraints on the coefficients appearing in (5).
Theorem 2.1. The operator P in equation (5) defines a Poisson bracket by equa-
tion (4) if and only if
(A) aij = −aji ,
(B) ∇ka
ij = b
(C) air
bjkr − 2c
= akr
bijr − 2c
(D) ∇ is flat (zero torsion, zero curvature) ,
(E) c
(k,l)
− aprc
Proof. [14] states that, by virtue of being Hamiltonian, the operator (5) can be put
in the form
P ij = aij
, (7)
by a change of coordinates ui = ui(ũ), and that for an operator of this shorter form
to be Hamiltonian is equivalent to the three conditions
(a) aij = −aji ,
(b) aij ,k = b
(c) airbjkr = a
jrbkir .
We first assume that P is a Poisson bracket, so there exists the special coordinates
in which P takes the form (7) and (a)-(c) hold. By reversing the change of variables
as ũi = ũi(u), conditions (A)-(C) of Theorem 2.1 are Mokhov’s three conditions
converted to tensorial identities. That ∇ is flat follows from its Christoffel symbols,
Γkij = −airc
j , being zero in the u coordinates.
4 JAMES T. FERGUSON
The formula in condition (E) is derived from the transformation rules above. In
changing from flat coordinates ui to coordinates ũi they give:
∂2ũj
∂uq∂us
∂ũ(k
∂ũl)
bpqr +
∂3ũj
∂uq∂ur∂us
∂2ũj
∂uq∂ur
∂ũk∂ũl
apq ,
∂ũk∂ũs
apq ,
∂2ũj
∂uq∂ur
apq ,
where the last line has used the identity
∂2ũi
∂ur∂us
∂ũj∂ũk
= 0 ,
which is a differential consequence of ∂ũ
= δij .
∂2ũi
∂up∂us
∂2ũj
∂ur∂uq
∂3ũj
∂uq∂ur∂us
∂2ũj
∂uq∂ur
∂ũk∂ũl
∂2ũj
∂uq∂ur
from which we see
(k,l)
∂2ũi
∂up∂us
∂2ũj
∂ur∂uq
∂ũ(l
∂ũk)
apq .
This last term can be seen to be
ãprc̃
(k c̃
Conversely, if (A)-(E) hold, the flatness of ∇ asserts the existence of coordinates
in which c
= 0, and condition (E) then asserts that c
= 0 in these coordinates.
If we take, as a simple case, an operator P as in (5) with b
constants,
and assume c
to be defined by (E), then P is Hamiltonian if and only if aij =
uk + A
0 where A
0 are constants with A
, Airl c
r = A
cikr ,
Air0 c
r = A
r and c
rk + cikr c
If we take an algebraA with basis {e1, . . . , en}, n = dimM, and use c
and A
define a multiplication, ◦ , and skew-symmetric bilinear form, 〈·, ·〉, by ei◦ej = cijr e
and 〈ei, ej〉 = A
0 , then we may rewrite these conditions as
ei ◦ ej − ej ◦ ei = Aijr e
(I ◦ J) ◦K = −(I ◦K) ◦ J , (8)
Λ(I, J,K) = Λ(J, I,K) , (9)
and 〈I, J ◦K〉 = 〈J, I ◦K〉 ,
for all I, J,K ∈ A, where Λ is the associator of ◦ : Λ(I, J,K) = (I◦J)◦K−I◦(J◦K).
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 5
Algebras satisfying conditions (8) and (9) have appeared before in [18], in the
context of linear hydrodynamic Hamiltonian operators taking values in a completely
odd superspace, where the following definition was proposed:
Definition 2.2. An algebra (A, ◦) satisfying conditions (8) and (9) is called a
Fermionic Novikov algebra.
In [1] Fermionic Novikov algebras in dimensions 2-5 were studied, and the listing
therein provides a source of examples of Hamiltonian operators of degree two.
Example 2.3.
0 0 0 a
0 0 −a −b− (t− 1)u1
0 a 0 c− u2
−a b+ (t− 1)u1 −c+ u2 0
0 0 0 0
0 0 0 u1x
0 0 −u1x 0
0 τu1x u
0 0 0 0
0 0 0 0
0 0 0 (u1x)
0 0 −(u1x)
0 0 0 0
0 0 0 u1x
0 0 −u1xx 0
0 τu1xx u
is Hamiltonian for all values of the constants a, b, c and τ with a 6= 0. This is the
most general Hamiltonian operator associated in the manner discussed above to the
algebra designated (44)τ in [1].
Returning to the general Hamiltonian operator (5), it can be seen from conditions
(B) and (E) in Theorem 2.1 that the coefficients b
and c
in (5) are completely
determined by aij and c
. Thus the Hamiltonian operator on L(M) is represented
uniquely on M by only these latter two objects.
Theorem 2.4. There is a one-to-one correspondence between Hamiltonian opera-
tors of the form (5) on L(M) and pairs (a,∇) on M consisting of a non-degenerate
bivector aij and a torsion-free connection ∇ satisfying two conditions: firstly, that
the curvature of ∇ vanishes, and secondly,
air∇ra
jk = ajr∇ra
ki . (10)
The Christoffel symbols, Γkij , of ∇ are related to c
= −airΓ
. We then
= ∇ka
ij + 2c
− aprc
With this, we may verify the following facts [17],[14]:
Corollary 2.5. For P in (5) a Hamiltonian operator we have
1. Γ is the symmetric part of Γ̄,
2. Let T̄ kij = Γ̄
ij − Γ̄
ji be the torsion of ∇̄. Then T̄ijk = airT̄
jk is skew
symmetric and the forms T̄ = 1
T̄ijkdu
i ∧ duj ∧ duk and a = 1
aijdu
i ∧ duj
are related by 3T̄ = da.
Proof. We begin by noting that equation (10) is equivalent to the condition
∇kaij = ∇iajk (11)
on the two-form aij .
6 JAMES T. FERGUSON
In terms of covariant Christoffel symbols, Theorem 2.4 gives
Γ̄kij =
akr∇raij + Γ
ij , (12)
from which it is clear that Γ̄k
= Γkij .
We therefore also have
∇kaij = Γ̄ijk − Γijk ,
where Γ̄ijk = airΓ̄
jk and Γijk = airΓ
jk. Because ∇ is torsion-free we have
T̄ijk = Γ̄ijk − Γ̄ikj ,
= Γ̄ijk − Γijk − Γ̄ikj + Γikj ,
∇kaij −
∇jaik ,
= ∇kaij ,
= ∇[kaij] ,
(da)ijk .
Lemma 2.6. For a Hamiltonian operator of the form (5), the following three state-
ments, presented in both covariant and contravariant forms, are equivalent:
1. The 2-form a is closed (and so symplectic), or equivalently aij satisfies
equation (3) (and so defines a Poisson bracket on M by equation (2));
2. ∇ka
ij = 0, i.e. ∇kaij = 0;
, i.e. Γkij = Γ̄
Proof. We see, from the characterisation of Hamiltonian operators given in Theorem
aij is Poisson ⇐⇒ airajk,r + a
jraki,r + a
kraij,r = 0
⇐⇒ air∇ra
jk + ajr∇ra
ki + akr∇ra
ij = 0
⇐⇒ 3akr∇ra
ij = 0
⇐⇒ ∇ka
ij = 0 ,
Lemma 2.6 therefore tells us that in the special case where the leading coefficient
in P is the inverse of a symplectic form, the pair (a,∇) defining P can be thought
of as containing the symplectic form aij , and a torsionless connection compatible
with it (in the sense that ∇a = 0); that is, a symplectic connection. More precisely
(see e.g. [3]):
Definition 2.7. A symplectic connection on a symplectic manifold (M,ω) is a
smooth connection ∇ which is torsion-free and compatible with the symplectic form
ω, i.e.
∇XY −∇YX − [X,Y ] = 0
(∇ω) (X,Y, Z) = X(ω(Y, Z))− ω(∇XY, Z)− ω(Y,∇Y Z) = 0 ,
where X,Y and Z are vector fields on M .
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 7
In local coordinates {xi}, introducing Christoffel symbols Γkij for ∇ and writing
ω = 1
ωijdx
i ∧ dxj , the conditions for ∇ to be a symplectic connection read Γkij =
Γkji, as usual, and
∇kωij =
− Γrkiωrj − Γ
kjωir = 0 . (13)
This definition is analogous to that of the Levi-Civita connection of a pseudo-
Riemannian metric, however there is an important difference in that the Levi-Civita
connection is uniquely specified by its metric. From the compatibility condition (13)
it can be seen that if Γkij are the Christoffel symbols of a symplectic connection for
ω, then the connection with Christoffel symbols Γ̃kij = Γ
ij + ω
krSrij is a sym-
plectic connection if and only if the tensor Sijk is completely symmetric. In [10]
a symplectic manifold with a specified symplectic connection is called, in light of
[9], a Fedosov manifold. Here we call the pair (ω,∇) of a symplectic form and a
symplectic connection a Fedosov structure on M, and call the structure flat if ∇ is
flat.
In the discussion of Hamiltonian operators it is convenient to work with con-
travariant quantities. We call
= −ωirΓ
the contravariant Christoffel symbols of the symplectic connection.
Result 2.8. The compatibility of ∇ and ω is equivalent to
Result 2.9. ∇ being torsion-free is equivalent to ωirΓjkr = ω
jrΓikr .
The curvature of ∇,
slt = ∂sΓ
lt − ∂lΓ
st + Γ
lt − Γ
can be expressed in terms of contravariant quantities by raising indices as
= ωisωjtRkslt .
This gives
Result 2.10.
= ωir
r − ∂rΓ
+ Γijr Γ
l + Γ
Having introduced symplectic connections, we are now in a position to interpret
the following Darboux theorem for Hamiltonian operators of degree 2:
Theorem 2.11. [17] Given a Hamiltonian operator
P ij = aij
x + c
where aij is non-degenerate, then P can be put in the constant form P ij = ωij
(where ω is a constant matrix) by a change of target space coordinates {ui} if and
only if aij is closed. The coordinates in which this happens are flat coordinates for
the connection Γkij = −girc
j which can be chosen, using a linear substitution, to
be canonical coordinates for the symplectic form aij = ωij .
In arbitrary coordinates operators satisfying the conditions of Theorem 2.11 have
the form
P ij = ωij
x + Γ
ukxx (14)
8 JAMES T. FERGUSON
where ωij is the inverse of a symplectic form, c
(k,l)
− ωprΓ
, and Γ
the contravariant Christoffel symbols of a flat symplectic connection compatible
with ω. This class of operators on L(M) is therefore in one-to-one correspondence
with flat Fedosov structures on M .
3. Flat Pencils of Fedosov Structures
In this section we consider pairs of Hamiltonian operators of the form (14):
1 = ω
+ 2Γ1
x + Γ1
ukxx ,
2 = ω
+ 2Γ2
x + Γ2
The first fact to establish is that if P1 and P2 are compatible then all elements
of the pencil, Pλ = P1 + λP2, remain in the class (14).
Theorem 3.1. If P1 and P2 are compatible then ω
1 and ω
2 form a finite-dimensional
bi-Hamiltonian structure on the target space.
Proof. Pλ could have the general form
x + cλ
ukxx ,
but clearly bλ
= 2Γ1
+ 2λΓ2
and cλ
+ λΓ2
, so bλ
= 2cλ
, and
hence, by Lemma 2.6, a
satisfies the Jacobi identity (3) for all λ. �
So we write
+ 2Γλ
x + Γλ
ukxx .
An immediate corollary of Theorem 3.1 is that the tensor Lij = ω
1 ω2rj has
vanishing Nijenhuis torsion.
3.1. Multiplication of covectors. As in [7], we proceed to understand the com-
patibility conditions on P1 and P2 in terms of the algebraic properties of a tensorial
multiplication of covectors on M .
Definition 3.2. Using the tensors
∆sjk = ω
r − ω
i = ω2is∆
sjk ,
we define a multiplication ◦ of covectors on M by
(α ◦ β)i = αjβk∆
Theorem 3.3. The compatibility of P1 and P2 is equivalent to
(I, J ◦K)2 = (J, I ◦K)2 , (15)
and (I ◦ J) ◦K = 0 , (16)
for all covectors I, J,K on M . Here (·, ·)2 is the skew-symmetric bilinear form on
T ∗M induced by ω
2 , i.e. (I, J)2 = IrJsω
2 . The compatibility also implies
= ∇2k∆
. (17)
Because of Theorem 3.1, we phrase the compatibility of P1 and P2 in terms of
Fedosov structures on M , and break the above theorem into stages:
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 9
Definition 3.4. Two flat Fedosov structures (ω1,∇
1) and (ω2,∇
2), where ∇1 and
∇2 have contravariant Christoffel symbols Γ1
and Γ2
respectively, are said to be
(i) almost compatible if and only if (ωλ,∇
λ) is a Fedosov structure for all λ,
where the connection ∇λ is given by Γλ
+ λΓ2
(ii) almost compatible and flat if and only if they are almost compatible, and
in addition the curvature of ∇λ vanishes for all λ .
(iii) compatible if and only if they are almost compatible and flat, and cλ
(k,l)
− ωλprΓλ
satisfies cλ
+ λc2
for all λ.
The compatibility of two flat Fedosov structures on M is equivalent to the com-
patibility of the associated Poisson brackets on L(M).
We now turn to the two Fedosov strucutres defined by P1 and P2, and to the
pair (ωλ,∇
λ) defined by Pλ. From the linearity of Result 2.8 in the contravari-
ant symbols it can be seen that ωλ is automatically ∇
λ-constant, so the almost
compatibility of (ω1,∇
1) and (ω2,∇
2) is equivalent to ∇λ being torsion free, i.e. to
In flat coordinates for ∇2, this condition reduces to
r = ω
r . (18)
Note that we already have
ωir1 Γ1
r = ω
r . (19)
Lemma 3.5. If (ω1,∇
1) and (ω2,∇
2) are almost compatible, then the flatness of
∇λ is equivalent to either, and hence both, of
s − ∂sΓ1
= 0 (20)
and Γ1
l + Γ1
= 0 (21)
in the flat coordinates for ∇2.
Proof. The contravariant curvature of Γλ is
= ωirλ
r − ∂sΓλ
l + Γλ
s − ∂sΓ1
+ ωis1
s − ∂sΓ2
l + Γ1
l + Γ1
+λ2R2
which in flat coordinates for Γ2
reads
= ωir1
r − ∂rΓ1
l + Γ1
+λωis2
s − ∂sΓ1
The vanishing of the order λ term is equivalent to equation (20), and with this the
vanishing of the λ-independent term is equivalent to (21). �
Lemma 3.6. If (ω1,∇
1) and (ω2,∇
2) are almost compatible then the condition
(k,l)
− ωλprΓλ
reads, in the flat coordinates for ∇2,
l − Γ1
= 0 . (22)
10 JAMES T. FERGUSON
Proof. For an arbitrary Fedosov structure (ω,∇) the object c
(k,l)
−ωprΓ
can be converted into a quadratic expression in contravariant quantities as
= ωskΓ
(k,l)
Γsip Γ
Γsjp . (23)
This has similarities to the formula for covariant curvature obtained in Result 2.10;
only certain signs have changed. Indeed, if we define a quantity c
dxr =
(∇∂k∇∂l +∇∂l∇∂k) dx
then c
= ωirc
We have two ways of expanding ωskλ cλ
, corresponding to whether we choose
first to substitute it into equation (23), or to expand the pencil quantities. We work
in flat coordinates for ∇2; in these, c2
also vanishes. First expanding the pencil
we have
1 + λω
= ωsk1 c1
+ λωsk2 c1
whilst (23) gives
ωskλ cλ
= ωskλ Γλ
(k,l)
ωsk1 + λω
(k,l)
The order 1 terms merely express equation (23) for P1. Equality of the order λ
terms is equivalent to Γ1
(k,l)
and so to
ωsk1 Γ1
(k,l)
= ωsk1 c1
= ωsk1 Γ1
(k,l)
Proof of Theorem 3.3. Using equation (18) in Definition 3.2 it can be seen that in
the flat coordinates for ∇2 we have ∆
. Thus we may regard equations
(18),(20),(21) and (22) as identities on ∆
; the result is Theorem 3.3. �
The condition imposed by equation (21) for an almost compatible and flat pair
of Fedosov structures on the mutliplication ◦ is (I ◦ J) ◦ K = −(I ◦ K) ◦ J , i.e.
the first condition (8) satisfied by the multiplication of a Fermionic Novikov alge-
bra. In general (9) is not satisfied even for compatible Fedosov structures, however
we do have, for two flat Fedosov structures, (ω1,∇
1), (ω2,∇
2), which are almost
compatible,
= ∆ijr ∆
∆ikr .
So, in particular, if ∆
is constant in the flat coordinates for ∇2, almost compatible
and flat Fedosov structures will define a Fermionic Novikov algebra structure on
the covectors of M .
In [1] it emerged that examples of such algebras which do not also satisfy the
‘Bosonic’ relation (I ◦J) ◦K = (I ◦K) ◦J , and hence (I ◦J) ◦K = 0, are relatively
rare. ∇2-constant multiplications arising from pairs of Fedosov structures which
are almost compatible and flat, but not compatible, such as that given in Example
3.10 below, are in this class.
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 11
3.2. The pencil in flat coordinates. We now turn our consideration to the form
the pencil takes in the flat coordinates for ∇2. From the elements of the proof of
Theorem 3.3 we have
1 + λω
+ 2Γ1
x + Γ1
ukxx . (24)
The Jacobi identity for Pλ (without assuming P1 and P2 are Hamiltonian them-
selves) is equivalent to the constraints
(i) ω
2 is constant and antisymmetric,
(ii) ω
1 is antisymmetric,
(iii) ωir1 Γ1
r = ω
(iv) ω
(v) ωir2 Γ1
r = ω
(vi) Γ1
(vii) Γ1
l = 0.
Proposition 3.7. In a fixed coordinate system {ui} (the flat coordinates for Γ2),
given a constant non-degenerate 2-form ω
2 and a vector field B = B
r∂r satisfying
ωis2 B
,s − ω
,pr =
,s − ω
,pr (25)
,sl = 0 (26)
then the prescription
1 = −(LBω2)
ij = ωir2 B
,r − ω
= ωir2 B
satisfies the constraints (i)-(vii). Further, all solutions of (i)-(vii) have this form.
Proof. Equations (25) and (26) are the quadratic constraints, ωir1 Γ1
r = ω
and Γ1
l = 0 respectively. That ω1 and Γ1 satisfy the (linear) constraints (iv),
(v) and (vi) is an immediate consequence of their definition.
Using the Poincare lemma together with the symmetries expressed in conditions
(vi) and (v), we have the existence of a vector field satisfying Γ1
= ωir2 A
j ,rk . With
this condition (iv) gives ω
1 = −(LAω2)
ij+cij , where cij is a constant antisymmetric
matrix. We may now introduce a vector field B with Bi = Ai + 1
xsw2src
ri which
satisfies ω
1 = −LBω
2 and Γ1
= ωir2 B
Since ω2 is a symplectic form, its symmetries are precisely (locally) Hamiltonian
vector fields. Therefore, if ω2 and ω1 are given, the requirement that ω
1 = −LBω
fixes the non-Hamiltonian part of B. Then the condition Γ1
= ωir2 B
fixes the
Hamiltonian to within a quadratic function. From the point of view of the multipli-
cation of covectors from Section 3.1, the Hamiltonian affects only the commutative
part of ◦, thus the anti-commutative part is fixed by ω
1 and ω
With consideration of the transformation rules (6), one can phrase Proposition
3.7 as the existence of a vector field B such that
1 = −LBω
= −LBΓ2
. (27)
12 JAMES T. FERGUSON
We can also calculate from (6) the correct interpretation of the Lie derivative for
an object of type c
, namely:
= Xrc
−X i,rc
−Xj,rc
kl +X
+Xr,lc
+Xr,klc
birk −
birl −X
air .
If we work in the flat coordinates for Γ2, so that the components c2
= 0, we
have for our pencil
−LBc2
= +ωir2 B
= (ωir2 B
),l ,
Now, in the flat coordinates for ∇2 we have the relation c1
. The linearity
of the transformation rules shows that the Lie derivative of c2
should be an object
of the same type as c1
. Thus we have, in addition to (27),
= −LBc2
One may understand these three infinitesimal relations between the coefficients
of P1 and P2 as averring the existence on L(M) of an evolutionary vector field
B̂ = Bi(u(x))
∂ui(x)
+ . . .
such that
1 = −LB̂P
We now turn our attention to some examples of pairs of Fedosov structures,
using the framework of Proposition 3.7.
Example 3.8. Two-dimensional pencils. Without loss of generality we take
where u1 and u2 are a flat coordinate system for ∇2.
We take
B = f(u1, u2)
+ g(u1, u2)
and from it calculate ω1 and Γ1 according to (27). In particular
ω1 = (f,1 + g,2)ω2 ,
from which it follows immediately that (ω1,∇
1) and (ω2,∇
2) are almost compatible.
They are almost compatible and flat if and only if h = f + λg satisfies the
homogeneous Monge-Ampere Equation h212 − h11h22 = 0 for all λ.
They are compatible if and only if a = f + λg and b = f + µg satisfy
a12b12 − a11b22 = 0
for all λ, µ.
For instance, one may recover the three two-dimensional Fermionic Novikov al-
gebras of [1] as constant multiplications via
(T1) f = u1, g = 0 ,
(T2) f = u1, g = (u1)2 ,
(T3) f = (u1)2, g = 0 .
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 13
Example 3.9. Commutative algebras. In the case in which ω1 is constant in the
flat coordinates for ∇2, we have, by condition (iv),
so that the multiplication ◦ is commutative.
In particular if
ω1 = ω2 = ω =
then the non-Hamiltonian part of B is
To this we may add a Hamiltonian vector field, giving
Since ω1 = ω2, equation (25) is immediate. Equation (26) becomes
H,ijr ω
H,skl = 0 ,
where the indices i, j, k, l, r, s account for both q and p variables.
A solution to this is H = f(x1, x2, . . . , xn), where each xi is either pi or q
i; only
one from each pair of conjugate variables features in H.
It is not hard to see that Proposition 3.7 can be modified to describe almost
compatible and flat pairs of Fedosov structures. Specifically, we replace equation
(26) by the expression corresponding to Γ1
l = Γ1
, namely:
,sl = B
ωrs2 B
,si . (28)
Example 3.10. The Fedosov structures specified by
= 0 ,
+ 2q1q2
+ q1p2
and ω
1 = −LBω
2 and Γ1
= −LBΓ2
are almost compatible and flat, but not
compatible.
The non-zero components of ω1 and ◦ are
{q1, p1}1 = {q2, p2}1 = 3q1 ,
{q2, p1}1 = 2q2 ,
{p2, p1}1 = p2 ,
dq2 ◦ dp2 = dq1 ,
dp1 ◦ dq1 = −3dq1 ,
dp1 ◦ dq2 = −2dq2 ,
dp1 ◦ dp2 = −dp2 ,
dp2 ◦ dq2 = −2dq1 .
14 JAMES T. FERGUSON
Thus, the products
(dp1 ◦ dq2) ◦ dp2 = −2dq1
and (dp1 ◦ dp2) ◦ dq2 = 2dq1
violate equation (16) but not (8). Note that ◦ also satisfies (9) and thus defines a
Fermionic Novikov algebra which is not ‘Bosonic’.
3.3. ωN manifold with Potential. The tangent bundle T ∗Q of a manifold Q is
naturally equipped with a symplectic form, and thus cotangent bundles form the
basic set of examples of symplectic manifolds. One may hope to find examples
of finite-dimensional bi-Hamiltonian structures on cotangent bundles by exploiting
the existence of additional structures on the underlying manifolds. The main object
used to do this is a (1, 1)-tensor Lij on Q whose Nijenhuis torsion is zero. Such an
object was utilised by Benenti [2] to demonstrate the separability of the geodesic
equations on a class of Riemannian manifolds. This result was later interpreted
in [12] in terms of a bi-Hamiltonian structure on T ∗Q which was extended to a
degenerate Poisson pencil on T ∗Q× R.
To obtain Fedosov structures we require more than just a tensor Lij on Q with
vanishing Nijenhuis torsion; we also need a means of specifying the connections.
If Q is equipped with a torsion-free connection ∇̃, then the Nijenhuis torsion of a
(1, 1)-tensor Lij can be written as
jk = L
j∇̃sL
k − L
k∇̃sL
j − L
s∇̃jL
k + L
s∇̃kL
If there exists a vector field, A, on Q such that Lij = ∇̃jA
i then
N ijk = (∇̃jA
s)(∇̃s∇̃kA
i)− (∇̃kA
s)(∇̃s∇̃jA
i)− (∇̃sA
i)(RsjkrA
where Rijkl is the curvature tensor of ∇̃.
So, if ∇̃ is flat then the vanishing of the Nijenhuis tensor of L = ∇̃A is equivalent
to the identity
(∇̃jA
s)(∇̃s∇̃kA
i) = (∇̃kA
s)(∇̃s∇̃jA
i) . (29)
Proposition 3.11. Given a manifold Q endowed with a flat connection ∇̃ and a
vector field A satisfying (29), the cotangent bundle T ∗Q is endowed with a compati-
ble pair of Fedosov structures, (ω1,∇
1) and (ω2,∇
2), as follows: ω2 is the canonical
Poisson bracket on T ∗Q.
The connection ∇2 on T ∗Q is the horizontal lift [19] of the connection ∇̃ on Q;
i.e. the Christoffel symbols Γ2
ij of ∇
2 are zero in the coordinates induced on T ∗Q
by the flat coordinates for ∇̃.
(ω1,∇
1) is calculated from (ω2,∇
2) according to the prescription of Proposition
3.7, where the vector field B is the horizontal lift of A to T ∗Q.
Proof. Let {q1, . . . , qn} be flat coordinates for ∇̃ onQ, and C = {q1, . . . , qn, p1, . . . , pn}
be the induced coordinates on T ∗Q. Then
The space of sections of the cotangent bundle of T ∗Q, Ω, naturally splits into
P = span{dpi} and Q = span{dq
i}. For Γ1
= ωir2 B
to be non-zero requires k
to represent a variable qk, and i to represent a pi variable. Thus Ω ◦ Ω ⊆ Q and
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 15
Q ◦ Ω = {0}, meaning that (Ω ◦ Ω) ◦ Ω = {0}. So the relation (26), Γ1
l = 0,
is satisfied.
1 has only one kind of non-zero component, ω
,i, so the expression
ωir1 Γ1
r has only one non-zero case:
qr = A
,rj ,
which is seen to be symmetric in i and j by condition (29), which in the flat
coordinates qi reads
As,j A
i,sk = A
s,k A
i,sj .
Example 3.12. If the eigenvalues of L : TQ → TQ are functionally independent
in some neighbourhood then they may be used as coordinates, and L takes the form
⊗ dui .
In this case we may set A =
(ui)2 ∂
, and have ∇̃ defined by vanishing
Christoffel symbols in these coordinates.
This gives, writing vi as the conjugate coordinate to u
i on T ∗Q,
= −1 ,
and all other Christoffel symbols zero.
4. Bi-Hamiltonian Structures in Degrees 1 and 2
We now consider a pair of operators, P1 and P2 in which P1 is a Hamiltonian
operator of hydrodynamic type and P2 is of second order, i.e. :
1 = g
ij(u)
(u)ukx ,
2 = a
x + c
ukxx ,
where gij is the inverse of a flat metric gij on M and Γ
= −girΓ
where the Γkij
are the Christoffel symbols of the Levi-Civita connection of g. We also assume that
2 is antisymmetric, so that a
ij = −aji, b
and c
(k,l)
The motivation [8] for studying such pairs of operators comes not from regarding
them as separate Hamiltonian operators, but from thinking of P
2 as a first order
(dispersive) deformation of P
1 into some non-homogeneous Hamiltonian operator
P ij = P
1 +εP
2 +O(ε
2). Thus, in such a pair, it is sensible to regard the geometry
1 as being more intrinsic than any associated to P
We choose to work in flat coordinates for g so that gij is constant and Γ
Direct calculation of the Jacobi identity for P ij in these coordinates yields
16 JAMES T. FERGUSON
Theorem 4.1. P2 is an infinitesimal deformation of P1, i.e. P
ij = P
1 + εP
O(ε2) satisfies the Jacobi identity to order ε, if and only if
(I) gircjkr + g
jrcikr = 0 ,
(II) c
(k,l)
(III) girc
= gjr(cikl,r − c
r,l) ,
(IV) gir(ajk,r − c
r ) + g
jr(aki,r − c
r ) + g
kr(aij,r − c
r ) = 0
in the flat coordinates for gij.
By introducing the tensor T
= airΓ
is it easy to convert conditions (I),
(III) and (IV) to arbitrary coordinates, whilst condition (II) becomes
− crik Γ
− cril Γ
+ T ijr Γ
kl + T
Γirl + T
Γirk .
To consider a bi-Hamiltonian structure involving operators P
1 and P
2 one need
only add conditions (C), (D) and (E) of Theorem 2.1 to Theorem 4.1, however,
condition (II) above allows (E) to be replaced by cijr c
l = c
Example 4.2. As discussed in section 2, P2 with b
constant and aij non-
degenerate is Hamiltonian if and only if aij = A
uk + A
0 with A
0 is constant, c
are the structure constants of a Fermionic Novikov algebra
(A, ◦), and A
0 defines a skew-symmetric bilinear form on A satisfying 〈I, J ◦K〉 =
〈J, I ◦K〉.
If we ask that P2 satisfies the above constancy conditions in the flat coordinates
for gij, then, defining an inner product on A by (ei, ej) = gij, we have that the
compatibility of P1 and P2 is equivalent to the additional constraints:
(I ◦ J) ◦K = (I ◦K) ◦ J ,
(I, J ◦K) = −(J, I ◦K)
(I, [J,K]) + (J, [K, I]) + (K, [I, J ]) = 0 ,
where [I, J ] = I ◦J−J ◦I is the commutator of ◦, which is a Lie bracket by equation
For example, if we take the algebra (A = span{e1, e2, e3, e4}, ◦) where the only
non-zero products are e3 ◦ e3 = e1 and e4 ◦ e3 = e2 then we may take as our
symplectic form and metric
[ωij ] =
0 0 a b
0 0 b c
−a −b 0 d− u2
−b −c −d+ u2 0
[gij ] =
0 0 0 e
0 0 −e 0
0 −e f g
e 0 g h
 ,
for any choice of the constants a, b, c, d, e, f, g, h such that e 6= 0 and b2 6= ac.
This algebra, essentially (57)−1, is the only algebra in [1] of dimension 2 or 4
which admits non-degenerate forms (·, ·) and 〈·, ·〉 satisfying the above compatibility
conditions with ◦, other than the trivial case in which all products are zero, i.e.
in which the Hamiltonian operators share the same flat connection, and so are
simultaneously constant.
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 17
Proposition 4.3. If P2 is an infinitesimal deformation of P1 then there exists a
tensor field Aij such that
aij = girAjr − g
jrAir ,
= 2gisA
− gjrAik,r − g
= gisA
− gisA
(k,l)s
= gisA
− gisA
in flat coordinates for gij. Further, any (1,1)-tensor field Aij produces an infinites-
imal deformation of P1 by the above formulae.
Proof. Using the non-degeneracy of gij , we introduce objects θkij and φij by
= girθ
aij = girgjsφrs .
Then condition (I) of Theorem 4.1 is equivalent to θkij = −θ
ji, and so we regard θ
as a family of 2-forms θk indexed by k.
Condition (III) is equivalent to θkjl,i = θ
il,j − θ
ij,l, so that dθ
k = 0 for each k.
This allows us to introduce a family of 1-forms ψk such that
ij = (dψ
k)ij = ψ
i,j − ψ
j,i .
Each ψk can be adjusted by the addition of the exterior derivative, dfk, of some
function fk without affecting the value of θkij .
Writing αij = φij − gjrψ
i + gjrψ
k, we find that condition (IV) is equivalent to
the closedness of the 2-form αij , upon substituting φij and ψ
j for a
ij and c
. Thus
we may introduce a 1-form h with components hi such that αij = hi,j − hj,i, and
φij = gjrψ
i − gjrψ
j + hi,j − hj,i .
If we now let Aij = ψ
j + (g
irhr),j then we have θ
ij = A
i,j − A
j,i and φij =
i − girψ
j , so that the two equations a
ij = girAjr − g
jrAir and c
= girA
are satisfied. The remaining to equations follow easily from c
For the converse, it is easy to check that conditions (I)-(IV) of Theorem 4.1
follow from (30) for any tensor field Aij . �
As with Proposition 3.7, Proposition 4.3 may be understood as asserting the
existence of an evolutionary vector field
e = Aij (u(x)) u
∂ui(x)
+ . . .
satisfying P2 = −LeP1 whenever P2 is an infinitesimal deformation of P1. This is
therefore not a surprising result; in [11] Getzler showed the triviality of infinitesimal
deformations of Hydrodynamic type Poisson brackets. With this, Proposition 4.3
can be looked upon as a proof of Theorem 4.1.
There is a freedom in Aij of A
j 7→ A
j + g
irf,rj for some function f , which does
not affect the coefficients of P2. This corresponds to adjusting e by a Hamiltonian
vector field, e 7→ e+ P1(δf).
If, with reference to Lemma 2.6, we impose the additional constraint on (30)
that b
then we have the potentiality condition gjrA
k,i = girA
k,j , so that
there exists a 1-form Bk such that
j = g
Bj,r . (31)
18 JAMES T. FERGUSON
In this case aij = girgjr(Br,s − Bs,r) = g
irgjr(dB)rs and the freedom A
j 7→ A
girf,rj is B 7→ B + df . This means that B can be determined purely from g
ij and
aij , and thus there is no freedom in the choice of c
and c
. In fact we may write
explicitly
= gjsgkr
(k,l)
, (32)
and with this, P2 is an infinitesimal deformation of P1 if and only if
girajk,r + g
jraki,r + g
kraij,r = 0 . (33)
Corollary 4.4. Given a flat metric g and a symplectic form ω, there is at most one
choice of flat symplectic connection ∇ such that the degree 2 Hamiltonian operator
specified by (ω,∇) is compatible with the hydrodynamic operator specified by g.
Clearly, if this connection exists it is given by(32), so this definition must be
checked against Theorem 2.1 to verify
2 = ω
x + c
is Hamiltonian. Since equation (33) is a consequence of the antisymmetry of P2,
compatibility with the Hydrodynamic operator follows immediately.
We conclude this section with an example of this type.
Example 4.5. The Kaup-Broer system [16],
u1xx + 2u
x + 2u
−u2xx + 2(u
1u2)x
is described by the pair of compatible Hamiltonian operators
u1 2u2
0 u1x
0 u2x
Scaling x 7→ εx, t 7→ εt splits P2 into P
2 + εP
2 where
u1 2u2
0 u1x
0 u2x
Since P2 = P
2 + εP
2 is Hamiltonian for all ε, P
2 and P
2 constitute a bi-
Hamiltonian structure of the type considered above. A set of flat coordinates for the
metric in P
ũ1 = u1 ,
ũ2 =
4u2 − (u1)2 ,
in which
(ũ2)2
0 −ũ2x
0 ũ1x
(ũ2)3
0 (ũ2x)
0 −ũ1xũ
(ũ2)2
0 −ũ2xx
0 ũ1xx
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 19
So in this situation we have, for the 1-form in (31),
dũ2 .
5. Conclusions
In section 3 an approach was taken based upon the methods of [7] to study com-
patible pairs of Hamiltonian operators of degree 2 which satisfy the conditions of
the relevant Darboux theorem, Theorem 2.11. As for Hydrodynamic Poisson pen-
cils, the compatibility could be reduced to algebraic constraints on a multiplication
of covectors. Driving this was the ability to reduce a given Hamiltonian operator
on L(M) to a flat Fedosov structure (ω,∇) on M , which are natural symplectic
analogues of the pair consisting of a flat metric and its Levi-Civita connection which
determines a Hydrodynamic Poisson bracket.
To extend such a results to pairs of arbitrary degree 2 Hamiltonian operators,
one must consider the pair (a,∇) of Theorem 2.4. The condition (10), whilst
atypical, expresses a familiar concept; in almost-symplectic geometry, it is common
to consider connections such that the covariant derivative of the almost-symplectic
form is zero, but which have torsion; if the torsion of such a connection is skew-
symmetric then its symmetric part satisfies (10). Equation (12) provides the means
of going from the symmetric connection to the compatible connection with skew-
torsion. The only formula missing above necessary to the study of arbitrary bi-
Hamiltonian structures of degree 2 is an expression for the contravariant curvature
of the connection defined by c
, which is, in the presence of Theorem 2.1’s condition
= air(c
) + cijr c
l + c
− (bijr − 2c
l + c
One may use (B) to replace the components of b
in this expression with those of
and the derivatives of aij . However, one sees that the compatibility conditions
do not naturally become algebraic constraints on ∆
, and the relevancy of such an
approach is undermined. It is interesting to note, however, that equation (23) still
holds (with Γ
), so that ◦ defined by ∆
still satisfies (I ◦J)◦K = (I ◦K)◦J ,
and that it is the ‘Fermionic’ condition (I ◦ J) ◦K = −(I ◦K) ◦ J which is altered.
The proof of Proposition 3.7 is easily adapted to confirm the existence of a vector
field B realising P1 = −LBP2 whenever P1, of the form (5) is an infinitesimal defor-
mation of P2 as a Hamiltonian operator, provided b1
= 2c1
. A simple calculation
of LBP2 for arbitrary B shows that b1
= 2c1
is also a necessary condition. Thus
we have determined the trivial deformations of a degree 2 Hamiltonian operator
admitting a constant form, which are themselves of degree 2. Clearly a different
approach is necessary to understand deformations of higher degrees. For the case
of operators not satisfying the constraints of Theorem 2.11, it is not immediately
obvious what conditions, if any, will guarantee the triviality of a deformation; ow-
ing to the different form the contravariant curvature tensor takes, the condition
is absent. Owing to the lack of a constant form, the methods of [8]
in ascertaining the triviality of higher degree deformations, if applicable, will be
somewhat more complicated.
Finally, there is a certain artificiality to the examples of compatible Fedosov
structures presented in section 3. Given Theorem 3.1’s assertion that underlying a
pair of compatible Fedosov structures is a finite-dimensional bi-Hamiltonian struc-
ture, the question is raised asking which finite-dimensional bi-Hamiltonian struc-
tures admit symplectic connections forming almost compatible, almost compatible
and flat, or compatible Fedosov structures? It would be interesting to exhibit a
20 JAMES T. FERGUSON
pair of compatible Fedosov structures in which the flat coordinates for one of the
connections are in some sense physical.
Acknowledgements
The author would like to thank Ian Strachan for suggesting this project, and the
Carnegie Trust for the Universities of Scotland for the scholarship under which this
work was conducted.
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Marcel Dekker Inc., New York, 1973. Pure and Applied Mathematics, No. 16.
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, U.K.
E-mail address: j.ferguson@maths.gla.ac.uk
http://arxiv.org/abs/math/0108160
	1. Introduction
	2. Hamiltonian Operators of Degree 2
	3. Flat Pencils of Fedosov Structures
	3.1. Multiplication of covectors
	3.2. The pencil in flat coordinates
	3.3. N manifold with Potential
	4. Bi-Hamiltonian Structures in Degrees 1 and 2
	5. Conclusions
	Acknowledgements
	References

  Bi-Hamiltonian structures involving Hamiltonian operators of degree 2 are
studied. Firstly, pairs of degree 2 operators are considered in terms of an
algebra structure on the space of 1-forms, related to so-called Fermionic
Novikov algebras. Then, degree 2 operators are considered as deformations of
hydrodynamic type Poisson brackets.

Introduction
Hamilton’s equations for a finite-dimensional system with position coordinates
qi and associated momenta pi,
are understood geometrically as describing the flow of a vector field XH which is
associated with the Hamiltonian function H(q1, . . . , qn, p1 . . . , pn) by the formula
XH(f) = {f,H}, where {·, ·} is the Poisson bracket:
{f, g} =
. (1)
More generally, one defines a Poisson bracket on an n-dimensional manifoldM as
a map C∞(M)× C∞(M) → C∞(M), (f, g) 7→ {f, g}, satisfying, for any functions
f, g, h on M :
(1) antisymmetry: {f, g} = −{g, f} ,
(2) linearity: {af + bg, h} = a{f, h}+ b{g, h} for any constants a, b ,
(3) product rule: {fg, h} = f{g, h}+ g{f, h} ,
(4) Jacobi identity: {{f, g}, h}+ {{g, h}, f}+ {{h, f}, g} = 0 .
The conditions 1-3 identify {·, ·} as a bivector: a rank two, antisymmetric, con-
travariant tensor field ω on M . It can therefore be represented, by introducing
coordinates {ui} on M , as a matrix of coefficients ωij , giving
ω = ωij
{f, g} = ωij
. (2)
The Jacobi identity places the following constraint on the components of ω:
+ ωjr
+ ωkr
= 0 . (3)
If the matrix ωij is non-degenerate, we may introduce its inverse ωij , satisfying
rj = δ
i . The Jacobi identity for ω
ij is equivalent to the closedness of ωij .
Date: April 11, 2007.
http://arxiv.org/abs/0704.1401v1
2 JAMES T. FERGUSON
We refer to a closed non-degenerate two-form as a symplectic form, and a mani-
fold equipped with one as a symplectic manifold. Darboux’s theorem asserts that
on any 2n-dimensional symplectic manifold there exists a set of local coordinates
{q1, . . . , qn, p1 . . . , pn} in which the Poisson bracket takes the form (1); i.e. the
components of ωij , and so those of ωij , are constant.
One may also introduce Poisson brackets on infinite-dimensional manifolds. The
loop space of a finite-dimensional manifold M , L(M), is the space of smooth maps
u : S1 →M . Poisson brackets relating Hamiltonians to flows in L(M) will therefore
act on functionals mapping L(M) → R. In [5],[6] Dubrovin and Novikov studied
the so-called Poisson brackets of differential-geometric type, which are of the form
{f, g} =
dx (4)
where ui are coordinates on the target space M , and x is the coordinate on S1.
P ij is a matrix of differential operators (in d
), with no explicit dependence on x,
which is assumed to be polynomial in the derivatives uix, u
xx, . . . . If {·, ·} defines a
Poisson bracket on the loop space then P is referred to as a Hamiltonian operator.
There is a grading on such operators, preserved by diffeomorphisms of M , given
by assigning degree 1 to d
, and degree n to the nth x-derivative of each field ui. An
important class is the hydrodynamic type Poisson brackets, which are homogeneous
of degree 1:
P ij = gij(u)
(u)ukx .
According to the programme set out by Novikov [15], differential-geometric type
Poisson brackets on L(M) should be studied in terms of finite-dimensional differ-
ential geometry on the target space M . When expanded as a polynomial in d
the field derivatives, the coefficients, which are functions of the fields ui alone, can
often be naturally related to known objects of differential geometry, or else used to
define new ones. In the hydrodynamic case, for instance, with gij non-degenerate,
P is Hamiltonian if and only if gij is a flat metric on M and Γkij = −girΓ
j are the
Christoffel symbols of its Levi-Civita connection.
In [7] Dubrovin considered the geometry of bi-Hamiltonian structures of Hy-
drodynamic operators, that is pairs of such operators compatible in the sense of
[13], that every linear combination of them also determines a Poisson bracket. In
particular, he introduced a multiplication of covectors on M and expressed the
compatibility of the operators in terms of a quadratic relations on this algebra.
This paper is principally concerned with Hamiltonian operators which are ho-
mogeneous of degree 2. Section 2 presents the differential geometry of such op-
erators, and in particular relates the subclass which can be put into a constant
form by a change of coordinates on M to symplectic connections. Section 3 then
considers pairs of operators from this subclass, and the algebraic constraints their
compatibility places upon the associated multiplication. In section 4 inhomoge-
neous bi-Hamiltonian structures consisting of a degree 1 and a degree 2 operator
are studied.
2. Hamiltonian Operators of Degree 2
We begin with a review of known results on Hamiltonian operators of degree 2:
ij = aij
x + c
xx, (5)
in which the matrix aij is assumed to be non-degenerate. Such operators have been
considered already in, for example, [17], [14], [4], [15], in which the (conditional)
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 3
Darboux theorem has been discussed. In preparation for the bi-Hamiltonian theory
we present these results without the use of special coordinates.
Under the change of coordinates ũi = ũi(up) the coefficients in P ij transform as
bpqr − 2
∂ũk∂ũs
apq ,
cpqr −
∂ũk∂ũs
apq ,
cpqrs +
∂ũk∂ũl
∂2ũj
∂uq∂us
∂ũ(k
∂ũl)
bpqr +
∂3ũj
∂uq∂ur∂us
∂2ũj
∂uq∂ur
∂ũk∂ũl
, (6)
where the brackets denote symmetrisation. So in particular aij transforms as a
rank 2 contravariant tensor on the target space and b
and c
are related to
Christoffel symbols of connections by b
= −2airΓ̄
and c
= −airΓ
. Call
these connections ∇̄ and ∇ respectively.
The transformation rules for c
are not determined uniquely by those for P ,
since (5) sees only the part symmetric in k and l. To fix c
, we always assume the
antisymmetric part is zero. Denote by aij the inverse of a
ij defined by aira
rj = δ
The condition that the operation defined in (4) is skew-symmetric and satisfies
the Jacobi identity places constraints on the coefficients appearing in (5).
Theorem 2.1. The operator P in equation (5) defines a Poisson bracket by equa-
tion (4) if and only if
(A) aij = −aji ,
(B) ∇ka
ij = b
(C) air
bjkr − 2c
= akr
bijr − 2c
(D) ∇ is flat (zero torsion, zero curvature) ,
(E) c
(k,l)
− aprc
Proof. [14] states that, by virtue of being Hamiltonian, the operator (5) can be put
in the form
P ij = aij
, (7)
by a change of coordinates ui = ui(ũ), and that for an operator of this shorter form
to be Hamiltonian is equivalent to the three conditions
(a) aij = −aji ,
(b) aij ,k = b
(c) airbjkr = a
jrbkir .
We first assume that P is a Poisson bracket, so there exists the special coordinates
in which P takes the form (7) and (a)-(c) hold. By reversing the change of variables
as ũi = ũi(u), conditions (A)-(C) of Theorem 2.1 are Mokhov’s three conditions
converted to tensorial identities. That ∇ is flat follows from its Christoffel symbols,
Γkij = −airc
j , being zero in the u coordinates.
4 JAMES T. FERGUSON
The formula in condition (E) is derived from the transformation rules above. In
changing from flat coordinates ui to coordinates ũi they give:
∂2ũj
∂uq∂us
∂ũ(k
∂ũl)
bpqr +
∂3ũj
∂uq∂ur∂us
∂2ũj
∂uq∂ur
∂ũk∂ũl
apq ,
∂ũk∂ũs
apq ,
∂2ũj
∂uq∂ur
apq ,
where the last line has used the identity
∂2ũi
∂ur∂us
∂ũj∂ũk
= 0 ,
which is a differential consequence of ∂ũ
= δij .
∂2ũi
∂up∂us
∂2ũj
∂ur∂uq
∂3ũj
∂uq∂ur∂us
∂2ũj
∂uq∂ur
∂ũk∂ũl
∂2ũj
∂uq∂ur
from which we see
(k,l)
∂2ũi
∂up∂us
∂2ũj
∂ur∂uq
∂ũ(l
∂ũk)
apq .
This last term can be seen to be
ãprc̃
(k c̃
Conversely, if (A)-(E) hold, the flatness of ∇ asserts the existence of coordinates
in which c
= 0, and condition (E) then asserts that c
= 0 in these coordinates.
If we take, as a simple case, an operator P as in (5) with b
constants,
and assume c
to be defined by (E), then P is Hamiltonian if and only if aij =
uk + A
0 where A
0 are constants with A
, Airl c
r = A
cikr ,
Air0 c
r = A
r and c
rk + cikr c
If we take an algebraA with basis {e1, . . . , en}, n = dimM, and use c
and A
define a multiplication, ◦ , and skew-symmetric bilinear form, 〈·, ·〉, by ei◦ej = cijr e
and 〈ei, ej〉 = A
0 , then we may rewrite these conditions as
ei ◦ ej − ej ◦ ei = Aijr e
(I ◦ J) ◦K = −(I ◦K) ◦ J , (8)
Λ(I, J,K) = Λ(J, I,K) , (9)
and 〈I, J ◦K〉 = 〈J, I ◦K〉 ,
for all I, J,K ∈ A, where Λ is the associator of ◦ : Λ(I, J,K) = (I◦J)◦K−I◦(J◦K).
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 5
Algebras satisfying conditions (8) and (9) have appeared before in [18], in the
context of linear hydrodynamic Hamiltonian operators taking values in a completely
odd superspace, where the following definition was proposed:
Definition 2.2. An algebra (A, ◦) satisfying conditions (8) and (9) is called a
Fermionic Novikov algebra.
In [1] Fermionic Novikov algebras in dimensions 2-5 were studied, and the listing
therein provides a source of examples of Hamiltonian operators of degree two.
Example 2.3.
0 0 0 a
0 0 −a −b− (t− 1)u1
0 a 0 c− u2
−a b+ (t− 1)u1 −c+ u2 0
0 0 0 0
0 0 0 u1x
0 0 −u1x 0
0 τu1x u
0 0 0 0
0 0 0 0
0 0 0 (u1x)
0 0 −(u1x)
0 0 0 0
0 0 0 u1x
0 0 −u1xx 0
0 τu1xx u
is Hamiltonian for all values of the constants a, b, c and τ with a 6= 0. This is the
most general Hamiltonian operator associated in the manner discussed above to the
algebra designated (44)τ in [1].
Returning to the general Hamiltonian operator (5), it can be seen from conditions
(B) and (E) in Theorem 2.1 that the coefficients b
and c
in (5) are completely
determined by aij and c
. Thus the Hamiltonian operator on L(M) is represented
uniquely on M by only these latter two objects.
Theorem 2.4. There is a one-to-one correspondence between Hamiltonian opera-
tors of the form (5) on L(M) and pairs (a,∇) on M consisting of a non-degenerate
bivector aij and a torsion-free connection ∇ satisfying two conditions: firstly, that
the curvature of ∇ vanishes, and secondly,
air∇ra
jk = ajr∇ra
ki . (10)
The Christoffel symbols, Γkij , of ∇ are related to c
= −airΓ
. We then
= ∇ka
ij + 2c
− aprc
With this, we may verify the following facts [17],[14]:
Corollary 2.5. For P in (5) a Hamiltonian operator we have
1. Γ is the symmetric part of Γ̄,
2. Let T̄ kij = Γ̄
ij − Γ̄
ji be the torsion of ∇̄. Then T̄ijk = airT̄
jk is skew
symmetric and the forms T̄ = 1
T̄ijkdu
i ∧ duj ∧ duk and a = 1
aijdu
i ∧ duj
are related by 3T̄ = da.
Proof. We begin by noting that equation (10) is equivalent to the condition
∇kaij = ∇iajk (11)
on the two-form aij .
6 JAMES T. FERGUSON
In terms of covariant Christoffel symbols, Theorem 2.4 gives
Γ̄kij =
akr∇raij + Γ
ij , (12)
from which it is clear that Γ̄k
= Γkij .
We therefore also have
∇kaij = Γ̄ijk − Γijk ,
where Γ̄ijk = airΓ̄
jk and Γijk = airΓ
jk. Because ∇ is torsion-free we have
T̄ijk = Γ̄ijk − Γ̄ikj ,
= Γ̄ijk − Γijk − Γ̄ikj + Γikj ,
∇kaij −
∇jaik ,
= ∇kaij ,
= ∇[kaij] ,
(da)ijk .
Lemma 2.6. For a Hamiltonian operator of the form (5), the following three state-
ments, presented in both covariant and contravariant forms, are equivalent:
1. The 2-form a is closed (and so symplectic), or equivalently aij satisfies
equation (3) (and so defines a Poisson bracket on M by equation (2));
2. ∇ka
ij = 0, i.e. ∇kaij = 0;
, i.e. Γkij = Γ̄
Proof. We see, from the characterisation of Hamiltonian operators given in Theorem
aij is Poisson ⇐⇒ airajk,r + a
jraki,r + a
kraij,r = 0
⇐⇒ air∇ra
jk + ajr∇ra
ki + akr∇ra
ij = 0
⇐⇒ 3akr∇ra
ij = 0
⇐⇒ ∇ka
ij = 0 ,
Lemma 2.6 therefore tells us that in the special case where the leading coefficient
in P is the inverse of a symplectic form, the pair (a,∇) defining P can be thought
of as containing the symplectic form aij , and a torsionless connection compatible
with it (in the sense that ∇a = 0); that is, a symplectic connection. More precisely
(see e.g. [3]):
Definition 2.7. A symplectic connection on a symplectic manifold (M,ω) is a
smooth connection ∇ which is torsion-free and compatible with the symplectic form
ω, i.e.
∇XY −∇YX − [X,Y ] = 0
(∇ω) (X,Y, Z) = X(ω(Y, Z))− ω(∇XY, Z)− ω(Y,∇Y Z) = 0 ,
where X,Y and Z are vector fields on M .
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 7
In local coordinates {xi}, introducing Christoffel symbols Γkij for ∇ and writing
ω = 1
ωijdx
i ∧ dxj , the conditions for ∇ to be a symplectic connection read Γkij =
Γkji, as usual, and
∇kωij =
− Γrkiωrj − Γ
kjωir = 0 . (13)
This definition is analogous to that of the Levi-Civita connection of a pseudo-
Riemannian metric, however there is an important difference in that the Levi-Civita
connection is uniquely specified by its metric. From the compatibility condition (13)
it can be seen that if Γkij are the Christoffel symbols of a symplectic connection for
ω, then the connection with Christoffel symbols Γ̃kij = Γ
ij + ω
krSrij is a sym-
plectic connection if and only if the tensor Sijk is completely symmetric. In [10]
a symplectic manifold with a specified symplectic connection is called, in light of
[9], a Fedosov manifold. Here we call the pair (ω,∇) of a symplectic form and a
symplectic connection a Fedosov structure on M, and call the structure flat if ∇ is
flat.
In the discussion of Hamiltonian operators it is convenient to work with con-
travariant quantities. We call
= −ωirΓ
the contravariant Christoffel symbols of the symplectic connection.
Result 2.8. The compatibility of ∇ and ω is equivalent to
Result 2.9. ∇ being torsion-free is equivalent to ωirΓjkr = ω
jrΓikr .
The curvature of ∇,
slt = ∂sΓ
lt − ∂lΓ
st + Γ
lt − Γ
can be expressed in terms of contravariant quantities by raising indices as
= ωisωjtRkslt .
This gives
Result 2.10.
= ωir
r − ∂rΓ
+ Γijr Γ
l + Γ
Having introduced symplectic connections, we are now in a position to interpret
the following Darboux theorem for Hamiltonian operators of degree 2:
Theorem 2.11. [17] Given a Hamiltonian operator
P ij = aij
x + c
where aij is non-degenerate, then P can be put in the constant form P ij = ωij
(where ω is a constant matrix) by a change of target space coordinates {ui} if and
only if aij is closed. The coordinates in which this happens are flat coordinates for
the connection Γkij = −girc
j which can be chosen, using a linear substitution, to
be canonical coordinates for the symplectic form aij = ωij .
In arbitrary coordinates operators satisfying the conditions of Theorem 2.11 have
the form
P ij = ωij
x + Γ
ukxx (14)
8 JAMES T. FERGUSON
where ωij is the inverse of a symplectic form, c
(k,l)
− ωprΓ
, and Γ
the contravariant Christoffel symbols of a flat symplectic connection compatible
with ω. This class of operators on L(M) is therefore in one-to-one correspondence
with flat Fedosov structures on M .
3. Flat Pencils of Fedosov Structures
In this section we consider pairs of Hamiltonian operators of the form (14):
1 = ω
+ 2Γ1
x + Γ1
ukxx ,
2 = ω
+ 2Γ2
x + Γ2
The first fact to establish is that if P1 and P2 are compatible then all elements
of the pencil, Pλ = P1 + λP2, remain in the class (14).
Theorem 3.1. If P1 and P2 are compatible then ω
1 and ω
2 form a finite-dimensional
bi-Hamiltonian structure on the target space.
Proof. Pλ could have the general form
x + cλ
ukxx ,
but clearly bλ
= 2Γ1
+ 2λΓ2
and cλ
+ λΓ2
, so bλ
= 2cλ
, and
hence, by Lemma 2.6, a
satisfies the Jacobi identity (3) for all λ. �
So we write
+ 2Γλ
x + Γλ
ukxx .
An immediate corollary of Theorem 3.1 is that the tensor Lij = ω
1 ω2rj has
vanishing Nijenhuis torsion.
3.1. Multiplication of covectors. As in [7], we proceed to understand the com-
patibility conditions on P1 and P2 in terms of the algebraic properties of a tensorial
multiplication of covectors on M .
Definition 3.2. Using the tensors
∆sjk = ω
r − ω
i = ω2is∆
sjk ,
we define a multiplication ◦ of covectors on M by
(α ◦ β)i = αjβk∆
Theorem 3.3. The compatibility of P1 and P2 is equivalent to
(I, J ◦K)2 = (J, I ◦K)2 , (15)
and (I ◦ J) ◦K = 0 , (16)
for all covectors I, J,K on M . Here (·, ·)2 is the skew-symmetric bilinear form on
T ∗M induced by ω
2 , i.e. (I, J)2 = IrJsω
2 . The compatibility also implies
= ∇2k∆
. (17)
Because of Theorem 3.1, we phrase the compatibility of P1 and P2 in terms of
Fedosov structures on M , and break the above theorem into stages:
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 9
Definition 3.4. Two flat Fedosov structures (ω1,∇
1) and (ω2,∇
2), where ∇1 and
∇2 have contravariant Christoffel symbols Γ1
and Γ2
respectively, are said to be
(i) almost compatible if and only if (ωλ,∇
λ) is a Fedosov structure for all λ,
where the connection ∇λ is given by Γλ
+ λΓ2
(ii) almost compatible and flat if and only if they are almost compatible, and
in addition the curvature of ∇λ vanishes for all λ .
(iii) compatible if and only if they are almost compatible and flat, and cλ
(k,l)
− ωλprΓλ
satisfies cλ
+ λc2
for all λ.
The compatibility of two flat Fedosov structures on M is equivalent to the com-
patibility of the associated Poisson brackets on L(M).
We now turn to the two Fedosov strucutres defined by P1 and P2, and to the
pair (ωλ,∇
λ) defined by Pλ. From the linearity of Result 2.8 in the contravari-
ant symbols it can be seen that ωλ is automatically ∇
λ-constant, so the almost
compatibility of (ω1,∇
1) and (ω2,∇
2) is equivalent to ∇λ being torsion free, i.e. to
In flat coordinates for ∇2, this condition reduces to
r = ω
r . (18)
Note that we already have
ωir1 Γ1
r = ω
r . (19)
Lemma 3.5. If (ω1,∇
1) and (ω2,∇
2) are almost compatible, then the flatness of
∇λ is equivalent to either, and hence both, of
s − ∂sΓ1
= 0 (20)
and Γ1
l + Γ1
= 0 (21)
in the flat coordinates for ∇2.
Proof. The contravariant curvature of Γλ is
= ωirλ
r − ∂sΓλ
l + Γλ
s − ∂sΓ1
+ ωis1
s − ∂sΓ2
l + Γ1
l + Γ1
+λ2R2
which in flat coordinates for Γ2
reads
= ωir1
r − ∂rΓ1
l + Γ1
+λωis2
s − ∂sΓ1
The vanishing of the order λ term is equivalent to equation (20), and with this the
vanishing of the λ-independent term is equivalent to (21). �
Lemma 3.6. If (ω1,∇
1) and (ω2,∇
2) are almost compatible then the condition
(k,l)
− ωλprΓλ
reads, in the flat coordinates for ∇2,
l − Γ1
= 0 . (22)
10 JAMES T. FERGUSON
Proof. For an arbitrary Fedosov structure (ω,∇) the object c
(k,l)
−ωprΓ
can be converted into a quadratic expression in contravariant quantities as
= ωskΓ
(k,l)
Γsip Γ
Γsjp . (23)
This has similarities to the formula for covariant curvature obtained in Result 2.10;
only certain signs have changed. Indeed, if we define a quantity c
dxr =
(∇∂k∇∂l +∇∂l∇∂k) dx
then c
= ωirc
We have two ways of expanding ωskλ cλ
, corresponding to whether we choose
first to substitute it into equation (23), or to expand the pencil quantities. We work
in flat coordinates for ∇2; in these, c2
also vanishes. First expanding the pencil
we have
1 + λω
= ωsk1 c1
+ λωsk2 c1
whilst (23) gives
ωskλ cλ
= ωskλ Γλ
(k,l)
ωsk1 + λω
(k,l)
The order 1 terms merely express equation (23) for P1. Equality of the order λ
terms is equivalent to Γ1
(k,l)
and so to
ωsk1 Γ1
(k,l)
= ωsk1 c1
= ωsk1 Γ1
(k,l)
Proof of Theorem 3.3. Using equation (18) in Definition 3.2 it can be seen that in
the flat coordinates for ∇2 we have ∆
. Thus we may regard equations
(18),(20),(21) and (22) as identities on ∆
; the result is Theorem 3.3. �
The condition imposed by equation (21) for an almost compatible and flat pair
of Fedosov structures on the mutliplication ◦ is (I ◦ J) ◦ K = −(I ◦ K) ◦ J , i.e.
the first condition (8) satisfied by the multiplication of a Fermionic Novikov alge-
bra. In general (9) is not satisfied even for compatible Fedosov structures, however
we do have, for two flat Fedosov structures, (ω1,∇
1), (ω2,∇
2), which are almost
compatible,
= ∆ijr ∆
∆ikr .
So, in particular, if ∆
is constant in the flat coordinates for ∇2, almost compatible
and flat Fedosov structures will define a Fermionic Novikov algebra structure on
the covectors of M .
In [1] it emerged that examples of such algebras which do not also satisfy the
‘Bosonic’ relation (I ◦J) ◦K = (I ◦K) ◦J , and hence (I ◦J) ◦K = 0, are relatively
rare. ∇2-constant multiplications arising from pairs of Fedosov structures which
are almost compatible and flat, but not compatible, such as that given in Example
3.10 below, are in this class.
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 11
3.2. The pencil in flat coordinates. We now turn our consideration to the form
the pencil takes in the flat coordinates for ∇2. From the elements of the proof of
Theorem 3.3 we have
1 + λω
+ 2Γ1
x + Γ1
ukxx . (24)
The Jacobi identity for Pλ (without assuming P1 and P2 are Hamiltonian them-
selves) is equivalent to the constraints
(i) ω
2 is constant and antisymmetric,
(ii) ω
1 is antisymmetric,
(iii) ωir1 Γ1
r = ω
(iv) ω
(v) ωir2 Γ1
r = ω
(vi) Γ1
(vii) Γ1
l = 0.
Proposition 3.7. In a fixed coordinate system {ui} (the flat coordinates for Γ2),
given a constant non-degenerate 2-form ω
2 and a vector field B = B
r∂r satisfying
ωis2 B
,s − ω
,pr =
,s − ω
,pr (25)
,sl = 0 (26)
then the prescription
1 = −(LBω2)
ij = ωir2 B
,r − ω
= ωir2 B
satisfies the constraints (i)-(vii). Further, all solutions of (i)-(vii) have this form.
Proof. Equations (25) and (26) are the quadratic constraints, ωir1 Γ1
r = ω
and Γ1
l = 0 respectively. That ω1 and Γ1 satisfy the (linear) constraints (iv),
(v) and (vi) is an immediate consequence of their definition.
Using the Poincare lemma together with the symmetries expressed in conditions
(vi) and (v), we have the existence of a vector field satisfying Γ1
= ωir2 A
j ,rk . With
this condition (iv) gives ω
1 = −(LAω2)
ij+cij , where cij is a constant antisymmetric
matrix. We may now introduce a vector field B with Bi = Ai + 1
xsw2src
ri which
satisfies ω
1 = −LBω
2 and Γ1
= ωir2 B
Since ω2 is a symplectic form, its symmetries are precisely (locally) Hamiltonian
vector fields. Therefore, if ω2 and ω1 are given, the requirement that ω
1 = −LBω
fixes the non-Hamiltonian part of B. Then the condition Γ1
= ωir2 B
fixes the
Hamiltonian to within a quadratic function. From the point of view of the multipli-
cation of covectors from Section 3.1, the Hamiltonian affects only the commutative
part of ◦, thus the anti-commutative part is fixed by ω
1 and ω
With consideration of the transformation rules (6), one can phrase Proposition
3.7 as the existence of a vector field B such that
1 = −LBω
= −LBΓ2
. (27)
12 JAMES T. FERGUSON
We can also calculate from (6) the correct interpretation of the Lie derivative for
an object of type c
, namely:
= Xrc
−X i,rc
−Xj,rc
kl +X
+Xr,lc
+Xr,klc
birk −
birl −X
air .
If we work in the flat coordinates for Γ2, so that the components c2
= 0, we
have for our pencil
−LBc2
= +ωir2 B
= (ωir2 B
),l ,
Now, in the flat coordinates for ∇2 we have the relation c1
. The linearity
of the transformation rules shows that the Lie derivative of c2
should be an object
of the same type as c1
. Thus we have, in addition to (27),
= −LBc2
One may understand these three infinitesimal relations between the coefficients
of P1 and P2 as averring the existence on L(M) of an evolutionary vector field
B̂ = Bi(u(x))
∂ui(x)
+ . . .
such that
1 = −LB̂P
We now turn our attention to some examples of pairs of Fedosov structures,
using the framework of Proposition 3.7.
Example 3.8. Two-dimensional pencils. Without loss of generality we take
where u1 and u2 are a flat coordinate system for ∇2.
We take
B = f(u1, u2)
+ g(u1, u2)
and from it calculate ω1 and Γ1 according to (27). In particular
ω1 = (f,1 + g,2)ω2 ,
from which it follows immediately that (ω1,∇
1) and (ω2,∇
2) are almost compatible.
They are almost compatible and flat if and only if h = f + λg satisfies the
homogeneous Monge-Ampere Equation h212 − h11h22 = 0 for all λ.
They are compatible if and only if a = f + λg and b = f + µg satisfy
a12b12 − a11b22 = 0
for all λ, µ.
For instance, one may recover the three two-dimensional Fermionic Novikov al-
gebras of [1] as constant multiplications via
(T1) f = u1, g = 0 ,
(T2) f = u1, g = (u1)2 ,
(T3) f = (u1)2, g = 0 .
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 13
Example 3.9. Commutative algebras. In the case in which ω1 is constant in the
flat coordinates for ∇2, we have, by condition (iv),
so that the multiplication ◦ is commutative.
In particular if
ω1 = ω2 = ω =
then the non-Hamiltonian part of B is
To this we may add a Hamiltonian vector field, giving
Since ω1 = ω2, equation (25) is immediate. Equation (26) becomes
H,ijr ω
H,skl = 0 ,
where the indices i, j, k, l, r, s account for both q and p variables.
A solution to this is H = f(x1, x2, . . . , xn), where each xi is either pi or q
i; only
one from each pair of conjugate variables features in H.
It is not hard to see that Proposition 3.7 can be modified to describe almost
compatible and flat pairs of Fedosov structures. Specifically, we replace equation
(26) by the expression corresponding to Γ1
l = Γ1
, namely:
,sl = B
ωrs2 B
,si . (28)
Example 3.10. The Fedosov structures specified by
= 0 ,
+ 2q1q2
+ q1p2
and ω
1 = −LBω
2 and Γ1
= −LBΓ2
are almost compatible and flat, but not
compatible.
The non-zero components of ω1 and ◦ are
{q1, p1}1 = {q2, p2}1 = 3q1 ,
{q2, p1}1 = 2q2 ,
{p2, p1}1 = p2 ,
dq2 ◦ dp2 = dq1 ,
dp1 ◦ dq1 = −3dq1 ,
dp1 ◦ dq2 = −2dq2 ,
dp1 ◦ dp2 = −dp2 ,
dp2 ◦ dq2 = −2dq1 .
14 JAMES T. FERGUSON
Thus, the products
(dp1 ◦ dq2) ◦ dp2 = −2dq1
and (dp1 ◦ dp2) ◦ dq2 = 2dq1
violate equation (16) but not (8). Note that ◦ also satisfies (9) and thus defines a
Fermionic Novikov algebra which is not ‘Bosonic’.
3.3. ωN manifold with Potential. The tangent bundle T ∗Q of a manifold Q is
naturally equipped with a symplectic form, and thus cotangent bundles form the
basic set of examples of symplectic manifolds. One may hope to find examples
of finite-dimensional bi-Hamiltonian structures on cotangent bundles by exploiting
the existence of additional structures on the underlying manifolds. The main object
used to do this is a (1, 1)-tensor Lij on Q whose Nijenhuis torsion is zero. Such an
object was utilised by Benenti [2] to demonstrate the separability of the geodesic
equations on a class of Riemannian manifolds. This result was later interpreted
in [12] in terms of a bi-Hamiltonian structure on T ∗Q which was extended to a
degenerate Poisson pencil on T ∗Q× R.
To obtain Fedosov structures we require more than just a tensor Lij on Q with
vanishing Nijenhuis torsion; we also need a means of specifying the connections.
If Q is equipped with a torsion-free connection ∇̃, then the Nijenhuis torsion of a
(1, 1)-tensor Lij can be written as
jk = L
j∇̃sL
k − L
k∇̃sL
j − L
s∇̃jL
k + L
s∇̃kL
If there exists a vector field, A, on Q such that Lij = ∇̃jA
i then
N ijk = (∇̃jA
s)(∇̃s∇̃kA
i)− (∇̃kA
s)(∇̃s∇̃jA
i)− (∇̃sA
i)(RsjkrA
where Rijkl is the curvature tensor of ∇̃.
So, if ∇̃ is flat then the vanishing of the Nijenhuis tensor of L = ∇̃A is equivalent
to the identity
(∇̃jA
s)(∇̃s∇̃kA
i) = (∇̃kA
s)(∇̃s∇̃jA
i) . (29)
Proposition 3.11. Given a manifold Q endowed with a flat connection ∇̃ and a
vector field A satisfying (29), the cotangent bundle T ∗Q is endowed with a compati-
ble pair of Fedosov structures, (ω1,∇
1) and (ω2,∇
2), as follows: ω2 is the canonical
Poisson bracket on T ∗Q.
The connection ∇2 on T ∗Q is the horizontal lift [19] of the connection ∇̃ on Q;
i.e. the Christoffel symbols Γ2
ij of ∇
2 are zero in the coordinates induced on T ∗Q
by the flat coordinates for ∇̃.
(ω1,∇
1) is calculated from (ω2,∇
2) according to the prescription of Proposition
3.7, where the vector field B is the horizontal lift of A to T ∗Q.
Proof. Let {q1, . . . , qn} be flat coordinates for ∇̃ onQ, and C = {q1, . . . , qn, p1, . . . , pn}
be the induced coordinates on T ∗Q. Then
The space of sections of the cotangent bundle of T ∗Q, Ω, naturally splits into
P = span{dpi} and Q = span{dq
i}. For Γ1
= ωir2 B
to be non-zero requires k
to represent a variable qk, and i to represent a pi variable. Thus Ω ◦ Ω ⊆ Q and
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 15
Q ◦ Ω = {0}, meaning that (Ω ◦ Ω) ◦ Ω = {0}. So the relation (26), Γ1
l = 0,
is satisfied.
1 has only one kind of non-zero component, ω
,i, so the expression
ωir1 Γ1
r has only one non-zero case:
qr = A
,rj ,
which is seen to be symmetric in i and j by condition (29), which in the flat
coordinates qi reads
As,j A
i,sk = A
s,k A
i,sj .
Example 3.12. If the eigenvalues of L : TQ → TQ are functionally independent
in some neighbourhood then they may be used as coordinates, and L takes the form
⊗ dui .
In this case we may set A =
(ui)2 ∂
, and have ∇̃ defined by vanishing
Christoffel symbols in these coordinates.
This gives, writing vi as the conjugate coordinate to u
i on T ∗Q,
= −1 ,
and all other Christoffel symbols zero.
4. Bi-Hamiltonian Structures in Degrees 1 and 2
We now consider a pair of operators, P1 and P2 in which P1 is a Hamiltonian
operator of hydrodynamic type and P2 is of second order, i.e. :
1 = g
ij(u)
(u)ukx ,
2 = a
x + c
ukxx ,
where gij is the inverse of a flat metric gij on M and Γ
= −girΓ
where the Γkij
are the Christoffel symbols of the Levi-Civita connection of g. We also assume that
2 is antisymmetric, so that a
ij = −aji, b
and c
(k,l)
The motivation [8] for studying such pairs of operators comes not from regarding
them as separate Hamiltonian operators, but from thinking of P
2 as a first order
(dispersive) deformation of P
1 into some non-homogeneous Hamiltonian operator
P ij = P
1 +εP
2 +O(ε
2). Thus, in such a pair, it is sensible to regard the geometry
1 as being more intrinsic than any associated to P
We choose to work in flat coordinates for g so that gij is constant and Γ
Direct calculation of the Jacobi identity for P ij in these coordinates yields
16 JAMES T. FERGUSON
Theorem 4.1. P2 is an infinitesimal deformation of P1, i.e. P
ij = P
1 + εP
O(ε2) satisfies the Jacobi identity to order ε, if and only if
(I) gircjkr + g
jrcikr = 0 ,
(II) c
(k,l)
(III) girc
= gjr(cikl,r − c
r,l) ,
(IV) gir(ajk,r − c
r ) + g
jr(aki,r − c
r ) + g
kr(aij,r − c
r ) = 0
in the flat coordinates for gij.
By introducing the tensor T
= airΓ
is it easy to convert conditions (I),
(III) and (IV) to arbitrary coordinates, whilst condition (II) becomes
− crik Γ
− cril Γ
+ T ijr Γ
kl + T
Γirl + T
Γirk .
To consider a bi-Hamiltonian structure involving operators P
1 and P
2 one need
only add conditions (C), (D) and (E) of Theorem 2.1 to Theorem 4.1, however,
condition (II) above allows (E) to be replaced by cijr c
l = c
Example 4.2. As discussed in section 2, P2 with b
constant and aij non-
degenerate is Hamiltonian if and only if aij = A
uk + A
0 with A
0 is constant, c
are the structure constants of a Fermionic Novikov algebra
(A, ◦), and A
0 defines a skew-symmetric bilinear form on A satisfying 〈I, J ◦K〉 =
〈J, I ◦K〉.
If we ask that P2 satisfies the above constancy conditions in the flat coordinates
for gij, then, defining an inner product on A by (ei, ej) = gij, we have that the
compatibility of P1 and P2 is equivalent to the additional constraints:
(I ◦ J) ◦K = (I ◦K) ◦ J ,
(I, J ◦K) = −(J, I ◦K)
(I, [J,K]) + (J, [K, I]) + (K, [I, J ]) = 0 ,
where [I, J ] = I ◦J−J ◦I is the commutator of ◦, which is a Lie bracket by equation
For example, if we take the algebra (A = span{e1, e2, e3, e4}, ◦) where the only
non-zero products are e3 ◦ e3 = e1 and e4 ◦ e3 = e2 then we may take as our
symplectic form and metric
[ωij ] =
0 0 a b
0 0 b c
−a −b 0 d− u2
−b −c −d+ u2 0
[gij ] =
0 0 0 e
0 0 −e 0
0 −e f g
e 0 g h
 ,
for any choice of the constants a, b, c, d, e, f, g, h such that e 6= 0 and b2 6= ac.
This algebra, essentially (57)−1, is the only algebra in [1] of dimension 2 or 4
which admits non-degenerate forms (·, ·) and 〈·, ·〉 satisfying the above compatibility
conditions with ◦, other than the trivial case in which all products are zero, i.e.
in which the Hamiltonian operators share the same flat connection, and so are
simultaneously constant.
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 17
Proposition 4.3. If P2 is an infinitesimal deformation of P1 then there exists a
tensor field Aij such that
aij = girAjr − g
jrAir ,
= 2gisA
− gjrAik,r − g
= gisA
− gisA
(k,l)s
= gisA
− gisA
in flat coordinates for gij. Further, any (1,1)-tensor field Aij produces an infinites-
imal deformation of P1 by the above formulae.
Proof. Using the non-degeneracy of gij , we introduce objects θkij and φij by
= girθ
aij = girgjsφrs .
Then condition (I) of Theorem 4.1 is equivalent to θkij = −θ
ji, and so we regard θ
as a family of 2-forms θk indexed by k.
Condition (III) is equivalent to θkjl,i = θ
il,j − θ
ij,l, so that dθ
k = 0 for each k.
This allows us to introduce a family of 1-forms ψk such that
ij = (dψ
k)ij = ψ
i,j − ψ
j,i .
Each ψk can be adjusted by the addition of the exterior derivative, dfk, of some
function fk without affecting the value of θkij .
Writing αij = φij − gjrψ
i + gjrψ
k, we find that condition (IV) is equivalent to
the closedness of the 2-form αij , upon substituting φij and ψ
j for a
ij and c
. Thus
we may introduce a 1-form h with components hi such that αij = hi,j − hj,i, and
φij = gjrψ
i − gjrψ
j + hi,j − hj,i .
If we now let Aij = ψ
j + (g
irhr),j then we have θ
ij = A
i,j − A
j,i and φij =
i − girψ
j , so that the two equations a
ij = girAjr − g
jrAir and c
= girA
are satisfied. The remaining to equations follow easily from c
For the converse, it is easy to check that conditions (I)-(IV) of Theorem 4.1
follow from (30) for any tensor field Aij . �
As with Proposition 3.7, Proposition 4.3 may be understood as asserting the
existence of an evolutionary vector field
e = Aij (u(x)) u
∂ui(x)
+ . . .
satisfying P2 = −LeP1 whenever P2 is an infinitesimal deformation of P1. This is
therefore not a surprising result; in [11] Getzler showed the triviality of infinitesimal
deformations of Hydrodynamic type Poisson brackets. With this, Proposition 4.3
can be looked upon as a proof of Theorem 4.1.
There is a freedom in Aij of A
j 7→ A
j + g
irf,rj for some function f , which does
not affect the coefficients of P2. This corresponds to adjusting e by a Hamiltonian
vector field, e 7→ e+ P1(δf).
If, with reference to Lemma 2.6, we impose the additional constraint on (30)
that b
then we have the potentiality condition gjrA
k,i = girA
k,j , so that
there exists a 1-form Bk such that
j = g
Bj,r . (31)
18 JAMES T. FERGUSON
In this case aij = girgjr(Br,s − Bs,r) = g
irgjr(dB)rs and the freedom A
j 7→ A
girf,rj is B 7→ B + df . This means that B can be determined purely from g
ij and
aij , and thus there is no freedom in the choice of c
and c
. In fact we may write
explicitly
= gjsgkr
(k,l)
, (32)
and with this, P2 is an infinitesimal deformation of P1 if and only if
girajk,r + g
jraki,r + g
kraij,r = 0 . (33)
Corollary 4.4. Given a flat metric g and a symplectic form ω, there is at most one
choice of flat symplectic connection ∇ such that the degree 2 Hamiltonian operator
specified by (ω,∇) is compatible with the hydrodynamic operator specified by g.
Clearly, if this connection exists it is given by(32), so this definition must be
checked against Theorem 2.1 to verify
2 = ω
x + c
is Hamiltonian. Since equation (33) is a consequence of the antisymmetry of P2,
compatibility with the Hydrodynamic operator follows immediately.
We conclude this section with an example of this type.
Example 4.5. The Kaup-Broer system [16],
u1xx + 2u
x + 2u
−u2xx + 2(u
1u2)x
is described by the pair of compatible Hamiltonian operators
u1 2u2
0 u1x
0 u2x
Scaling x 7→ εx, t 7→ εt splits P2 into P
2 + εP
2 where
u1 2u2
0 u1x
0 u2x
Since P2 = P
2 + εP
2 is Hamiltonian for all ε, P
2 and P
2 constitute a bi-
Hamiltonian structure of the type considered above. A set of flat coordinates for the
metric in P
ũ1 = u1 ,
ũ2 =
4u2 − (u1)2 ,
in which
(ũ2)2
0 −ũ2x
0 ũ1x
(ũ2)3
0 (ũ2x)
0 −ũ1xũ
(ũ2)2
0 −ũ2xx
0 ũ1xx
FLAT PENCILS OF SYMPLECTIC CONNECTIONS 19
So in this situation we have, for the 1-form in (31),
dũ2 .
5. Conclusions
In section 3 an approach was taken based upon the methods of [7] to study com-
patible pairs of Hamiltonian operators of degree 2 which satisfy the conditions of
the relevant Darboux theorem, Theorem 2.11. As for Hydrodynamic Poisson pen-
cils, the compatibility could be reduced to algebraic constraints on a multiplication
of covectors. Driving this was the ability to reduce a given Hamiltonian operator
on L(M) to a flat Fedosov structure (ω,∇) on M , which are natural symplectic
analogues of the pair consisting of a flat metric and its Levi-Civita connection which
determines a Hydrodynamic Poisson bracket.
To extend such a results to pairs of arbitrary degree 2 Hamiltonian operators,
one must consider the pair (a,∇) of Theorem 2.4. The condition (10), whilst
atypical, expresses a familiar concept; in almost-symplectic geometry, it is common
to consider connections such that the covariant derivative of the almost-symplectic
form is zero, but which have torsion; if the torsion of such a connection is skew-
symmetric then its symmetric part satisfies (10). Equation (12) provides the means
of going from the symmetric connection to the compatible connection with skew-
torsion. The only formula missing above necessary to the study of arbitrary bi-
Hamiltonian structures of degree 2 is an expression for the contravariant curvature
of the connection defined by c
, which is, in the presence of Theorem 2.1’s condition
= air(c
) + cijr c
l + c
− (bijr − 2c
l + c
One may use (B) to replace the components of b
in this expression with those of
and the derivatives of aij . However, one sees that the compatibility conditions
do not naturally become algebraic constraints on ∆
, and the relevancy of such an
approach is undermined. It is interesting to note, however, that equation (23) still
holds (with Γ
), so that ◦ defined by ∆
still satisfies (I ◦J)◦K = (I ◦K)◦J ,
and that it is the ‘Fermionic’ condition (I ◦ J) ◦K = −(I ◦K) ◦ J which is altered.
The proof of Proposition 3.7 is easily adapted to confirm the existence of a vector
field B realising P1 = −LBP2 whenever P1, of the form (5) is an infinitesimal defor-
mation of P2 as a Hamiltonian operator, provided b1
= 2c1
. A simple calculation
of LBP2 for arbitrary B shows that b1
= 2c1
is also a necessary condition. Thus
we have determined the trivial deformations of a degree 2 Hamiltonian operator
admitting a constant form, which are themselves of degree 2. Clearly a different
approach is necessary to understand deformations of higher degrees. For the case
of operators not satisfying the constraints of Theorem 2.11, it is not immediately
obvious what conditions, if any, will guarantee the triviality of a deformation; ow-
ing to the different form the contravariant curvature tensor takes, the condition
is absent. Owing to the lack of a constant form, the methods of [8]
in ascertaining the triviality of higher degree deformations, if applicable, will be
somewhat more complicated.
Finally, there is a certain artificiality to the examples of compatible Fedosov
structures presented in section 3. Given Theorem 3.1’s assertion that underlying a
pair of compatible Fedosov structures is a finite-dimensional bi-Hamiltonian struc-
ture, the question is raised asking which finite-dimensional bi-Hamiltonian struc-
tures admit symplectic connections forming almost compatible, almost compatible
and flat, or compatible Fedosov structures? It would be interesting to exhibit a
20 JAMES T. FERGUSON
pair of compatible Fedosov structures in which the flat coordinates for one of the
connections are in some sense physical.
Acknowledgements
The author would like to thank Ian Strachan for suggesting this project, and the
Carnegie Trust for the Universities of Scotland for the scholarship under which this
work was conducted.
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Department of Mathematics, University of Glasgow, Glasgow G12 8QW, U.K.
E-mail address: j.ferguson@maths.gla.ac.uk
http://arxiv.org/abs/math/0108160
	1. Introduction
	2. Hamiltonian Operators of Degree 2
	3. Flat Pencils of Fedosov Structures
	3.1. Multiplication of covectors
	3.2. The pencil in flat coordinates
	3.3. N manifold with Potential
	4. Bi-Hamiltonian Structures in Degrees 1 and 2
	5. Conclusions
	Acknowledgements
	References
