LOCALLY CONFORMAL SYMPLECTIC STRUCTURES AND THEIR GENERALIZATIONS FROM THE POINT OF VIEW OF

LIE ALGEBROIDS

ROMAN KADOBIANSKI, JAN KUBARSKI,

Abstract. We study locally conformal symplectic structures and their generalizations from the point of view of transitive Lie algebroids. To consider l.c.s. structures and their generalizations we use Lie algebroids with trivial adjoint Lie algebra bundle M × ℝ and M × 𝔤. We observe that important l.c.s’s notions can be translated on the Lie algebroid’s language. We generalize l.c.s. structures to 𝔤-l.c.s. structures in which we can consider an arbitrary finite dimensional Lie algebra 𝔤 instead of the commutative Lie algebra ℝ.

1. l.c.s structures from the point of view of Lie algebroids

We study locally conformal symplectic structures and their generalizations from the point of view of transitive Lie algebroids. We recall that an l.c.s. structure on a manifold M is a pair (ω, Ω) of differentiable forms on M such that

(1) ω is a real closed 1-form on M,

(2) Ω is a real non-degenerated 2-form fulfilling the property

dΩ = −ω ∧ Ω.

From the non-degeneracy of Ω follows that M has even dimension.

By a transitive Lie algebroid on a manifold M [P1] we mean a system (A, [[·, ·]], #_{A}) consisting of a vector bundle A over M and mappings [[·, ·]] : Sec A×Sec A → Sec A, #_{A} : A → T M, such that

(a) (Sec A, [[·, ·]]) is a real Lie algebra,

(b) #_{A}, called an anchor, is an epimorphism of vector bundles,

(c) Sec #_{A} : Sec A → 𝔛 (M) , ξ ↦ #_{A} ∘ ξ, is a homomorphism of Lie algebras,

(d) [[ξ, f · η]] = f · [[ξ, η]] + (#_{A} ∘ ξ) (f) · η, ξ, η ∈ Sec A, f ∈ Ω^{0} (M) = C^{∞} (M) .

The axiom (c) follows from the remaining ones, see [H], [B-K-W].

It follows that ggg := ker #_{A} is a LAB (Lie algebra bundle), called the adjoint of A. The Lie algebra ggg_{x} is called the structure Lie algebra at x. The exact sequence

0 → ggg → A ⟶^{#_{A}} T M → 0

is called the Atiyah sequence of A, while any splitting λ : T M → A, #_{A} ∘λ = id_{T M}, is a connection in A. The following geometric objects give rise to transitive Lie algebroids:

— Lie groupoids,

— principal fibre bundles,

— vector bundles,

— transversely complete foliations,

— nonclosed Lie subgroups.

We add that differential groupoids (non-transitive, in general), Poisson and Jacobi manifolds as well as any infinitesimal action of a Lie algebra on a manifold produce nontransitive Lie algebroids. The image of the anchor always is an integrable singular (or regular) foliation [P2], [F], and the restriction of the Lie algebroid to any leaf of this foliation is a transitive Lie algebroid.

To consider l.c.s. structures and their generalizations we use Lie algebroids with trivial adjoint Lie algebra bundle ggg = M × 𝔤.

From the general theorem concerning the form of any transitive Lie algebroids (Mackenzie [M], Kubarski [K1]) we have:

• Each transitive Lie algebroid on M with a trivial adjoint bundle ggg ≅ M ×ℝ

is isomorphic to

A = T M × ℝ

with #_{A} = pr_{1} : T M × ℝ → T M as the anchor and the bracket [[·, ·]] in Sec A is defined via some flat covariant derivative ∇ in M × ℝ and a 2form Ω ∈ Ω^{2} (M) fulfilling the Bianchi identity ∇Ω = 0 in the following way

[[(X, f) , (Y, g)]] = ([X, Y] , ∇_{X}g − ∇_{Y} f − Ω (X, Y)) .

We recall that a covariant derivative ∇ in a vector bundle ξ determines a standard operator d_{∇} : Ω^{∗} (M; ξ) → Ω^{∗} (M; ξ) and d_{∇}θ is sometimes denoted by ∇θ. If ∇ is flat then (d_{∇})^{2} = 0 and it determines the cohomology space H_{∇} (M; ξ) in the obvious way.

Each flat covariant derivative in ggg = M × ℝ is of the form

∇_{X}f = ∂_{X}f + ω (X) · f

where ω is a closed differentiable 1-form on M. Then the differential operator d_{∇} is denoted rather by d_{ω} [G-L], [H-R]. We have

d_{ω} (θ) = dθ + ω ∧ θ,

and write H_{ω} (M) := H_{d_{ω}} (M) .

The condition ∇Ω = 0 is then equivalent to dΩ = −ω ∧ Ω.

Hence any transitive Lie algebroid with the trivial adjoint bundle ggg = M × ℝ is determined by the following data:

(*) a closed 1-form ω and a 2-form Ω such that dΩ = −ω ∧ Ω.

The Lie algebroid obtained in this way will be denoted by

(T M × ℝ, ω, Ω) .

Lemma 1.1. A connection λ : T M → T M×ℝ in the Lie algebroid A = (T M × ℝ, ω, Ω) is of the form λ (X) = (X, η (X)) for a 1-form η ∈ Ω^{1} (M) . The curvature form Ω^{λ} (X, Y) = [[λX, λY]] − λ [X, Y] of the connection λ is equal to

(1.1)  Ω^{λ} = d_{ω} (η) − Ω = dη + ω ∧ η − Ω.

According to (*) the pair (ω, Ω) determining the above Lie algebroid is precisely a locally conformal symplectic structure (l.c.s. structure, for short) on the manifold M provided that the 2-form Ω is non-degenerate. Therefore our transitive Lie algebroids T M × ℝ determined by (ω, Ω) are natural generalizations of the locally conformal symplectic structures. For an l.c.s. structure (ω, Ω) , following (*), the form Ω represents the cohomology class [Ω] ∈ H_{ω}^{2} (M) which is called the Lichnerowicz class of the l.c.s structure (ω, Ω) [B2]. If the 1-form ω is exact the l.c.s. structure is called globally conformal symplectic structure. The property that an l.c.s. structure is global can be equivalently expressed in the language of Lie algebroids.[K-K-K-W], [K-M]. For this purpose we recall that a transitive Lie algebroid (A, [[·, ·]], #_{A}) is called invariantly oriented [K3] if there is specified a nonsingular cross section ε of the bundle ⋀^{n} ggg, ggg := ker #_{A} and n = rankggg, which is invariant with respect to the adjoint representation of A in ⋀^{n} ggg, equivalently, if ggg is orientable and the modular class of the Lie algebroid is zero ([E-L-W], [K-M]). We add that for a transitive Lie algebroid the modular class is equal to the characteristic class of the top-power of the adjoint representation ad_{A} . The structure Lie algebras ggg_{x} of the invariantly oriented Lie algebroid are unimodular.

A cross-section ε of the bundle ⋀^{n} ggg is invariant if and only if, in any open subset U ⊂ M on which ε is of the form ε_{|U} = (h_{1} ∧ … ∧ h_{n})_{|U} , h_{i} ∈ Secggg, we have, for all ξ ∈ Sec A,

∑_{i=1}^{n} (h_{1} ∧ … ∧ [[ξ, h_{i}]] ∧ … ∧ h_{n})_{|U} = 0.

In the case A = (T M × ℝ, ω, Ω) we have n = 1 and ggg =M × ℝ and a positive function ε ∈ C^{∞} (M) = Sec (M × ℝ) is invariant if and only if ε is ∇-constant, ∇ε = 0 [K3, Lemma 6.2.1]. The condition ∇ε = 0 is equivalent to ω = d (− ln (ε)) .

Theorem 1.1. Let (ω, Ω) be an l.c.s. structure on an arbitrary m-dimensional connected manifold (oriented or not) The following conditions are equivalent:

(a) the l.c.s. structure (ω, Ω) is globally conformal symplectic structure (i.e. [ω] = 0).

(b) the associated Lie algebroid A = (T M × ℝ, ω, Ω) is invariantly oriented,

(c) H_{∂_{A}^{or},c}^{m+1} (A, or (M)) ≠ 0,

(d) H_{∂_{A}^{or},c}^{m+1} (A, or (M)) = ℝ, and the pairing

H^{j} (A) × H_{∂_{A}^{or},c}^{m+1−j} (A, or (M)) → H_{∂_{A}^{or},c}^{m+1} (A, or (M)) ≅ ℝ

is non-degenerate, i.e. H^{j} (A) ≅ (H_{∂_{A}^{or},c}^{m+1−j} (A, or (M)) )^{∗}.

Proof. (a) ⟺ (b) see [K-K-K-W], (b) ⟺ (c) ⟺ (d) see [K-M]. □

Remark 1.1. (1) For an orientable manifold M the conditions (c) and (d) are equal to:

(c’) H_{c}^{m+1} (A) ≠ 0, (d’) H_{c}^{m+1} (A) = ℝ, and the pairing

H^{j} (A) × H_{c}^{m+1−j} (A) → H_{c}^{m+1} (A) ≅ ℝ

is non-degenerate, i.e. H^{j} (A) ≅ (H_{c}^{m+1−j} (A)) .

(2) ∂_{A}^{or} is the canonical representation of A in the orientation bundle or(M), (∂_{A}^{or})_{γ} (σ) = (∂^{or})_{#_{A}(γ)} (σ) , γ ∈ A, σ ∈ Γ (or (M)) . ∂^{or} is the canonical flat structure of the orientation bundle or (M) [B-T].

(3) Each representation ∇ of a Lie algebroid A in a vector bundle ξ (i.e. a homomorphism of a Lie algebroid A in the Lie algebroid A (ξ) of the vector bundle ξ [K2], [M]) determines a standard differential operator d_{∇} : Ω (A; ξ) → Ω (A; ξ) and H_{∇} (A; ξ) is the space of cohomology of the complex (Ω (A; ξ) , d_{∇}) . Local trivializations of A (𝔣) are constucted in the following way: Let ψ : U × V → p^{−1} [U] = 𝔣_{|U} be a local trivialization of a vector bundle 𝔣; V is the typical fibre. Consider the trivial Lie algebroid T U × End (V). For a cross-section σ ∈ Sec 𝔣, denote by σ_{ψ} the V -valued function U ∋ x ↦ ψ_{x}^{−1} (σ (x)) ∈ V. The mapping

ψ̄ : T U × End (V) ⟶ A (𝔣)_{|U}
ψ̄ (v, a) (σ) = ψ_{x} (v (σ_{ψ}) + a (σ_{ψ} (x))) ,

(v ∈ T_{x}U, x ∈ U, a ∈ End (V) , σ ∈ Sec 𝔣) is an isomorphism of Lie algebroids [K2].

(4) The associated Lie algebra bundle of the considered Lie algebroid A = (T M × ℝ, ω, Ω) is the trivial line bundle ggg =M × ℝ. Therefore, the top group of cohomology H_{∂_{A}^{or},c}^{m+1} (A, or (M)) can be written (analogously to real coefficients, see [K-K-K-W]) as follows

H_{∂_{A}^{or},c}^{m+1} (A; or (M)) = H_{d_{∂^{−ω} ⊗∂^{or}}}^{m} (M; or (M)) = H_{(∂^{or})^{−ω}}^{m} (M; or (M)) .

Then the equivalence (a) ⟺ (c) follows trivially, since

H_{(∂^{or})^{−ω}}^{m} (M; or (M)) ≠ 0 ⟺ [−ω] = 0,

see [K-M].

Two l.c.s. structures (ω, Ω) and (ω′, Ω′) on a manifold M are called conformally equivalent if

Ω′ = \frac{1}{a}Ω, ω′ = ω + \frac{da}{a},

for a nowhere vanishing function a on M (non-singular for short).

If two l.c.s. structures (ω′, Ω′) and (ω, Ω) on a manifold M are conformally equivalent then the associated Lie algebroids A′ = (T M × ℝ, ω′, Ω′) and (T M × ℝ, ω, Ω) are isomorphic via the mapping

\begin{matrix}H : (T M × ℝ, ω′, Ω′) ⟶ (T M × ℝ, ω, Ω) \\ H (X, f) = (X, a · f) \end{matrix}

where a ∈ C^{∞} (M) is a non-singular smooth function. The isomorphism H : A′ → A of the above form will be called a conformal isomorphism.

We add that the general form of a homomorphism H : T M × ℝ → T M × ℝ of vector bundles commuting with anchors #_{A} = pr_{1} is as follows

(**)  H (X, f) = H_{η,a} (X, f) := (X, η (X) + a · f) ,

for η ∈ Ω^{1} (M) and a ∈ C^{∞} (M) .

Proposition 1.1. (A) The following conditions are equivalent:

(1) H_{η,a} is a homomorphism of Lie algebroids,

(2) (a) ∇η = Ω − a · Ω′, (b) ∇_{X} (a · f) = a · ∇′_{X}f,

(3) (a) d_{ω} (η) = dη + ω ∧ η = Ω − a · Ω′, (b) a · (ω′ − ω) = da.

The homomorphism H_{η,a} is an isomorphism of Lie algebroids if and only if a is non-singular. Conditions (1), (2), (3) are then equivalent to

(4) (a) Ω′ = \frac{1}{a} · (Ω − d_{ω} (η)) ,

(b) ω′ = ω + d (ln |a|) .

(B) For an arbitrary Lie algebroid A′ = (T M × ℝ, ω′, Ω′) and data (η, a) where η ∈ Ω^{1} (M) and a is a non-singular function, the differential forms ω = ω′ − d (ln |a|) , Ω = a · Ω′ + d_{ω} (η) fulfil the condition dΩ = −ω ∧ Ω, i.e. the data (ω, Ω) determines a Lie algebroid A = (T M × ℝ, ω, Ω) and H_{η,a} : A′ → A given by (**) is an isomorphism of Lie algebroids.

Proof. Easy calculation. □

Clearly H_{η′,a′} ∘ H_{η,a} = H_{η′+a′·η,a′·a}, (H_{η,a})^{−1} = H_{− \frac{η}{a} , \frac{1}{a}} . In particular,

H_{η,a} = H_{η,1} ∘ H_{0,a},

see the diagram

[figure: A′ = (T M × ℝ, ω′, Ω′); (T M × ℝ, ω, Ω) = A; H_{η,a}; (T M × ℝ, ω, a · Ω′); H_{0,a}; H_{η,1}]

It means that if A′ is isomorphic to A then there exists a Lie algebroid A′′ = (T M × ℝ, ω, Ω′′), Ω′′ = a · Ω′, conformally isomorphic to A′, i.e. such that [A] , [A′′] ∈ Opext (T M, ∇, M × ℝ) = the set of isomorphic classes of Lie algebroids having the same representation ∇ (a flat covariant derivative ∇).

Let (ω′, Ω′) and (ω, Ω) be l.c.s. structures. We observe that the isomorphism H_{η,a} : A′ → A given by (**) is equivalent to conformal equivalence of the associated l.c.s. structures if and only if η = 0.

How we can formulate the problem of existence of l.c.s. structures? We have the simple

Proposition 1.2. Any Lie algebroid A′ = (T M × ℝ, ω′, Ω′) is isomorphic to A = (T M × ℝ, ω, Ω) with Ω non-degenerate (i.e. (ω, Ω) is an l.c.s. structure) if and only if there exists in A′ a connection for which the curvature tensor is non-degenerate.

Proof. Let H_{η,a} : A′ → A be an isomorphism of Lie algebroids

(1.2)  \begin{aligned}0 & ⟶ & M × ℝ & ⟶ & (T M × ℝ, ω′, Ω′) & ⟶_{⟵_{λ′}} & T M & ⟶ & 0 \\ & & ↓^{H_{η,a}^{+}} & & ↓^{H_{η,a}} & & ↓ \\ 0 & ⟶ & M × ℝ & ⟶ & (T M × ℝ, ω, Ω) & ⟶_{⟵_{λ}} & T M & ⟶ & 0 \end{aligned}

H_{η,a}^{+} (f) = a · f. For arbitrary connections λ′ and λ in A′ and A, respectively such that H_{η,a} ∘ λ′ = λ we have the following equality for curvature tensors

Ω^{λ} = H_{η,a}^{+} ∘ Ω^{λ′}.

Therefore, if Ω is nondegenerate and λ′ is a connection such that H_{η,a}∘λ′ = λ where λ (v) = (v, 0) , then Ω^{λ} = −Ω (see lemma 1.1) and, clearly, Ω^{λ′} is non-degenerate.

Conversely, if λ′ (X) = (X, η (X)) is any connection in A′ such that Ω^{λ′} is nondegenerate, then H_{−η,1} is an isomorphism of A′ on A := (T M × ℝ, ω′, −Ω^{λ′}) (see (1.1)) and (ω′, −Ω^{λ′}) is an l.c.s. structure. □

So, the problem of existing of l.c.s. structures can be precisely formulated as follows:

Problem 1.1. We introduce into the class of pairs (ω, Ω) fulfilling (*), i.e. dΩ = −ω ∧ Ω, the equivalence relation

r) (ω′, Ω′) ≈ (ω, Ω) ≡ the Lie algebroids A′ = (T M × ℝ, ω′, Ω′) and A = (T M × ℝ, ω, Ω) are isomorphic, i.e. there exists η ∈ Ω^{1} (M) and a ∈ C^{∞} (M) , a (x) ≠ 0 for all x ∈ M, such that (4a), (4b) hold: (4a) Ω′ =

\frac{1}{a} (Ω − dη − ω ∧ η) , (4b) ω′ = ω + \frac{da}{a}.

Let dim M be even. We can ask: Does there in every (in given) equivalence class [(ω′, Ω′)] exist (ω, Ω) being an l.c.s. structure; equivalently, does there in the Lie algebroid A′ = (T M × ℝ, ω′, Ω′) exist a connection with non-degenerate curvature tensor, i.e. equivalently, does there exist a 1-form η ∈ Ω^{1} (M) such that dη + ω ∧ η − Ω is a non-degenerate.

This problem has a local solution, see Proposition 2.5 below for more general situations.

We add that for a fixed closed form ω, i.e. a flat covariant derivative ∇_{X}f = ∂_{X}f + ω (X) · f in the trivial bundle M × ℝ, the classification of Lie algebroids of the form (T M × ℝ, ω, ·) up to isomorphism is as follows: for the class of isomorphic Lie algebroids Opext (T M, ∇, M × ℝ) we have [M]

Opext (T M, ∇, M × ℝ) ≅ H_{∇}^{2} (M; ℝ) , [(T M × ℝ, ω, Ω)] ↦ [Ω] .

A. Banyaga [B2] give examples of l.c.s. structures (ω, Ω) such that the Lichnerowicz class [Ω] is not trivial, [Ω] ≠ 0. For deformations and equivalence of l.c.s. structures see [B1].

To sum up we see that important l.c.s’s notions can be translated into the Lie algebroid’s language. We have the following table:

\underline{l.c.s.}

Lie algebroid

(M, ω, Ω) ≡ ω is closed, dΩ = −ω ∧ Ω.

A = T M × ℝ

with anchor #_{A} = pr_{1} : T M × ℝ → T M, with bracket [[(X, f) , (Y, g)]] = ([X, Y] , ∇_{X}g − ∇_{Y} f − Ω (X, Y)) where ∇_{X}g = ∂_{X}g + ω (X) · g ∇ is flat and ∇Ω = 0.

globally c.s. ≡ ω is exact

A is invariantly oriented

two l.c.s. structures (ω′, Ω′) and (ω, Ω) on M are conformally equivalent ≡ ω′ = ω + \frac{da}{a}, Ω′ = \frac{1}{a}Ω the corresponding Lie algebroids are isomorphic via H_{0,a} : T M × ℝ → T M × ℝ, H (X, f) = (X, a · f) a ∈ C^{∞} (M) , a (x) ≠ 0 for all x.

2. GENERALIZATIONS: 𝔤-l.c.s. structures and Lie algebroids

We generalize l.c.s. structures to 𝔤-l.c.s. structures in which we can consider an arbitrary finite dimensional Lie algebra 𝔤 instead of the commutative Lie algebra ℝ. From the general theorem on the form of Lie algebroids, mentioned above, we have [M], [K1];

Theorem 2.1. Each transitive Lie algebroid with a trivial adjoint bundle of Lie algebras M × 𝔤 is isomorphic to T M × 𝔤 with #_{A} = pr_{1} : T M × 𝔤 → T M as the anchor and the bracket

[[(X, σ) , (Y, η)]] = ([X, Y] , ∇_{X}η − ∇_{Y} σ + [σ, η] − Ω (X, Y))

in Sec A is defined via the following data (∇, Ω): a covariant derivative ∇ in the trivial vector bundle M × 𝔤 and a 2-form Ω ∈ Ω^{2} (M; 𝔤) fulfilling the conditions:

(1) R_{X,Y}^{∇} σ = − [Ω (X, Y) , σ] , R^{∇} being the curvature tensor of ∇,

(2) ∇_{X} [σ, η] = [∇_{X}σ, η] + [σ, ∇_{X}η] , σ, η ∈ C^{∞} (M; 𝔤),

(3) ∇Ω = 0.

The Lie algebroid obtained in the above way via the data (∇, Ω) fulfilling (1)÷(3) above will be denoted here by

(2.1)  (T M × 𝔤, ∇, Ω) .

The form −Ω is the curvature form of the connection λ : T M → T M × 𝔤, λ (v) = (v, 0) , in this Lie algebroid (T M × 𝔤, ∇, Ω).

0 → M × 𝔤 → T M × 𝔤 ⟶_{⟵_{λ}} T M → 0.

More generally, the curvature form of an arbitrary connection λ (X) = (X, η (X)) , η ∈ Ω^{1} (M; 𝔤) , is given by

(2.2)  Ω^{λ} (X, Y) = (∇η) (X, Y) + [ηX, ηY] − Ω (X, Y) .

We write the covariant derivative ∇ in the trivial bundle M × 𝔤 in the form

∇_{X}σ = ∂_{X}σ + ω (X) (σ)

for a 1-form ω ∈ Ω^{1} (M; End 𝔤) . Then ∇θ = d_{∇}θ = d_{dR}θ + ω ∧ θ. The curvature tensor R^{∇} of ∇ is equal to

R_{X,Y}^{∇} σ = dω (X, Y) (σ) + [ω (X) , ω (Y)] (σ) .

Theorem 3.31 Chapter IV from [M] classifies all transitive Lie algebroids having a given coupling Ξ. For the Lie algebroid (2.1) we have,

Ξ : T M → OutDo[(M × 𝔤)] = T M × Der (𝔤) /ad (𝔤) ,
Ξ (v) = (v, [a_{v}]) ,

where a_{v} (σ) = ∇_{v}σ̃ − v \binom{}{σ̃} , σ̃ : M → 𝔤, σ̃ (x) ≡ σ ∈ 𝔤,

(2.3)  Opext (T M, Ξ, M × 𝔤) ≅ H_{ρ^{Ξ}}^{2} (M, Z𝔤)

where Z𝔤 is the center of 𝔤 and ρ^{Ξ} : T M → T M × End (Z𝔤) is the central representation ρ^{Ξ} (v) = (v, a_{v}) for Ξ.

Proposition 2.1. The conditions (1)-(3) characterizing the data (∇, Ω) determining the Lie algebroid (T M × 𝔤, ∇, Ω) can be expressed as follows • the condition (1) is equivalent to

dω (X, Y) (σ) + [ω (X) , ω (Y)] (σ) = − [Ω (X, Y) , σ] ,

• the condition (2) is equivalent to ω_{x} ∈ Der (𝔤) , i.e. ω_{x} is a differentiation of the Lie algebra 𝔤,

• the condition (3) is equivalent to

dΩ = −ω ∧ Ω

(the values of forms ω and Ω are multiplied with respect to the 2-linear homomorphism End 𝔤 × 𝔤 → 𝔤, (a, σ) ↦ a (σ) .

Definition 2.1. The pair (∇, Ω) determining the above Lie algebroid (T M × 𝔤, ∇, Ω) will be called 𝔤-locally conformal symplectic structure (𝔤-l.c.s. structure, for short) on the manifold provided that the 2-form Ω is non-degenerate in the following sense: for each point x ∈ M the mapping

(2.4)  T_{x}M → L (T_{x}M, 𝔤) , v ↦ Ω_{x} (v, ·) ,

is a monomorphism.

It is easy to see that if the mapping (2.4) is a monomorphism at a point x then it is a monomorphism at every point near x.

We notice that if dim 𝔤 ≥ 2 there is no dimensional obstructions to the existence of an non-degenerate tensors:

Lemma 2.1. For arbitrary vector spaces V and 𝔤 such that dim 𝔤 ≥ 2 there exists a 2-linear skew-symmetric non-degenerate tensor Ω ∈ Ω^{2} (V; 𝔤) .

Proof. Let (e_{1}, ..., e_{n}) be a basis of 𝔤. If dim V is even, then there exists a real 2-linear skew-symmetric non-degenerate tensor, say Ω_{0}. The form Ω := Ω_{0} · e_{1} ∈ Ω^{2} (V; 𝔤) is non-degenerate. If dim V = 2k + 1 and (v_{1}, ..., v_{2k+1}) is a basis of V and u^{1}, ..., u^{2k+1} is a dual basis, then put

Ω_{0} = u^{1} ∧ u^{2} + ... + u^{2k−1} ∧ u^{2k},
Ω_{1} = u^{2k} ∧ u^{2k+1}.

The form Ω := Ω_{0} · e_{1} + Ω_{1} · e_{2} is non-degenerate. □

Definition 2.2. A 𝔤-l.c.s. structure is called globally conformal symplectic structure if the associated Lie algebroid (T M × 𝔤, ∇, Ω) is invariantly oriented.

Theorem 2.2. Let (∇, Ω) be a 𝔤-l.c.s. structure on an arbitrary m-dimensional connected manifold (oriented or not), dim 𝔤 = n.Write ∇_{X}σ = ∂_{X}σ + ω (X) (σ) for ω ∈ Ω^{1} (M; End 𝔤). The following conditions are equivalent:

(a) The Lie algebroid (T M × 𝔤, ∇, Ω) is invariantly oriented (i.e. (∇, Ω) is a globally conformal symplectic structure),

(b) 𝔤 is unimodular and tr ω is an exact form. [Let e_{1}, ..., e_{n} be a basis of 𝔤. For a non-singular function f ∈ C^{∞} (M) the element ε = f · e_{1} ∧ ... ∧ e_{n} is an invariant cross-section if and only if tr ω = d (− ln |f|)],

(c) the modular class of A = (T M × 𝔤, ∇, Ω) is zero, m_{A} = 0,

(d) H_{∂_{A}^{or},c}^{m+n} (A, or (M)) ≠ 0,

(e) H_{∂_{A}^{or},c}^{m+n} (A, or (M)) = ℝ, and the pairing

H^{j} (A) × H_{∂_{A}^{or},c}^{m+n−j} (A, or (M)) → H_{∂_{A}^{or},c}^{m+n} (A, or (M)) ≅ ℝ

is non-degenerate, i.e. H^{j} (A) ≅ (H_{∂_{A}^{or},c}^{m+n−j} (A, or (M)) )^{∗}.

Proof. (a) ⟺ (b) The very easy proof will be omitted.

(a) ⟺ (c) ⟺ (d) ⟺ (e) see [K-M]. □

Theorem 2.3. If the Lie algebra 𝔤 is semisimple, then each 𝔤-l.c.s. structure is globally c.s. structure.

Proof. According to Theorem 7.2.3 from [K1] (see independently (2.3)) for the trivial LAB ggg = M × 𝔤 there exists exactly one, up to isomorphism, a transitive Lie algebroid A with the adjoint LAB ggg = M × 𝔤. Therefore, A must be isomorphic to the trivial Lie algebroid A = T M × 𝔤 with the data (∂, 0) . This Lie algebroid is invariantly oriented: ε (x) ≡ ε_{o} ∈ ⋀^{n} 𝔤 is an invariant cros-section. □

Let (e_{1}, ..., e_{n}) be a basis of 𝔤 with the structure constants c_{ij}^{k}. The covariant derivative ∇ determines a matrix of 1-forms ω_{i}^{j} ∈ Ω^{1} (M) by

∇_{X}e_{i} = ∑_{j} ω_{i}^{j} (X) e_{j}.

Analogously we have a collection of 2-forms Ω^{j} by

Ω_{X,Y} = ∑_{j} Ω_{X,Y}^{j} e_{j}.

We interpret the data (1)÷(3) concerning (∇, Ω) in the terms of the matrix ω_{i}^{j} and the collection Ω^{j} and the structure constants c_{ij}^{k}.

Proposition 2.2. (A) The conditions (1)-(3) characterizing the data (∇, Ω) determining the Lie algebroid (T M × 𝔤, ∇, Ω) can be expressed as follows.

• The condition (1) is equivalent to

− ∑_{j} Ω_{X,Y}^{j} · c_{j,i}^{r} = dω_{i}^{r} (X, Y) − ∑_{j} (ω_{i}^{j} (X) ω_{j}^{r} (Y) − ω_{i}^{j} (Y) ω_{j}^{r} (X) ),

• the condition (2) is equivalent to

∑_{k} c_{ij}^{k} · ω_{k}^{r} (X) = ∑_{k} (ω_{i}^{k} (X) c_{kj}^{r} − ω_{j}^{k} (X) c_{ki}^{r}) ,

• the condition (3) is equivalent to dΩ^{j} = − ∑_{i} Ω^{i} ∧ ω_{i}^{j}. (B) For an abelian Lie algebra 𝔤 = ℝ^{n} (i.e. c_{ij}^{k} = 0) the conditions above are

equivalent to

• dω (X, Y) = −ω (X) ∘ ω (Y) + ω (Y) ∘ ω (X) (equivalently dω_{i}^{r} (X, Y) =

∑_{j} (ω_{i}^{j} (X) ω_{j}^{r} (Y) − ω_{i}^{j} (Y) ω_{j}^{r} (X))),

• dΩ^{j} = − ∑_{i} Ω^{i} ∧ ω_{i}^{j}.

Two 𝔤-l.c.s. structures (∇′, Ω′) , (∇, Ω) on a manifold M will be called 𝔤conformally equivalent if the associated Lie algebroids are isomorphic via an isomorphism of the special form (called 𝔤-conformal) H (X, σ) = (X, a (σ)) for some mapping a : M → Aut (𝔤) . Then the equivalent relations between the data (∇, Ω) and (∇′, Ω′) are as follows:

• Ω′ = a^{−1} ∘ Ω,

• a ∘ ∇′_{X} (σ) = ∇_{X} (a ∘ σ) . We use the notation a ∘ σ for the cross-section defined by (a ∘ σ)_{x} = a_{x} (σ_{x}) . Writing ∇′ and ∇ with using 1-forms ω′, ω ∈ Ω^{1} (M; End 𝔤) (as above) the last

condition can be equivalently written in the form

ω (X) ∘ a = −∂_{X}a + a ∘ ω′ (X) .

In the terms of the matrices ω_{i}^{j′} and ω_{i}^{j} this condition is equivalent to

∑_{j} ω_{i}^{′j} (X) · a_{j}^{k} − ∑_{j} a_{i}^{j} · ω_{j}^{k} (X) = ∂_{X} (a_{i}^{k}) .

The general form of a homomorphism H : T M × 𝔤 → T M × 𝔤 commuting with anchors pr_{1} is as follows

(2.5)  H (X, σ) = H_{η,a} (X, σ) = (X, η (X) + a ∘ σ)

for η ∈ Ω^{1} (M; 𝔤) , a ∈ C^{∞} (M, End 𝔤) . Consider two Lie algebroids

A′ = (T M × 𝔤, ∇′, Ω′) and A = (T M × 𝔤, ∇, Ω)

Proposition 2.3. The following conditions are equivalent.

(1) H is a homomorphism of Lie algebroids H : A′ → A,

(2) (a) a_{x} is a homomorphism of Lie algebras, (b) (∇η) (X, Y) + [η (X) , η (Y)] = (Ω − aΩ′) (X, Y) ,

(c) a ∘ ∇′_{X}σ = ∇_{X} (a ∘ σ) + [η (X) , a ∘ σ] ,

(3) For the basis e_{1}, ..., e_{n} and the matrix a_{i}^{j} defined by a (e_{i}) = ∑_{j} a_{i}^{j} (e_{j}) (a) a_{x} is a homomorphism of Lie algebras,

(b) dη^{k} (X, Y) − (∑_{i} η^{i} ∧ ω_{i}^{k}) (X, Y) + ∑_{i,j} η^{i} (X) · η^{j} (Y) · c_{ij}^{k} = = (Ω^{k} − ∑_{i} Ω^{′i} · a_{i}^{k}) (X, Y) ,

(c) ∑_{j} ω_{i}^{′j} (X) · a_{j}^{k} = ∑_{j} a_{i}^{j} · ω_{j}^{k} (X) + ∂_{X}a_{i}^{k} + ∑_{j,s} η^{j} (X) · a_{i}^{s} · c_{js}^{k}.

The homomorphism H_{η,a} is an isomorphism of Lie algebroids if and only if a_{x} is an isomorphism of Lie algebras.

Proof. Straightforward calculations. □

If (∇′, Ω′) and (∇, Ω) are 𝔤-l.c.s structures and A′ and A are corresponding Lie algebroids, then the isomorphism H_{η,a} given by (2.5) is equivalent to conformal equivalence of the associated 𝔤-l.c.s structures (∇′, Ω′) and (∇, Ω) if and only if η = 0.

Analogously, we can put the problem of existence of l.c.s. structures. We have firstly the simple

Proposition 2.4. Any Lie algebroid A′ = (T M × 𝔤, ∇′, Ω′) is isomorphic to A = (T M × 𝔤, ∇, Ω) with Ω non-degenerate (i.e. (∇, Ω) is a 𝔤-l.c.s. structure) if and only if there exists in A′ a connection for which the curvature tensor is nondegenerate.

Problem 2.1. We introduce into the class of pairs (∇, Ω) fulfilling (1)-(3) from Theorem 2.1, the equivalence relation

r𝔤) (∇′, Ω′) ≈ (∇, Ω) ≡

≡ the Lie algebroids A′ = (T M × 𝔤, ∇′, Ω′) and A = (T M × 𝔤, ∇, Ω) are isomorphic,

i.e. there exist η ∈ Ω^{1} (M; 𝔤) , a ∈ C^{∞} (M, Aut 𝔤) such that (2b) and (2c), from Prop. 2.3 holds: (∇η) (X, Y)+[η (X) , η (Y)] = (Ω − aΩ′) (X, Y) and a ∘ ∇′_{X}σ = ∇_{X} (a ∘ σ) + [η (X) , a ∘ σ] .

We can ask: does there in every (in given) equivalence class [(∇′, Ω′)] exist (∇, Ω) being a 𝔤-l.c.s. structure; equivalently, does there in the Lie algebroid A′ = (T M × 𝔤, ∇′, Ω′) exist a connection with non-degenerate curvature tensor, i.e. equivalently, does there exists a 1-form η ∈ Ω^{1} (M; 𝔤) such that the 2-form (∇η) (X, Y) + [ηX, ηY] − Ω (X, Y) is a non-degenerate.

For 𝔤 = ℝ we obtain Problem 1.1 and we need to assume that dim M is even.

Proposition 2.5. The above problem has a local solution.

Proof. Let a : T_{x_{0}}M × T_{x_{0}}M → 𝔤 be an arbitrary non-degenerate 2-linear skewsymmetric tensor (for dim 𝔤 ≥ 2 see Lemma 2.1). We can locally extend Ω_{x_{0}} + a to a closed 2-form Φ and find by the Poincaré lemma a 1-form η such that dη = Φ; therefore that (dη)_{x_{0}} = Ω_{x_{0}} + a. Slightly modifying η we can assume that η_{x_{0}} = 0, indeed, locally there is a closed 1-form θ such that θ_{x_{0}} = η_{x_{0}}, so η−θ is zero at x_{0} and d (η − θ)_{x_{0}} = (dη)_{x_{0}} . Clearly (∇η)_{x_{0}} (X, Y) + [η_{x_{0}}X, η_{x_{0}}Y] − Ω_{x_{0}} (X, Y) = a (X, Y) so the curvature tensor Ω^{λ} of the connection λ (X) = (X, η (X)) , see (2.2), is a non-degenerate near x_{0}. □

Problem 2.2. It would be interesting to investigate the group of all compactly supported diffeomorphisms of M that preserve the 𝔤-l.c.s. structure up to 𝔤-conformal equivalence (analogously as was given for usual l.c.s. structures by Haller and Rybicki [H-R]).

We add that two extreme cases: (1) 𝔤 commutative (for example 𝔤 = ℝ) and (2) 𝔤 semisimple, are quite different. In the second case all Lie algebroids of the form (T M × 𝔤, ∇, Ω) (i.e. with the trivial adjoint Lie algebra M × 𝔤 ) are isomorphic, clearly to the trivial one T M × 𝔤 with the structure given by the data (∂, 0) . We add that not each isomorphism is 𝔤-conformal. This Lie algebroid is invariantly oriented.

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Roman Kadobianski: National Technical University of Ukraine, ”Kiev Polytechnic Institute”, UKRAINE, prosp. Peremogy 37, Kiev 03056

Jan Kubarski: Institute of mathematics, Technical University of Łódź, PL-90-924 Łódź, Al. Politechniki 11, POLAND