HOMOTOPY AND GEOMETRY

BANACH CENTER PUBLICATIONS, VOLUME 45 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1998

ALGEBROID NATURE OF THE CHARACTERISTIC CLASSES OF FLAT BUNDLES

JAN KUBARSKI Institute of Mathematics, Technical University of Łodz PL-90-924, Łodz, Al.Politechniki 11, POLAND E-mail: kubarski@ck-sg.p.lodz.pl

Abstract. Two following homotopic notions are important in many domains of differential geometry:

— homotopic homomorphisms between principal bundles (and between other objects),

— homotopic subbundles.

They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be — in a natural way — expressed in the language of Lie algebroids. Besides, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism [K4], or the subject of this paper, the characteristic homomorphism for flat bundles) are invariants of Lie algebroids of these bundles. This enables one to construct the characteristic homomorphism of a flat regular Lie algebroid, measuring the incompatibility of the flat structure with a given subalgebroid. For two given Lie subalgebroids, these homomorphisms are equivalent if the Lie subalgebroids are homotopic. Some new examples of applications of this characteristic homomorphism to a transitive case (for TC-foliations) and to a non-transitive case (for a principal bundle equipped with a partial flat connection) are pointed to (Ex. 3.1). An example of a transitive Lie algebroid of a TC-foliation, which leads to the nontrivial characteristic homomorphism is obtained.

1. Preliminaries.

1.1. Characteristic homomorphism of flat bundles. Consider

• a G-principal bundle P,

• a flat connection ω in P,

• an H-reduction P′ ⊂ P, H being a closed Lie subgroup of G.

Keywords: Lie algebroid, characteristic classes, flat bundle, TC-foliation 1991 Mathematics Subject Classification: Primary 57R20; Secondary 53C12, 55R40. The paper is in final form and no version of it will be published elsewhere.

Let 𝔥 and 𝔤 be Lie algebras of H and G, respectively. There is a characteristic homomorphism

Δ_{#P,P′,ω} : H^{⋆} (𝔤, H) ⟶ H_{dR} (M)

measuring the independence of ω and P′, which means to what extent ω is not contained in T P′ (for details see, for example [K-T]). One of the fundamental properties of Δ_{#P,P′,ω} is the independence of the choice of homotopic subbundle P′ :

if P′ ≈^{homotopic} P′′ ⊂ P, then Δ_{#P,P′,ω} = Δ_{#P,P′′,ω}.

This means that the nontriviality of Δ_{#P,P′} implies the impossibility of the homotopic changing of P′ to contain the connection ω.

We recall that the domain H^{⋆} (𝔤, H) , called the relative Lie algebra cohomology [K-T], is the cohomology space of the complex (⋀ (𝔤/𝔥)_{I}^{⋆} , d^{H}) ,

H^{⋆} (𝔤, H) = H^{⋆} (⋀ (𝔤/𝔥)_{I}^{⋆} , d^{H})

where ⋀ (𝔤/𝔥)_{I}^{⋆} is the space of invariant elements (with respect to the adjoint representation), whereas the differential d^{H} is defined in such a way that, for ψ ∈ ⋀^{k} (𝔤/𝔥)_{I}^{⋆} and w_{i} ∈ 𝔤,

⟨d^{H} (ψ) , [w_{i}] ∧ … ∧ [w_{k}]⟩ = ∑_{i<j} (−1)^{i+j} ⟨ψ, [[w_{i}, w_{j}]] ∧ [w_{i}] ∧ … i^{∧} … j^{∧} … ∧ [w_{k}]⟩

([w_{i}, w_{j}] is the bracket in the Lie algebra 𝔤, whereas [w] denotes the equivalence class in 𝔤/𝔥). The homomorphism Δ_{#P,P^{′,ω}} on the level of forms Δ_{⋆} : ⋀ (𝔤/𝔥)^{⋆} → Ω (M) can be defined as follows: Δ_{⋆} (ψ) for ψ ∈ ⋀^{k} (𝔤/𝔥)^{⋆} is the only k-form for which

π^{′⋆} (Δ_{⋆}ψ) = \frac{1}{k!} · i^{′⋆}⟨⋀^{k} s^{⋆} (ψ) , ω ∧ … ∧ ω⟩,

with i′ : P′ ↪ P, s : 𝔤 → 𝔤/𝔥, π′ : P′ → M, ω ∧ … ∧ ω ∈ Ω^{k}(P; ⋀^{k} 𝔤).

It turns out that the characteristic homomorphism of a flat bundle described above, is a notion of its Lie algebroid. A construction of its version on the category of regular Lie algebroids is the aim of this work. A simple example of a transitive Lie algebroid of the TC-foliation of left cosets of a Lie group by a nonclosed Lie subgroup having this homomorphism nontrivial is given (see Chap. 8).

1.2. The category of Lie algebroids. Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example: differential groupoids, principal bundles, vector bundles, actions of Lie groups on manifolds, transversally complete foliations, nonclosed Lie subgroups, Poisson manifolds, some complete closed pseudogroups, etc. The category of Lie algebroids is more elastic than the category of principal bundles. It enables one to generalize the characteristic homomorphisms in the direction of nontransitive objects.

1.2.1. Definitions and notations. By a Lie algebroid on a manifold M [P1], [P2], we mean a system A = (A, [[·, ·]], γ) consisting of a vector bundle A on M and mappings [[·, ·]] : Sec A×Sec A → Sec A, γ : A → T M, such that (1) (Sec A, [[·, ·]]) is an ℝ-Lie algebra,

(2) γ, called the anchor, is a homomorphism of vector bundles, (3) Sec γ : Sec A → 𝔛 (M) , ξ ↦ γ ∘ ξ, is a homomorphism of Lie algebras, (4) [[ξ, f · η]] = f · [[ξ, η]] + (γ ∘ ξ) (f) · η, f ∈ C^{∞} (M) . A Lie algebroid A is said to be transitive if γ is an epimorphism of vector bundles, and regular if γ is of constant rank. In the last situation, if Im γ = F ⊂ T M, then F is an involutive distribution and A is called a regular Lie algebroid over the foliated manifold (M, F) . In the sequel, we adopt the notions and the notations from [P1], [M1], [K4], among them, the adjoint bundle of Lie algebras ggg := Ker γ, the Atiyah sequence 0 → ggg → A → F → 0 and the notion of a connection.

A principal bundle P and a TC-foliation (M, E) determine transitive Lie algebroids A (P) [M1], [K3], [K8] and A (M, E) [MO1], respectively. A geometrically important object consisting of a given principal bundle P on M and a foliation F on the base M of P is a source of a nontransitive Lie algebroid A (P)^{F} on a foliated manifold (M, F) for which A (P)^{F} = γ^{−1} [F] , γ : A (P) → T M being the anchor of A (P) . Poisson manifolds are sources of nontransitive Lie algebroids [C-D-W].

Theorem 1.1. [K2, Th.1.1] If A′ ⊂ A (P) is a transitive Lie subalgebroid of the Lie algebroid A (P) of a principal bundle P = (P, π, M, G, ·) , then there exists a reduction P′ of P having A′ as its Lie algebroid. □

1.2.2. The Lie algebroid of a vector bundle, representations and invariant crosssections. With each vector bundle 𝔣 we associate a transitive Lie algebroid A (𝔣) being the Lie algebroid of the principal bundle L (𝔣) of all repers of 𝔣 [M1] or of the Lie groupoid GL (𝔣) of all linear isomorphisms between fibres, see for example, [N-V-Q], [KU], [M1].

Equivalently we can construct this Lie algebroid in such a way that the fibre A (𝔣)_{|x} over x ∈ M is equal to the space of all linear homomorphisms

l : Sec 𝔣 ⟶ 𝔣_{|x}

for which there exists a vector u ∈ T_{x}M such that l (f · ν) = f (x) · l (ν) + u (f) · ν (x) , f ∈ C^{∞} (M) , ν ∈ Sec 𝔣 (see [K4], [B]). The space of global cross-sections Sec A (𝔣) is equal to the space of all covariant differential operators in 𝔣 [M1].

By a representation of A on 𝔣 (both over a manifold M) we mean a strong homomorphism of regular Lie algebroids T : A → A (𝔣) [M1]. For a cross-section ξ ∈ Sec A, its image T ∘ ξ is a cross-section of A (𝔣) , therefore it determines a covariant differential operator 𝔏_{T ∘ξ}: Sec 𝔣 → Sec 𝔣. A cross-section ν ∈ Sec 𝔣 is called T -invariant if 𝔏_{T ∘ξ} (ν) = 0 for all ξ ∈ Sec A. The space of all T -invariant cross-sections of 𝔣 is denoted by (Sec 𝔣)_{I^{0}(T)} or, briefly, by (Sec 𝔣)_{I^{0}} .

Example 1.2. (1) The adjoint representation ad_{A} : A → A (ggg) of A on its adjoint bundle of Lie algebras ggg is defined in such a way that for ξ ∈ Sec A,

𝔏_{ad_{A} ∘ξ} (ν) = [[ξ, ν]], ν ∈ Secggg.

(2) A given representation T : A → A (𝔣) yields the representations on associated vector bundles 𝔣^{⋆}, ⋀^{k} 𝔣, ⋁^{k} 𝔣, etc [K4], denoted also by T.

In the context of transitive Lie algebroids, we have two fundamental facts:

• Two cross-sections of 𝔣 invariant with respect to a representation T of A on 𝔣, equal at one point, must be equal to each other globally [M1], [K9]. (Remark: therefore, for a regular Lie algebroid over a foliated manifold (M, F) these cross-sections equal to each other on the whole leaf of the foliation F, passing through this point.)

• Theorem Th. IV.1.19 [M1, p.195] gives the homotopic description of the space (Sec 𝔣)_{I^{o}(M)} of all invariant cross-sections with respect to a representation of a transitive Lie algebroid.

1.2.3. The Chern-Weil homomorphism. In [K4] the Chern-Weil homomorphism d_{A} : I_{A} → H_{F} (M) of a regular Lie algebroid A over (M, F) there is constructed, whose domain I_{A} is an algebra of ad_{A}-invariant cross-sections of ⋁^{k} ggg^{⋆}, k ∈ ℕ. For an integrable Lie algebroid A = A (P) , P being a connected principal bundle (the structured Lie group G may be disconnected !), this algebra is isomorphic to the space of invariant polynomials (⋁ 𝔤^{⋆})_{I}. We pay attention to the fact that this holds although in the Lie algebroid A (P) there is no direct information about the structure Lie group of P which may be disconnected and that a Lie algebroid is – in some sense – a simpler structure than a principal bundle (nonisomorphic principal bundles can possess isomorphic Lie algebroids, see [K3]). The case A = A (P)^{F} is considered in [K6]. Here, I_{A} contains elements of the form ∑ f^{i} · Γ_{i}, f^{i} being F -basic functions and Γ_{i} ∈ I_{A(P)}, i.e. contains the subalgebra F_{b} (M, F) · I_{A(P)}. In [K6] there is an example of P and F for which in the domain I_{A(P)^{F}} there are elements not belonging to F_{b} (M, F) · I_{A(P)}, called singular. For the principal bundle P of repers of a vector bundle 𝔣 and any foliation F on the base M, the restriction h_{A(P)^{F}} to F_{b} (M, F)·I_{A(P)} agrees with the construction of Moore-Schochet [M-S] of the characteristic homomorphism of a vector bundle 𝔣 over a foliated manifold. Therefore h_{A(P)^{F}} can be subtler than the one constructed in [M-S].

For the Lie algebroid A (G; H) over G/H̄ of the TC-foliation of left cosets of a connected Lie group G by a nonclosed Lie subgroup H [K4], the domain I_{A(G;H)} of its Chern-Weil homomorphism is isomorphic to ⋁ (𝔥̄/𝔥)^{⋆} , 𝔥, 𝔥̄ being the Lie algebras of H and its closure H̄. For any connected, compact and semisimple Lie group G, we have that h_{A(G;H)}^{2} : (𝔥̄/𝔥)^{⋆} → H^{2} (G/H̄) is a monomorphism; assuming (in addition) the simple connectedness of G, we obtain – according to the Almeida-Molino theorem [A-M] – a nonintegrable transitive Lie algebroid having the nontrivial Chern-Weil homomorphism.

1.2.4. Invariant cross-sections over ℝ × M. In [K4, Defs.1.1.5 and 2.3.1] there are notions of the inverse-image of a regular Lie algebroid f^{∧}A and the inverse-image of a representation f^{⋆}T via a morphism of foliated manifolds f : (M′, E′) → (M, E) , where A is any regular Lie algebroid over (M, E).

According to [K4, Th.2.4.4], the linear mapping f^{⋆} : Sec 𝔣 → Sec f^{⋆}𝔣, ν ↦ ν ∘ f, can be restricted to the space of cross-sections invariant under T and f^{⋆}T, respectively:

f_{I^{0}}^{⋆} : (Sec 𝔣)_{I^{0}(T)} ⟶ (Sec f^{⋆}𝔣)_{I^{0}(f^{⋆}T)} .

The following theorem plays a crucial role in all problems of the homotopic independence of the characteristic homomorphisms considered on the category of Lie algebroids.

Theorem 1.3. [K9](on invariant cross-sections over ℝ × M) Let B be a regular Lie algebroid over the foliated manifold (ℝ × M, T ℝ × E) and 𝔣 a vector bundle over ℝ × M, and T : A → A (𝔣) a given representation. Take t ∈ ℝ and the mapping f_{t} : M → ℝ × M, x ↦ (t, x). Then the restriction mapping f_{t}^{⋆} : Sec 𝔣 → Sec 𝔣_{|{t}×M} (𝔣_{|{t}×M} ≅ f_{t}^{⋆}𝔣) maps isomorphically the space of invariant cross-sections with respect to T onto the space of invariant cross-sections with respect to the restricted representation f_{t}^{⋆}T :

(f_{t}^{⋆})_{I^{0}} : (Sec 𝔣)_{I^{0}(T)} ⟶^{≅} (Sec 𝔣_{|{t}×M})_{I^{0}(f_{t}^{⋆}T)} . □

1.2.5. Cohomology with coefficients. Let A and 𝔣 be a Lie algebroid and a vector bundle, both over the same manifold, say M. Each element of

Ω_{A} (M; 𝔣) = ⨁^{q≥0} Ω_{A}^{q} (M; 𝔣) ,

where Ω_{A}^{q} (M; 𝔣) = Sec (⋀^{q} A^{⋆} ⨂ 𝔣) , will be called an A-differential form of degree q, with values in 𝔣; while, for the trivial vector bundle 𝔣 = M × ℝ, briefly: an A-differential form of degree q (or a real form). In the case A = T M (the usual differential forms on M), the space of A-differential forms with values in 𝔣 (analogously, of the space of real forms) will be denoted by Ω (M; 𝔣) (Ω (M), respectively). For an involutive C^{∞} constant dimensional distribution E on M, Ω_{E} (M; 𝔣) consists of the so-called tangential differential forms on the foliated manifold (M, E) [M-S], [K4].

Ω_{A} (M; 𝔣) is a graded module over Ω^{0} (M) = C^{∞} (M) and a module over the algebra Ω_{A} (M) of real forms.

Let 𝔣^{1}, …, 𝔣^{k}, 𝔣 be vector bundles over M. An arbitrary k-linear homomorphism of vector bundles φ : 𝔣^{1} × … × 𝔣^{k} → 𝔣 determines the mapping

φ_{⋆} : Ω_{A} (M; 𝔣^{1}) × … × Ω_{A} (M; 𝔣^{k}) ⟶ Ω_{A} (M; 𝔣)

defined by the standard formula

φ_{⋆} (Ψ_{1}, …, Ψ_{k}) (ξ_{1}, …, ξ_{m})  (1.1)
= \frac{1}{q_{1}!·...·q_{k}!} · ∑_{σ} sgn σ · φ(Ψ_{1}(ξ_{σ(1)}, …), …, Ψ_{k}(…, ξ_{σ(m)}))

in which m = ∑ q_{i}, q_{i} = the degree of Ψ_{i}.

For a given representation T : A → A (𝔣) of a Lie algebroid A on a vector bundle 𝔣, we have defined three operators ([MR], [K1], [M1])

ι_{ξ}, Θ_{ξ}^{T} , d^{T} : Ω_{A} (M; 𝔣) ⟶ Ω_{A} (M; 𝔣) , ξ ∈ Sec A,

called the substitution operator, the Lie derivative (with respect to ξ ), and the exterior derivative.

For real forms — considering the trivial representation T : A → A (M × ℝ) = T M × End (ℝ) defined by 𝔏_{T ∘ξ} (𝔣) = (γ ∘ ξ) (𝔣) for f ∈ Ω^{0} (M) and ξ ∈ Sec A or, equivalently, by T (v) = (γ (v) , 0)) — the operator d^{T} will be denoted by d^{A}. In particular, if A = E ⊂ T M is an involutive constant rank distribution on M, we obtain the standard operator of the exterior derivative d^{E} of tangential differential forms [M-S].

For arbitrary taken vector bundles 𝔣^{1}, …, 𝔣^{k}, 𝔣 over M and a k-linear homomorphism φ : 𝔣^{1} × … × 𝔣^{k} → 𝔣 and forms Ψ_{j} ∈ Ω_{A}^{q_{j}} (M; 𝔣^{j}) , we have

ι_{ξ} (φ_{⋆} (Ψ_{1}, …, Ψ_{k})) = ∑_{j=1}^{k} (−1)^{q_{1}+···+q_{j−1}} φ_{⋆} (Ψ_{1}, …, ι_{ξ}Ψ_{j}, …, Ψ_{k}) .  (1.2)

Let now T^{1}, …, T^{k}, T denote fixed representations of A on 𝔣^{1}, …, 𝔣^{k}, 𝔣, respectively, and assume that φ is Hom-invariant where Hom denotes the induced representation on the space of k-linear homomorphisms Hom^{k} (T^{1}, …, T^{k}; T ) defined by the formula

𝔏_{Hom ∘ξ} (φ) (ν^{1}, …, ν^{k}) = 𝔏_{T ∘ξ} (φ (ν^{1}, …, ν^{k})) − ∑_{i} φ (ν^{1}, …, 𝔏_{T^{i}∘ξ}ν^{i}, …, ν^{k}).

Then

d^{T} (φ_{⋆} (Ψ_{1}, …, Ψ_{k})) = ∑_{j=1}^{k} (−1)^{q_{1}+···+q_{j−1}} φ_{⋆}(Ψ_{1}, …, d^{T^{j}}Ψ_{j}, …, Ψ_{k}). (1.3)

A form Ψ ∈ Ω_{A} (M; 𝔣) is called a horizontal form if ι_{ν}Ψ = 0 for all ν ∈ Secggg. The space of horizontal forms is denoted by Ω_{A,i} (M; 𝔣) . According to (1.2) Ω_{A,i} (M; 𝔣) is a module over the algebra of real horizontal forms Ω_{A,i} (M) . Ω_{A,i} (M) is stable under d^{A}.

2. Homomorphisms ω^{∧} and Ω^{∨}. Let A = (A, [[·, ·]], γ) be an arbitrary regular Lie algebroid over a foliated manifold (M, E) , and λ : E → A any connection in A, i.e. any splitting of its Atiyah sequence [K4]:

[figure: 0; ggg; A; E; γ; 0.; λ]

The linear homomorphism of graded vector spaces γ_{⋆} : Ω_{E} (M; 𝔣) → Ω_{A} (M; 𝔣) defined by the formula γ_{⋆} (θ) (x; …, v_{i}, …) = θ (x; …, γv_{i}, …) , v_{i} ∈ A_{|x}, maps isomorphically Ω_{E} (M; 𝔣) onto the space of horizontal forms Ω_{A,i} (M; 𝔣) . The inverse mapping is λ_{⋆} : Ω_{A,i} (M; 𝔣) → Ω_{E} (M; 𝔣) defined by λ_{⋆} (Ψ) (x; …, w_{i}, …) = Ψ (x; …, λw_{i}, …) , w_{i} ∈ E_{|x}. For the trivial vector bundle f = M × ℝ, one can easily obtain the equality d^{E} = λ_{⋆} ∘ d^{A} ∘ γ_{⋆} which is equivalent to

γ_{⋆} ∘ d^{E} = d^{A} ∘ γ_{⋆}.  (2.1)

Let ω : A → ggg be the connection form of λ. The mapping H = id_{A} −ω : A → A is to be the horizontal projection of vectors from A. It determines the horizontal projection of forms H_{⋆} : Ω_{A} (M; 𝔣) → Ω_{A} (M; 𝔣) by H_{⋆} (Ψ) (x; …, v_{i}, …) = Ψ (x; …, Hv_{i}, …) . In [K4] there is defined the curvature tensor Ω_{b} ∈ Ω_{E}^{2} (M;ggg) of λ by Ω_{b} (X_{1}, X_{2}) = −ω ([[λ ∘ X_{1}, λ ∘ X_{2}]]) , X_{i} ∈ Sec E. We define the so-called curvature form of λ – needed in the sequel – as a horizontal 2-form Ω on the Lie algebroid A, with values in ggg, by the formula

Ω (ξ_{1}, ξ_{2}) = −ω ([[H ∘ ξ_{1}, H ∘ ξ_{2}]]) , ξ_{i} ∈ Sec A.

Below, the exterior derivative of forms on the Lie algebroid A, with values in ggg, [also in the associated vector bundles] with respect to the adjoint representation ad_{A} : A → A (ggg) [or induced ones] will be shortly defined by d^{g g g}.

Proposition 2.1. (The Maurer-Cartan equation)

Ω = d^{g g g}ω − \frac{1}{2} [ω, ω] .

(The form [ω, ω] is defined via formula (1.1) for the 2-linear homomorphism [·, ·] : ggg × ggg → ggg where [·, ·]_{|x} : ggg_{|x} × ggg_{|x} → ggg_{|x} is the Lie algebra structure of the isotropy Lie algebra ggg_{|x}.)

(Remark: The difference here, in comparison with the classical formula for principal bundles — the sign ”−” before the second component — has its roots in the fact that the Lie algebra of the structure Lie group in the principal bundle considered there is taken left, not right.)

P r o o f. Clearly, it is sufficient to notice that ι_{ν}Ω = 0 = ι_{ν} (d^{g g g}ω − \frac{1}{2} [ω, ω]) , ν ∈ Secggg, and H_{⋆} (Ω) = H_{⋆} (d^{g g g}ω − \frac{1}{2} [ω, ω]) .

2.1. Homomorphism ω^{∧}. For each point x ∈ M, the mapping

ρ : ggg_{|x}^{⋆} ⟶ A_{|x}^{⋆} = ⋀^{1} A_{|x}^{⋆} ⊂ ⋀ A_{|x}^{⋆}, w^{⋆} ⟼ w^{⋆} ∘ ω_{|x},

is linear and keeps the property ρ (w^{⋆}) ∧ ρ (w^{⋆}) = 0 for w^{⋆} ∈ ggg_{|x}^{⋆}. ⋀ A_{|x}^{⋆} is an associative algebra with unit element, therefore, by the universal property of the exterior algebra ⋀ A_{|x}^{⋆}, see [G, p.103], we obtain the existence and the uniqueness of a homomorphism of algebras of degree 0,

ω_{x}^{∧} : ⋀ggg_{|x}^{⋆} ⟶ ⋀ A_{|x}^{⋆},

extending ρ and such that ω_{x}^{∧} (1) = 1. Using the canonical duality between the exterior algebra over a vector space and over its dual [G, p.104] we have that

⟨ω_{x}^{∧} (ψ) , w_{1} ∧ … ∧ w_{k}⟩ = ⟨ψ, ω (x; w_{1}) ∧ … ∧ ω (x; w_{k})⟩

for ψ ∈ ⋀^{k} ggg_{|x}^{⋆} and w_{i} ∈ A_{|x}. We notice that if Ψ ∈ Sec ⋀^{k} ggg^{⋆}, then

ω^{∧} (Ψ) : M ⟶ ⋀^{k} A^{⋆}, x ⟼ ω_{x}^{∧} (Ψ (x)) ,

is a C^{∞} cross-section of ⋀^{k} A^{⋆}, i.e. ω^{∧} (Ψ) ∈ Ω_{A}^{k} (M) . Of course

ω^{∧} : ⨁^{k≥0} Sec ⋀^{k}ggg^{⋆} ⟶ Ω_{A} (M) , Ψ ⟼ ω^{∧} (Ψ) ,

is a homomorphism of algebras where the space ⨁^{k≥0} Sec ⋀^{k} ggg^{⋆} is equipped with the structure (Ψ_{1}, Ψ_{2}) ↦ Ψ_{1} ∧ Ψ_{2} for which Ψ_{1} ∧ Ψ_{2} is defined point by point. Define a C^{∞} 2-linear homomorphism of vector bundles ⟨·, ·⟩ : ⋀^{k} ggg^{⋆} × ⋀^{k} ggg → ℝ via the family of the canonical dualities ⟨·, ·⟩_{x} : ⋀^{k} ggg_{|x}^{⋆} × ⋀^{k} ggg_{|x} → ℝ. Looking at formula (1.1) and treating

Ψ ∈ Sec ⋀^{k} ggg^{⋆} as a 0-form on A, with values in ⋀^{k} ggg^{⋆}, we can easily assert

ω^{∧} (Ψ) = \frac{1}{k!}⟨Ψ, \underbrace{ω ∧ … ∧ ω}_{k times}⟩,  (2.2)

where ω ∧ … ∧ ω is defined by formula (1.1) for the k-linear homomorphism

∧ : ggg × … × ggg ⟶ ⋀^{k}ggg,

whereas ⟨Ψ, ω ∧ … ∧ ω⟩ — for the duality ⟨·, ·⟩. In view of (1.2) and of (2.2) above, we check

ι_{ν} (ω^{∧} (Ψ)) = ω^{∧} (ι_{ν} (Ψ)) , if ν ∈ Secggg.  (2.3)

2.2. Homomorphism Ω^{∨}. Let Ω ∈ Ω_{A}^{2} (M;ggg) be the curvature form of the connection λ under consideration. For each point x ∈ M, the mapping

μ : ggg_{|x}^{⋆} ⟶ ⋀^{2} A_{|x}^{⋆} ⊂ ⋀^{ev} A_{|x}^{⋆}, w^{⋆} ⟼ w^{⋆} ∘ Ω_{|x},

is linear and keeps the property μ (u^{⋆})∧μ (w^{⋆}) = μ (w^{⋆})∧μ (u^{⋆}) for u^{⋆}, w^{⋆} ∈ ggg_{|x}^{⋆}. ⋀^{ev} A_{|x}^{⋆} is an associative algebra with unit element, therefore, by the universal symmetric algebra property of ⋁ggg_{|x}^{⋆} [G, p.192], there exists a unique homomorphism of algebras of degree 0

Ω_{x}^{∨} : ⋁ggg_{|x}^{⋆} ⟶ ⋀^{ev} A_{|x}^{⋆}

extending μ and such that Ω_{x}^{∨} (1) = 1. Via the canonical dualities [G, pp.104, 193], the homomorphism Ω_{x}^{∨} is defined by the formula

⟨Ω_{x}^{∨} (Γ_{x}) , w_{1} ∧ … ∧ w_{2k}⟩  (2.4)
= \frac{1}{k!} · \frac{1}{2^{k}} · ∑_{σ} sgn σ · ⟨Γ, Ω (x; w_{σ(1)} ∧ w_{σ(2)}) ∨ … ∨ Ω (x; w_{σ(2k−1)} ∧ w_{σ(2k)})⟩

for Γ_{x} ∈ ⋁^{k} ggg_{|x}^{⋆} and w_{i} ∈ A_{|x}. Indeed, in view of the linearity with respect to Γ_{x} of both sides of the above equality, it is sufficient to check it on the simple tensors Γ_{x} = w_{1}^{⋆} ∨ … ∨ w_{k}^{⋆}, w_{i}^{⋆} ∈ ggg_{|x}^{⋆}.

⟨Ω_{x}^{∨} (w_{1}^{⋆} ∨ … ∨ w_{k}^{⋆}) , w_{1} ∧ … ∧ w_{2k}⟩
= ⟨Ω_{x}^{∨} (w_{1}^{⋆}) ∧ … ∧ Ω_{x}^{∨} (w_{k}^{⋆}) , w_{1} ∧ … ∧ w_{2k}⟩
= \frac{1}{2^{k}} · ∑_{σ} sgn σ · w_{1}^{⋆} (Ω (x; w_{σ(1)} ∧ w_{σ(2)})) · . . . · w_{k}^{⋆} (Ω (x; w_{σ(2k−1)} ∧ w_{σ(2k)}))
= \frac{1}{k!} · \frac{1}{2^{k}} · ∑_{σ} sgn σ · ∑_{τ} w_{1}^{⋆} (Ω (x; w_{σ(2·τ(1)−1)} ∧ w_{σ(2·τ(1))})) · . . .
. . . · w_{k}^{⋆} (Ω (x; w_{σ(2·τ(k)−1)} ∧ w_{σ(2·τ(k))}))
= \frac{1}{k!} · \frac{1}{2^{k}} · ∑_{σ} sgn σ · perm [⟨w_{i}^{⋆}, Ω (x; w_{σ(2j−1)} ∧ w_{σ(2j)})⟩; i, j ≤ k]
= \frac{1}{k!} · \frac{1}{2^{k}} · ∑_{σ} sgn σ · ⟨w_{1}^{⋆} ∨ … ∨ w_{k}^{⋆}, Ω (x; w_{σ(1)} ∧ w_{σ(2)}) ∨ …
… ∨ Ω (x; w_{σ(2k−1)} ∧ w_{σ(2k)})⟩.

Applying (2.4), we see that, for Γ ∈ Sec ⋁^{k} ggg^{⋆}, the cross-section Ω^{∨} (Γ) of ⋀^{2k} A^{⋆} defined by x ↦ Ω_{x}^{∨} (Γ (x)) is C^{∞}, i.e. Ω^{∨} (Γ) ∈ Ω_{A}^{2k} (M) . The space ⨁^{k≥0} Sec ⋁^{k} ggg^{⋆} forms an algebra in a standard way, and the mapping

Ω^{∨} : ⨁^{k≥0} Sec ⋁^{k}ggg^{⋆} ⟶ Ω_{A}^{ev} (M) , Γ ⟼ Ω^{∨} (Γ) ,

is a homomorphism of algebras. By simple calculations, we obtain

Ω^{∨} (Γ) = \frac{1}{k!} · ⟨Γ, \underbrace{Ω ∨ … ∨ Ω}_{k times}⟩ for Γ ∈ Sec ⋁^{k} ggg^{⋆}

(the forms Ω ∨ … ∨ Ω and ⟨Γ, Ω ∨ … ∨ Ω⟩ are defined by (1.1) for suitable multilinear homomorphisms).

It is well known that, in the vector space ⋀ggg_{|x}^{⋆}, the classical Chevalley-Eilenberg differential works, see, for example, [G-H-V, Vol.III, p.107]. For our purpose, we must slightly modify it by multiplying it by −1 (cf. Remark next to Prop.2.1), i.e. we adopt the following differential:

δ_{x} : ⋀ggg_{|x}^{⋆} ⟶ ⋀ggg_{|x}^{⋆}  (2.5)
⟨δ_{x} (ψ) , w_{0} ∧ … ∧ w_{k}⟩ = − ∑_{i<j} (−1)^{i+j} ⟨ψ, [w_{i}, w_{j}] ∧ … i^{∧} … j^{∧} … ∧ w_{k}⟩

for ψ ∈ ⋀^{k} ggg_{|x}^{⋆} (k ≥ 1), w_{i} ∈ ggg_{|x}, and δ_{x}ψ = 0 for ψ ∈ ⋀^{0} ggg_{|x}^{⋆}. δ_{x} is an antiderivation of degree +1 and, for an arbitrary k ≥ 0, the induced homomorphism of vector bundles

δ^{k} : ⋀^{k}ggg^{⋆} ⟶ ⋀^{k+1}ggg^{⋆}

is, obviously, of the C^{∞} class.

Applying the Maurer-Cartan equation, we get for w^{⋆} ∈ Secggg^{⋆}

Ω^{∨} (w^{⋆}) = ⟨w^{⋆}, d^{g g g}ω⟩ − ω^{∧} (δ (w^{⋆})) .  (2.6)

Define the mapping

K : ⨁^{k≥0} Sec ⋀^{k}ggg^{⋆} ⟶ Ω_{A} (M)

by the formula

K (Ψ) = \frac{1}{k!} · ⟨Ψ, d^{g g g}(\underbrace{ω ∧ … ∧ ω}_{k times})⟩ − ω^{∧} (δΨ) for Ψ ∈ Sec ⋀^{k} ggg^{⋆}.  (2.7)

Of course, by (2.6),

K (w^{⋆}) = Ω^{∨} (w^{⋆}) if w^{⋆} ∈ Secggg^{⋆}.  (2.8)

Proposition 2.2. The fundamental formulae for K :

(1) K (w_{1}^{⋆} ∧ … ∧ w_{k}^{⋆}) = ∑_{s=1}^{k} (−1)^{s−1} K (w_{s}^{⋆}) ∧ ω^{∧}(w_{1}^{⋆} ∧ … s^{∧} … ∧ w_{k}^{⋆})

for w_{s}^{⋆} ∈ Secggg^{⋆}.

(2) K (Ψ) = d^{A} (ω^{∧} (Ψ)) − ω^{∧} (δΨ) − \frac{1}{k!} · ⟨d^{g g g}Ψ, \underbrace{ω ∧ … ∧ ω}_{k times}⟩ for Ψ ∈ Sec ⋀^{k} ggg^{⋆}.

P r o o f. (1): Applying (1.3), we get

K (w_{1}^{⋆} ∧ … ∧ w_{k}^{⋆})

= \frac{1}{(k−1)!} · ⟨w_{1}^{⋆} ∧ … ∧ w_{k}^{⋆}, (d^{g g g}ω) ∧ ω ∧ … ∧ ω⟩

− ∑_{s=1}^{k} (−1)^{s−1} ω^{∧} (δw_{s}^{⋆}) ∧ ω^{∧} (w_{1}^{⋆} ∧ … s^{∧} … ∧ w_{k}^{⋆}) .

On the other hand,

∑_{s=1}^{k} (−1)^{s−1} K (w_{s}^{⋆}) ∧ ω^{∧} (w_{1}^{⋆} ∧ … s^{∧} … ∧ w_{k}^{⋆})

= ∑_{s=1}^{k} (−1)^{s−1} (⟨w_{s}^{⋆}, d^{g g g}ω⟩ − ω^{∧} (δw_{s}^{⋆})) ∧ ω^{∧} (w_{1}^{⋆} ∧ … s^{∧} … ∧ w_{k}^{⋆}) ,

therefore, it is sufficient to check the equality

\frac{1}{(k−1)!} · ⟨w_{1}^{⋆} ∧ … ∧ w_{k}^{⋆}, (d^{g g g}ω) ∧ ω ∧ … ∧ ω⟩
= ∑_{s=1}^{k} (−1)^{s−1} ⟨w_{s}^{⋆}, d^{g g g}ω⟩ ∧ ω^{∧} (w_{1}^{⋆} ∧ … s^{∧} … ∧ w_{k}^{⋆}) .

(2): By (1.3) (treating Ψ as a 0-form on the Lie algebroid A, with values in ⋀^{k} ggg^{⋆}), we have d^{A}⟨Ψ, ω ∧…∧ω⟩ = ⟨d^{g g g}Ψ, ω ∧…∧ω⟩+⟨Ψ, d^{g g g} (ω ∧ … ∧ ω)⟩. Therefore, by (2.2), K (Ψ) = d^{A} (ω^{∧} (Ψ)) − ω^{∧} (δΨ) − \frac{1}{k!} · ⟨d^{g g g}Ψ, ω ∧ … ∧ ω⟩.

Because of the fact that each cross-section Ψ ∈ Sec ⋀^{k} ggg^{⋆} is locally a sum of crosssections of the form w_{1}^{⋆} ∧ … ∧ w_{k}^{⋆} for w_{s}^{⋆} ∈ Secggg^{⋆}, we get

Corollary 2.3. If the connection λ considered is flat (i.e. Ω = 0), then, according to (2.8) and Prop.2.2 (1), we see that K ≡ 0, which means, by (2.7) and Prop.2.2(2), that

ω^{∧} (δΨ) = \frac{1}{k!} · ⟨Ψ, d^{g g g} (ω ∧ … ∧ ω)⟩ = d^{A} (ω^{∧} (Ψ)) − \frac{1}{k!} · ⟨d^{g g g}Ψ, ω ∧ … ∧ ω⟩.

R e m a r k 1. Keep the assumption Ω = 0. If Ψ ∈ Sec ⋀^{k} ggg^{⋆} is invariant with respect to the representation ad_{A} of A on ggg (equivalently, if d^{g g g}Ψ = 0 ), then d^{A} (ω^{∧}Ψ) = 0. Indeed, by Cor.2.3, we have d^{A} (ω^{∧}Ψ) = ω^{∧} (δΨ); but, for each point x ∈ M, the tensor Ψ (x) ∈ ⋀^{k} ggg_{|x}^{⋆} is invariant under the canonical representation of the Lie algebra ggg_{|x} on

⋀^{k} ggg_{|x}^{⋆} and such a tensor is a cycle [G-H-V, Vol.III, p.186], so (δΨ) (x) = δ_{x} (Ψ_{x}) = 0. Therefore, there exists a homomorphism of algebras

ω^{#} : ⨁^{k≥0} (Sec ⋀^{k}ggg^{⋆})_{I^{0}} ⟶ Z_{A} (M) ⟶ H_{A} (M) , Ψ ⟼ [ω^{∧} (Ψ)] .

3. A construction of the characteristic classes of flat regular Lie algebroids. Here we construct characteristic classes having the following property:

— the existence of nontrivial classes among them is a measure of the incompatibility of the flat structure of a given regular Lie algebroid A (over (M, E)) with a given subalgebroid B of A (also over (M, E)).

In the case of an integrable transitive Lie algebroid A = A (P) , P being any principal bundle, these classes agree with the so-called characteristic classes of the flat principal bundle P [K-T].

Consider in a given regular Lie algebroid (A, [[·, ·]], γ_{A}) over (M, E) two geometric structures:

(1) a flat connection λ : E → A,

(2) a subalgebroid B ⊂ A over (M, E) , see the following diagram

[figure: 0; ggg; A; E; γ_{A}; 0; ω; λ; 0; h h h; B; E; γ_{B}; 0; j]

Notice that h h h = ggg∩B (h h h := ker γ_{B}).

The system (A, λ, B) will then be called an FS-regular Lie algebroid (over (M, E)).

Example 3.1. (1) A triad (P, P′, ω) consisting of a principal bundle P, of an Hreduction P′ and a flat connection in P with a connection form ω determines an FStransitive Lie algebroid (A (P) , λ, A (P′)) (λ corresponds to ω). For the theory of flat principal bundles with given reductions, see [K-T].

(2) We recall that both a transitive Lie algebroid A = (A, [[·, ·]], γ_{A}) on M and an involutive distribution F ⊂ T M give rise to the regular Lie algebroid over (M, F) of the form A^{F} = γ_{A}^{−1} [F] ⊂ A, see [K4, s.1.1.3]. Consider now a triple (A, B, λ) consisting of a transitive Lie algebroid A on M, a transitive Lie subalgebroid B of A and a partially flat connection λ in A, namely over a given involutive distribution F ⊂ T M. The triple

(A^{F} , B^{F} , λ|F )

is an FS-regular Lie algebroid over (M, F).

(3) Let now the system (P, P′, ω) be given as in example (1) above with the difference that ω is assumed to be partially flat, say, over an involutive distribution F ⊂ T M. Such a system (named a foliated bundle) is investigated, for example, in [K-T]. It determines the (nontransitive) FS-regular Lie algebroid (A (P)^{F} , A (P′)^{F} , λ|F ) written above.

(4) Let (M, F) be any transversally complete foliation with the basic fibration π_{b} : M → W. Denote by A (M, F) its transitive Lie algebroid on W defined by P.Molino in 1977 [MO1]. Let F_{b} ⊂ T M be the vector bundle tangent to the basic foliation. Denote, as usual, by L (M, F) the space of foliate vector fields and by α : T M → Q = T M/F, β : Q → A (M, F) the canonical linear homomorphisms (see [K4]). It is easy to see that:

• There is a 1-1 correspondence between transitive Lie subalgebroids

B ⊂ A (M, F)

and involutive vector subbundles B̃ ⊂ T M such that

(1) F ⊂ B̃,

(2) F_{b} + B̃ = T M,

(3) the Lie algebra Sec(B̃) ∩ L (M, F) generates at each point x ∈ M, the entire space B̃_{|x}.

The correspondence B ↦ B̃ is established in such a way that

B̃_{|x} = α_{x}^{−1} [β_{x}^{−1} [B_{|π_{b}(x)}]] .

• There is a 1-1 correspondence between connections λ in A (M, F) and distributions C ⊂ T M such that

(a) F_{b} ∩ C = F,

(b) F_{b} + C = T M,

(c) Sec C ∩ L (M, F) generates at each point x ∈ M the entire vector space C_{|x}.

(The correspondence: λ ↦ C := α^{−1} [β^{−1} [Im λ]] .) Such a distribution C exists (because every Lie algebroid has a connection) and C is involutive if and only if the corresponding connection λ is flat. Each distribution C on M satisfying (a)÷(c) above is called an F-connection.

For the Lie algebroid A (G; H) of the TC-foliation of left cosets of a connected Lie group G by a nonclosed Lie subgroup H, conditions (3) and (c) above are equivalent to (3’) and (c’) below, respectively:

(3’) , (c’) B̃ (and C) is C^{∞} and H̄-right-invariant.

Thus, for a given TC-foliation (M, F) , the triad (A (M, F) , C, B̃) where C ⊂ T M is

an involutive distribution fulfilling (a), (b), (c) above, and B̃ ⊂ T M is an involutive distribution fulfilling (1), (2), (3) above, yields an FS-transitive Lie algebroid. Using the restriction A (M, F)^{E} (to a foliation E on the basic manifold W), we can obtain an FS-regular Lie algebroid over (W, E) .

We construct some characteristic classes of an FS-regular Lie algebroid (A, λ, B) , measuring the independence of λ and B, i.e. how far Im λ is not contained in B. The construction has a number of steps.

Let sss : ggg → ggg/h h h be the canonical projection. Applying (2.3), we can easily obtain that the form φ (Ψ) := ω^{∧}(⋀^{k} sss^{⋆} ∘ Ψ), Ψ ∈ Sec ⋀^{k} (ggg/h h h)^{⋆} , is h h h-horizontal, i.e., equivalently, its restriction to the subalgebroid B, j^{⋆}(ω^{∧}(⋀^{k} sss^{⋆} ∘ Ψ)), is horizontal. Therefore there exists a form ΔΨ ∈ Ω_{E}^{k} (M) such that

j^{⋆}(ω^{∧}(⋀^{k} sss^{⋆} ∘ Ψ)) = (γ_{B})_{⋆} (ΔΨ) .

Notice that if λ is a connection in B (i.e. Im λ ⊂ B), then ΔΨ = 0.

Put Δ : ⨁^{k≥0} Sec ⋀^{k} (ggg/h h h)^{⋆} → Ω_{E} (M) , Ψ ↦ ΔΨ. The mapping Δ being a superposition of homomorphisms of algebras, see the following diagram

[figure: Sec ⋀^{k} ggg^{⋆}; Ω_{A,h h h} (M); ω^{∧}; Sec ⋀^{k} (ggg/h h h)^{⋆}; Ω_{E} (M); Δ; φ; Ω_{B,i} (M); (γ_{B})_{⋆}; ≅; j^{⋆}]

(where Ω_{A, h h hh h h} (M) denotes the space of h h h-horizontal forms on A), is itself such a homomorphism.

Directly, Δ is defined by the formula

(ΔΨ) (x; w_{1} ∧ … ∧ w_{k}) = ⟨Ψ_{x}; [ω (x; w̃_{1})] ∧ … ∧ [ω (x; w̃_{k})]⟩

for w̃_{i} ∈ B_{|x} such that γ_{B} \binom{}{w̃_{i}} = w_{i}, w_{i} ∈ E_{|x}, x ∈ M.

Define a representation ad_{B,g g g}^{∧} : B → A (⋀^{k} (ggg/h h h)^{⋆}) by the formula

⟨𝔏_{ad_{B,g g g}^{∧} ∘ξ} (Ψ) , [ν_{1}] ∧ … ∧ [ν_{k}]⟩
= (γ_{B} ∘ ξ) ⟨Ψ, [ν_{1}] ∧ … ∧ [ν_{k}]⟩ − ∑_{j=1}^{k}⟨Ψ, [ν_{1}] ∧ … ∧ [[[ξ, ν_{j}]]] ∧ … ∧ [ν_{k}]⟩

for Ψ ∈ Sec ⋀^{k} (ggg/h h h)^{⋆} , ξ ∈ Sec B, and ν_{j} ∈ Secggg. The correctness of this definition follows from the fact that if one of ν_{j}^{,s} lies in h h h, then [[ξ, ν_{j}]] lies in h h h, too. Notice that ad_{B,g g g}^{∧} = ⋀^{k} (ad_{B,g g g})^{♮} [K4, 2.1.2-3] where ad_{B,g g g} : B → A (ggg/h h h) is a representation given by

𝔏_{ad_{B,g g g} ∘ξ} ([ν]) = [𝔏_{ad_{A} ∘ξ} (ν)] = [[[ξ, ν]]]

for ξ ∈ Sec B and ν ∈ Secggg.

In the space ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} of cross-sections invariant with respect to ad_{B,g g g}^{∧},

we introduce a differential δ̄ of degree +1 defined as follows: for Ψ ∈ (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} and ν_{j} ∈ Secggg, we put

⟨δ̄Ψ, [ν_{0}] ∧ … ∧ [ν_{k}]⟩ = − ∑_{i<j} (−1)^{i+j} ⟨Ψ, [[[ν_{i}, ν_{j}]]] ∧ [ν_{0}] ∧ … i^{∧} … j^{∧} … ∧ [ν_{k}]⟩. (3.1)

The correctness of this definition is obvious (by the invariance of Ψ and the equality γ_{B} ∘ ν_{j_{0}} = 0).

To see the invariance of δ̄Ψ, take ξ ∈ Sec B and ν_{j} ∈ Secggg. From the invariance of Ψ we get

(γ_{B} ∘ ξ) ⟨δ̄Ψ, [ν_{0}] ∧ … ∧ [ν_{k}]⟩

= − ∑_{i<j} (−1)^{i+j} (⟨Ψ, [[[ξ, [[ν_{i}, ν_{j}]]]]] ∧ [ν_{0}] ∧ … i^{∧} … j^{∧} … ∧ [ν_{k}]⟩+

+ ∑_{s, s∉{i,j}}⟨Ψ, [[[ν_{i}, ν_{j}]]] ∧ [ν_{0}] ∧ … ∧ [[[ξ, ν_{s}]]] ∧ … i^{∧} … j^{∧} … ∧ [ν_{k}]⟩)

= − ∑_{s} ( ∑_{s≠i<j≠s} (−1)^{i+j} ⟨Ψ, [[[ν_{i}, ν_{j}]]] ∧ [ν_{0}] ∧ … ∧ [[[ξ, ν_{s}]]] ∧ … i^{∧} … j^{∧} … ∧ [ν_{k}]⟩

+ ∑_{i<s} (−1)^{i+s} ⟨Ψ, [[[ν_{i}, [[ξ, ν_{s}]]]]] ∧ [ν_{0}] ∧ … i^{∧} … j^{∧} … ∧ [ν_{k}]⟩ +

+ ∑_{s<j} (−1)^{s+j} ⟨Ψ, [[[ξ, [[ν_{s}, ν_{j}]]]]] ∧ [ν_{0}] ∧ … i^{∧} … j^{∧} … ∧ [ν_{k}]⟩)

= ∑_{s}⟨δ̄Ψ, [ν_{0}] ∧ … ∧ [[[ξ, ν_{s}]]] ∧ … ∧ [ν_{k}]⟩.

It remains to notice that

(i) δ̄^{2} = 0,

(ii) δ̄ is an antiderivation of degree +1.

For this purpose, firstly, for an arbitrary point x ∈ M, we can define a space of tensors (⋀^{k}(ggg_{|x}/h h h_{|x})^{⋆})_{I^{0}} invariant with respect to the representation of the Lie algebra

h h h_{|x}, induced on ⋀^{k}(ggg_{|x}/h h h_{|x})^{⋆} by the representation ad_{x}^{⋆} of h h h_{|x} on (ggg_{|x}/h h h_{|x})^{⋆} defined as follows: ⟨ad_{x}^{⋆} (ν) (ψ) , [μ]⟩ = −⟨ψ, [[ν, μ]]⟩ for ν ∈ h h h_{|x}, ψ ∈ (ggg_{|x}/h h h_{|x})^{⋆} and μ ∈ ggg_{|x}. Secondly, we define an antiderivation

δ̄_{x} : (⋀^{k}(ggg_{|x}/ h h h_{|x})^{⋆})_{I^{0}} ⟶ (⋀^{k}(ggg_{|x}/ h h h_{|x})^{⋆})_{I^{0}}

of degree +1 as the one which on elements ψ of degree +1 equals ⟨δ̄_{x} (ψ) , [ν] ∧ [μ]⟩ = ⟨ψ, [[ν, μ]]⟩, ν, μ ∈ ggg_{|x}. It can easily be seen that if Ψ ∈ (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} , then Ψ_{x} ∈ (⋀^{k}(ggg_{|x}/h h h_{|x})^{⋆})_{I^{0}} , and (δ̄Ψ)_{x} = δ̄_{x} (Ψ_{x}) .

In consequence, δ̄ fulfils (i) and (ii) in an evident manner. Of course, these properties of δ̄ can also be checked directly.

Definition 3.2. The relative cohomology algebra of ggg with respect to B is defined as the cohomology algebra of the complex (⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} , δ̄)

H (ggg :B) := H^{⋆} (⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} , δ̄) .

Proposition 3.3. The mapping Δ restricted to the invariant cross-sections

Δ_{⋆} = Δ_{(A,λ,B)} : ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} ⟶ Ω_{E} (M) , Ψ ⟼ ΔΨ,

commutes with differentials δ̄ and d^{E}.

P r o o f. We need to prove the equality

Δ (δ̄Ψ) = d^{E} (ΔΨ)  (3.2)

for invariant cross-sections Ψ. The fact that (γ_{B})_{⋆} is a monomorphism implies that this equality is equivalent to (γ_{B})_{⋆} (Δ (δ̄Ψ)) = (γ_{B})_{⋆} (d^{E} (ΔΨ)) . But, by definition (γ_{B})_{⋆} (Δ (δ̄Ψ)) = j^{⋆}(ω^{∧}(⋀^{k+1} sss^{⋆} ∘ (δ̄Ψ))). On the other hand, applying (2.1) and the obvious fact d^{B} (j^{⋆}Ψ) = j^{⋆} (d^{A}Ψ) , we get

(γ_{B})_{⋆} (d^{E} (ΔΨ)) = d^{B} ((γ_{B})_{⋆} (ΔΨ)) = j^{⋆}(d^{A}(ω^{∧}(⋀^{k} sss^{⋆} ∘ Ψ))).

Therefore, to prove (3.2), it remains to check that the forms ω^{∧}(⋀^{k+1} sss^{⋆} ∘ (δ̄Ψ)) and d^{A}(ω^{∧}(⋀^{k} sss^{⋆} ∘ Ψ)) agree on the cross-sections of B. Let ξ_{0}, …, ξ_{k} ∈ Sec B; then (see (2.5))

⟨ω^{∧}( ⋀^{k+1} sss^{⋆} ∘ (δ̄Ψ)), ξ_{0} ∧ … ∧ ξ_{k}⟩

= − ∑_{i<j} (−1)^{i+j} ⟨Ψ, [[[ω (ξ_{i}) , ω (ξ_{j})]]] ∧ … i^{∧} … j^{∧} … ⟩

= − ∑_{i<j} (−1)^{i+j} ⟨⋀^{k} sss^{⋆} ∘ Ψ, [[ω (ξ_{i}) , ω (ξ_{j})]] ∧ … i^{∧} … j^{∧} … ⟩

= ⟨ω^{∧}(δ ∘ ⋀^{k} sss^{⋆} ∘ Ψ), ξ_{0} ∧ … ∧ ξ_{k}⟩.

On the other hand, by Prop.2.2(2) and the flatness of λ, we have

d^{A}(ω^{∧}(⋀^{k} sss^{⋆} ∘ Ψ)) = ω^{∧}(δ ∘ ⋀^{k} sss^{⋆} ∘ Ψ) + \frac{1}{k!} · ⟨d^{g g g}(⋀^{k} sss^{⋆} ∘ Ψ), \underbrace{ω ∧ … ∧ ω}_{k times}⟩.

So, it remains to notice that j^{⋆}⟨d^{g g g}(⋀^{k} sss^{⋆} ∘ Ψ), ω ∧ … ∧ ω⟩ = 0.

The above Proposition yields as a corollary Theorem 3.4. The mapping

Δ_{#} : H (ggg,B) ⟶ H_{E} (M) , [Ψ] ⟼ [Δ_{⋆}Ψ] ,

is a correctly defined homomorphism of algebras. □

Δ_{#} is called the characteristic homomorphism of the FS-regular Lie algebroid (A, λ, B) . Its image Im Δ_{#} ⊂ H_{E} (M) is a subalgebra of H_{E} (M) , called the characteristic algebra of the FS-regular Lie algebroid (A, λ, B) , whereas its elements — the characteristic classes of that algebroid.

The compatibility of λ with B implies the vanishing of Δ_{#} [of course, already on the level of forms]. Δ_{#} is then a measure of the incompatibility of λ with B.

4. Functoriality.

Definition 4.1. Let (A′, λ′, B′) and (A, λ, B) be two FS-regular Lie algebroids over (M′, E′) and (M, E), respectively. By a homomorphism

H : (A′, λ′, B′) ⟶ (A, λ, B)

between them we mean a homomorphism H : A′ → A of regular Lie algebroids, say over f : (M′, E′) → (M, E) , such that

(1) H ∘ λ′ = λ ∘ f_{⋆},

(2) H [B′] ⊂ B.

Notice that H′ = H|B′ : B′ → B is then a homomorphism of regular Lie algebroids, too, see the diagram:

[figure: A′; A; ggg′; ggg; H^{+}; ω′; ω; E′; E; H; γ_{A}; γ_{A′}; λ′; λ; B′; B; h h h′; h h h; H^{′+}; E′; E; H′; γ_{B}; γ_{B′}; 0; 0; 0; 0; 0; 0; f_{⋆}; 0; 0; f_{⋆}]

By the pullback of an FS-regular Lie algebroid (A, λ, B) over (M, E) via a mapping f : (M′, E′) → (M, E) we mean the FS-regular Lie algebroid (f^{∧}A, λ̄, f^{∧}B) where λ̄ is the pullback of the connection λ, see definition 3.2.1 from [K4].

Notice that pr_{2} : f^{∧}A = E′ ×_{(f_{⋆},γ)} A → A is a homomorphism of FS-regular Lie algebroids, called canonical. In view of the equality H̄ ∘ λ′ = λ̄, any homomorphism H : (A′, λ′, B′) → (A, λ, B) of FS-regular Lie algebroids can be represented in the form of a superposition of a strong homomorphism with the canonical one:

(A′, λ′, B′) ⟶^{H̄} (f^{∧}A, λ̄, f^{∧}B) ⟶^{pr_{2}} (A, λ, B) .

Let H : (A′, λ′, B′) → (A, λ, B) be a homomorphism of FS-regular Lie algebroids, see diagram above. We define the pullback

H^{+⋆} : Sec ⋀^{k} (ggg/h h h)^{⋆} ⟶ Sec ⋀^{k} (ggg′/h h h′)^{⋆}

by the formula

⟨H^{+⋆} (Ψ)_{x} , [w′_{1}] ∧ … ∧ [w′_{k}]⟩ = ⟨Ψ_{f(x)}, [H^{+} (w′_{1})] ∧ … ∧ [H^{+} (w′_{k})]⟩

where Ψ ∈ Sec ⋀^{k} (ggg/h h h)^{⋆} , x ∈ M, w′_{i} ∈ ggg′_{|x}.

Proposition 4.2. (1) H^{+⋆} maps the invariant cross-sections into the invariant ones.

(2) H^{+⋆} restricted to the invariant cross-sections commutes with the differentials δ̄′ and δ̄.

P r o o f. It is enough to prove the proposition in two cases of H: of a strong homomorphism and of the canonical one. A very easy proof of (1) and (2) for the first case will be omitted. Consider now the canonical homomorphism pr_{2} . Identify the vector bundles f^{⋆} (ggg/h h h) ≅ f^{⋆}ggg/f^{⋆}h h h. Then, of course, H^{+⋆}Ψ = f^{⋆}Ψ and, by standard calculations, we assert the following equality (cf. [K4, 2.3.2]):

f^{⋆} (ad_{B,g g g}) = ad_{f^{∧}B,f^{⋆}g g g}

which, together with f^{⋆} (⋀^{k} T ) = ⋀^{k} (f^{⋆}T) for any representation T (cf. [K4, 2.3.3]), yields

f^{⋆} (ad_{B,g g g}^{∧}) = f^{⋆}(⋀^{k} (ad_{B,g g g})^{♮}) = ⋀^{k} (ad_{f^{∧}B,f^{⋆}g g g})^{♮} = ad_{f^{∧}B,f^{⋆}g g g}^{∧} .

Proposition (2) needs now only standard calculations.

As a corollary we obtain that H^{+⋆} determines a homomorphism of algebras

H^{+#} : H (ggg,B) ⟶ H (ggg′,B′) .

Proposition 4.3. (The functoriality of Δ_{#} ). Let (A′, λ′, B′) and (A, λ, B) be two FS-regular Lie algebroids over (M′, E′) and (M, E) , respectively, and let

H : (A′, λ′, B′) ⟶ (A, λ, B)

be a homomorphism between them over f : (M′, E′) → (M, E) . Then the following diagram

[figure: H (ggg′, B′); H_{E′} (M′); Δ′_{#}; H (ggg, B); H_{E} (M); Δ_{#}; H^{+#}; f^{#}]

commutes.

P r o o f. It is sufficient to show the commutativity of the diagram on the level of forms; this means — the equality: (γ_{B′})_{⋆} (f^{⋆} (Δ_{⋆}Ψ)) = j^{′⋆}(ω^{′∧}(⋀^{k} sss^{′⋆} ∘ H^{+⋆}Ψ)) for an invariant Ψ.

R e m a r k 2. The problem of the equivalence of the characteristic homomorphisms Δ_{#} and Δ′_{#} for H being a homotopic equivalence is open. The problem reduces to the isomorphy of H^{+#}. A definition of the homotopy between homomorphisms of FSalgebroids can be formulated in the spirit of the definition of the homotopy from [K7].

5. The homotopic independence of Δ_{#} on a subalgebroid. Let (A, [[·, ·]], γ) be a given regular Lie algebroid with the Atiyah sequence 0 → ggg ↪A →^{γ} E → 0 and consider the Lie algebroid (T ℝ × A, [[·, ·]]′, id ×γ) – the product of the trivial Lie algebroid T ℝ with A (see [H-M], [K7]). Its Atiyah sequence is

[figure: 0; 0 × ggg; T ℝ × A; T ℝ × E; id × γ; 0.]

For the mapping f_{t} : M → ℝ × M, x ↦ (t, x) , take the pullback f_{t}^{∧} (T ℝ×A) . Notice that f_{t}^{∧} (T ℝ × A) = {(γ (w) , 0, w) ∈ E × (T ℝ×A) ; w ∈ A} , and that the homomorphism F_{t} : A → T ℝ×A, w ↦ (θ_{t}, w) , (θ_{t} being the null tangent vector at t ∈ ℝ) of regular Lie algebroids (see the proof of Th.4.3.1 in [K4]) is represented in the form of the canonical superposition

F_{t} : A ⟶^{\overline{F}_{t}} f_{t}^{∧} (T ℝ×A) ⟶^{pr_{2}} T ℝ×A  (5.1)

(see [K4, s.1.1]). It is not difficult to see that

F̄_{t} : A ⟶ f_{t}^{∧} (T ℝ×A) , w ⟼ (γ (w) , 0, w) ,

is an isomorphism of regular Lie algebroids.

Definition 5.1. Two Lie subalgebroids B_{0}, B_{1} ⊂ A (both over (M, E)) are said to be homotopic if there exists a Lie subalgebroid B ⊂ T ℝ×A over (ℝ × M, T ℝ × E) such that the isomorphism F̄_{t} maps B_{t} onto f_{t}^{∧} (B) for t = 0, 1 (equivalently, if, for v ∈ A, we have: v ∈ B_{t} ⇔ (θ_{t}, v) ∈ B). B is called joining B_{0} to B_{1}.

We compare the relation of homotopic subbundles of a principal bundle P with the relation of homotopic subalgebroids of A (P) . Let P = (P, π, M, G, ·) be a Gprincipal bundle over a manifold M. It determines a new G-principal bundle ℝ × P = (ℝ × P, id ×π, ℝ × M, G, ·′) with the action (t, z) ·′ a = (t, z · a) . For an arbitrary t ∈ ℝ, the mapping F_{t} : P → f_{t}^{⋆} (ℝ × P) , z ↦ (πz, (t, z)) , is an isomorphism of G-principal bundles. Take a Lie subgroup H ⊂ G (nonclosed and disconnected in general). Two H-reductions P_{t} ⊂ P, t = 0, 1, are said to be homotopic [K-T] if there exists an Hreduction P̄ ⊂ ℝ × P such that F_{t} maps P_{t} onto f_{t}^{⋆} \binom{}{P̄} for t = 0, 1. P̄ is called joining P_{0} to P_{1}. Notice that P̄ is determined uniquely by the family of H-reductions P_{t} = F_{t}^{−1} [f_{t}^{⋆} \binom{}{P̄}] , t ∈ ℝ [which follows from the observation: z ∈ P_{t} ⇔ (t, z) ∈ P̄].

If H is closed and P_{t} are defined by C^{∞} cross-sections σ_{t} : M → P/H for t = 0, 1, of the associated bundle P/H → M, then, P_{0} and P_{1} are homotopic if and only if σ_{0} and σ_{1} are homotopic in the usual sense (via cross-sections, of course).

Proposition 5.2. If P_{t} ↪^{i_{t}} P, t = 0, 1, are homotopic H-reductions of P, then the Lie subalgebroids B_{0} := d (i_{0}) [A (P_{0})] and B_{1} := d (i_{1}) [A (P_{1})] of A (P) are homotopic. The converse theorem is true provided that P_{t} and G are connected. More generally, if A (P_{0}) and A (P_{1}) are homotopic and P_{0} and P_{1} are connected [G can be disconnected], then there exists a ∈ G such that R_{a} [P_{0}] is homotopic to P_{1}.

P r o o f. Let P_{0,} P_{1} ⊂ P be two H-reductions of P. Assume that they are homotopic, and that P̄ ⊂ ℝ × P is a joining H-reduction. Then B := φ [A \binom{}{P̄}] ⊂ T ℝ × A (P) ,

φ : A (ℝ × P) = T (ℝ × P) /G ∋ [(v, w)] ⟼ (v, [w]) ∈ T ℝ × T P/G = T ℝ × A (P)

being the canonical isomorphism, is a Lie algebroid joining B_{0} to B_{1}. Indeed, one can easily see that F̄_{t} : A (P) → f_{t}^{∧} (T ℝ × A (P)) equals the superposition

A (P) ⟶^{dF_{t}} A (f_{t}^{⋆} (ℝ × P)) ≅ f_{t}^{∧} (A (ℝ × P)) ≅ f_{t}^{∧} (T ℝ × A (P))

and then maps B_{t} onto f_{t}^{∧} (B) for t = 0, 1.

Conversely, assume that the Lie subalgebroids B_{0} and B_{1} are homotopic, say, via a joining Lie subalgebroid B of T ℝ × A (P) . This means that F̄_{t} maps B_{t} onto f_{t}^{∧} (B) for t = 0, 1. Let P̄ ⊂ ℝ × P be the arbitrarily taken connected H-reduction corresponding to the Lie subalgebroid φ^{−1} [B] ⊂ A (ℝ × P) , see Th.1.1. Put P̃_{t} := F_{t}^{−1} [f_{t}^{⋆} \binom{}{P̄}] , t ∈ ℝ.

By its construction, {P̃_{t}, t ∈ ℝ} is a family of homotopic H-reductions. Of course, P_{t}

and P̃_{t} are, for t = 0, 1, two H-reductions corresponding to the same Lie subalgebroid B_{t}. If P_{t} is connected, then, according to the fact that P_{t} and P̃_{t} are integral manifolds of the same G-right invariant distribution on P, we notice that P̃_{t} = R_{g_{t}} [P_{t}] for a point g_{t} ∈ G. Therefore R_{g_{0}g_{1}^{−1}} [P_{0}] is homotopic to P_{1}. If, additionally, G is connected, R_{g_{0}g_{1}^{−1}} [P_{0}] is homotopic to P_{0}. Thus P_{0} and P_{1} are homotopic then.

For the further investigations, we fix

• a regular Lie algebroid A = (A, [[·, ·]], γ) over (M, E),

• a flat connection λ : E → A in it,

• two Lie subalgebroids B_{0}, B_{1} ⊂ A, both over (M, E), homotopic to each other via a joining Lie algebroid B ⊂ T ℝ × A.

λ determines a flat connection in T ℝ × A of the form id ×λ : T ℝ × E → T ℝ × A. This implies that the triad

(T ℝ × A, id ×λ, B)  (5.2)

is an FS-regular Lie algebroid. Besides, we have that

F_{t} : (A, λ, B_{t}) ⟶ (T ℝ × A, id ×λ, B)

is a homomorphism of FS-regular Lie algebroids.

Proposition 5.3. The characteristic homomorphisms Δ_{t#}, t = 0, 1, of FS-regular Lie algebroids (A, λ, B_{t}) are related to each other by the commutativity of the following diagram:

[figure: H (0 × ggg, B); H_{E} (M) .; Δ_{0#}; H (ggg, B_{0}); F_{0}^{+#}; H (ggg, B_{1}); F_{1}^{+#}; Δ_{1#}]

P r o o f. By the functoriality of the characteristic homomorphisms of FS-regular Lie algebroids, we get the commutative diagram

[figure: H (0 × ggg, B); H_{T ℝ×E} (ℝ × M); H (ggg, B_{0}); H_{E} (M); Δ_{0#}; F_{0}^{+#}; f_{0}^{#}; H (ggg, B_{1}); H_{E} (M); Δ_{#}; F_{1}^{+#}; f_{1}^{#}; Δ_{0#}]

where Δ_{#} is the characteristic homomorphism of (5.2). Since f_{0}^{#} = f_{1}^{#} (see the proof of Th.4.3.1 from [K4]) and f_{0}^{#} is an isomorphism (because f_{0} and f_{1} are homotopic in the category of foliated manifolds and each of them is a homotopic equivalence in this

category) therefore so f_{1}^{#} = (f_{0}^{#})^{−1} , which implies our proposition.

Notice that if F_{t}^{+#}, t = 0, 1, are isomorphisms, then Δ_{0#} and Δ_{1#} can be interpreted as equivalence homomorphisms in the sense of the following definition.

Definition 5.4. Let B_{0}, B_{1} ⊂ A be two Lie subalgebroids of a flat regular Lie algebroid A (all the three over (M, E)). We say that the characteristic homomorphisms Δ_{t#} : H (ggg, B_{t}) → H_{E} (M) , t = 0, 1, corresponding to B_{0} and B_{1}, respectively, are equivalent if there exists an isomorphism of algebras α : H (ggg, B_{0}) ⟶^{≅} H (ggg, B_{1}) such that Δ_{0#} = Δ_{1#} ∘ α.

Theorem 5.5. If B_{0} and B_{1} are homotopic, then Δ_{0#} and Δ_{1#} are equivalent.

P r o o f. Recall that F_{t} = pr_{2} ∘F̄_{t}, see (5.1). F̄_{t} is an isomorphism of FS-regular Lie algebroids, therefore

F̄_{t}^{+#} : H (f_{t}^{⋆} (0 × ggg) , f_{t}^{∧}B) ⟶^{≅} H (ggg, B_{t})

is an isomorphism of algebras. It remains to consider the homomorphism

pr_{2}^{+#} : H (0 × ggg, B) → H (f_{t}^{⋆} (0 × ggg) , f_{t}^{∧}B) .

Identifying (via the canonical isomorphism) the vector bundles

f_{t}^{⋆} (0 × ggg) /f_{t}^{⋆}h h h ≅ f_{t}^{⋆} (0 × ggg/h h h) ,

we get (cf. the proof of Prop.5.2.1 [K4])

• f_{t}^{∧} (ad_{B,0×g g gg g g}^{∧}) = ad_{f_{t}^{∧}B,f_{t}^{⋆}(0×g g gg g g)}^{∧},

• pr_{2}^{+#} : ⨁^{k≥0} (Sec ⋀^{k} (0 × ggg/h h h)^{⋆})_{I^{0}} → ⨁^{k≥0} (Sec ⋀^{k} f_{t}^{⋆} (0 × ggg/h h h)^{⋆})_{I^{0}} is the usual pullback Ψ ↦ f_{t}^{⋆}Ψ.

Theorem 5.5 follows now from Theorem 1.3.

6. Comparison with the characteristic classes of a flat principal fibre bundle. Given:

(a) a G-principal fibre bundle P = (P, π, M, G, ·) ,

(b) a flat connection in P with a connection form ω,

(c) a closed Lie subgroup H ⊂ G and an H-reduction P′ ⊂ P,

let 𝔤 and 𝔥 denote the Lie algebras of G and H, respectively. Of course, i : P′ ↪ P is an (H ↪ G)-homomorphism of principal bundles and its differential di : A (P′) → A (P) , see [K3], [M1], is a monomorphism of the corresponding transitive Lie algebroids, see the diagram

[figure: 0; ggg; A (P); T M; γ; 0; 0; h h h; A (P′); T M; γ′; 0; (di)^{+}; di]

Identify A (P′) with Im (di) and h h h with Im (di)^{+} . Then, for each z ∈ P|x^{′}, the isomor-

phism z^{∧} : 𝔤 → ggg_{|x}, v ↦ [A_{z⋆v}] (A_{z} : G → P, a ↦ za), see [K4, s.5.1], maps 𝔥 onto h h h_{|x}

and determines an isomorphism [z^{∧}] : 𝔤/𝔥 →^{≅} (ggg/h h h)_{|x} . It is worth recalling that z^{∧} is an isomorphism of Lie algebras provided that 𝔤 is the right Lie algebra of G, see [K3], [K4].

According to section 3, we have a representation ad_{A(P′),g g g} : A (P′) → A (ggg/h h h) such that 𝔏_{ad_{A(P′),g g g} ∘ξ} ([ν]) = [[[ξ, ν]]] , ξ ∈ Sec A (P′) , ν ∈ Secggg, and the representation induced

by it ad_{A(P′),g g g}^{∧} : A (P′) → A (⋀^{k} (ggg/h h h)^{⋆}) . Consider auxiliarily the induced representation Ad_{P′,g g g} of the principal bundle P′ on the 𝔤/𝔥-vector bundle ggg/h h h, defined by

Ad_{P′,g g g} : P′ ⟶ L (ggg/h h h) , z ⟼ [z^{∧}],

and the representation Ad_{P′,g g g}^{∧} : P′ → L (⋀^{k} (ggg/h h h)^{⋆}) induced by it (cf. [K4, 5.3.2]). By the same argument as in the proof of Th.5.4.3 in [K4], to see that ad_{A(P′),g g g} is the differential of Ad_{P′,g g g}, we must only notice an analogous fact concerning the representations of Lie algebras and of Lie groups: 𝔥 → End (g/h) , v ↦ [ad_{𝔤} (v)] , and H → GL (𝔤/𝔥) , h ↦ [Ad_{G} (h)] . By this, ad_{A(P′),g g g}^{∧} is the differential of Ad_{P′,g g g}^{∧} . Therefore, according to [K4, Props 5.5.2-3], we have a monomorphism

κ : (⋀ (𝔤/𝔥)^{⋆})_{I} ⟶^{≅} ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I} ⊂ ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}}

defined by the formula κ (ψ) (x) = Ad_{P′,g g g}^{∧} (z) (ψ) , z ∈ P|x^{′}, and being an isomorphism when P′ is connected. It is needful to verify that κ commutes with the differentials d^{H} and δ̄ acting on the spaces (⋀ (𝔤/𝔥)^{⋆})_{I} and ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} , respectively (notice that the spaces of cohomology of these are domains of the characteristic homomorphisms). The differential δ̄ in ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} is defined in section 3 above, whereas, the

differential d^{H} in (⋀ (𝔤/𝔥)^{⋆})_{I} must be defined by the formula

⟨d^{H} (ψ) , [w_{1}] ∧ … ∧ [w_{k}]⟩ = ∑_{i<j} (−1)^{i+j} ⟨ψ, [[w_{i}, w_{j}]] ∧ [w_{1}] ∧ … i^{∧} … j^{∧} … ⟩

w_{1}, …, w_{k} ∈ 𝔤; here [w_{i}, w_{j}] is the bracket in the left Lie algebra of G [we get it following the fact that this differential must be the one for which the canonical isomorphism G_{Ω^{⋆}(G/H)} ≅ (⋀ (𝔤/𝔥)^{⋆})_{I} (also (⋀ 𝔤^{⋆})^{H} ≅ (⋀ (𝔤/𝔥)^{⋆})_{I} ) should be an isomorphism of DG-algebras, see [K-T]]. The equality δ̄ ∘ κ^{k} = κ^{k+1} ∘ d^{H} may now be obtained immediately.

Theorem 6.1. The characteristic homomorphisms Δ_{#} : H (𝔤, H) → H_{dR} (M) of the triad (P, P′, ω) (see [K-T]) and Δ_{#} : H (ggg, A (P′)) → H_{dR} (M) of the FS-transitive Lie algebroid (A (P) , λ, A (P′)) (λ corresponds to ω) are related by the following commutative diagram:

[figure: Δ_{#}; H_{dR} (M) .; Δ_{#}; H (𝔤, H); H (ggg, A (P′)); κ_{#}]

P r o o f. We prove the commutativity of this diagram on the level of forms. For the purpose, consider the diagram

[figure: Ω_{A(P′),i} (M); Ω_{A(P′)} (M); Ω (M); Ω (P′); π^{′⋆}; ≅; ρ; γ′_{⋆}; Ω_{A(P),h h h} (M); (di)^{⋆}; (⋀ (𝔤/𝔥)^{⋆})_{I}; ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}}; κ; φ; Δ; Δ]

in which

• φ (Ψ) = (ω^{A})^{∧} (⋀^{k} sss^{⋆} ∘ Ψ) = \frac{1}{k!} · ⟨⋀^{k} sss^{⋆} ∘ Ψ, ω^{A} ∧ … ∧ ω^{A}⟩, ω^{A} : A (P) → ggg being the connection form corresponding to λ.

• Ω_{A(P),h h hh h h} (M) denotes the space of h h h-horizontal forms on A (P) ,

• ρ maps real forms on A (P′) into right-invariant forms on P′, Θ ↦ Θ̃, for Θ̃ see [K3].

We recall that, for ψ ∈ (⋀^{k} (𝔤/𝔥)^{⋆})_{I}, the form Δ (ψ) ∈ Ω^{k} (M) is defined uniquely

in such a way that π^{′⋆} (Δ (ψ)) = \frac{1}{k!} · i^{⋆}⟨⋀^{k} s^{⋆} (ψ) , ω ∧ … ∧ ω⟩ where i : P′ ↪ P, whereas s : 𝔤 → 𝔤/𝔥 and π′ : P′ → M are the canonical projections. On the other hand, Δ (Ψ) for Ψ ∈ (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} is given as a form for which γ′_{⋆} (Δ (Ψ)) = (di)^{⋆} (φ (Ψ)) .

Therefore, to end the proof, we need to assert the equality i^{⋆}⟨⋀^{k} s^{⋆} (ψ) , ω ∧ … ∧ ω⟩ = ρ ∘ (di)^{⋆} (⟨⋀^{k} sss^{⋆} (κψ) , ω^{A} ∧ … ∧ ω^{A}⟩) only (very easy calculations are omitted).

7. The tangential characteristic classes of a partially flat principal bundle. Consider now Ex.3.1(2), i.e. a triple (A, B, λ) consisting of a transitive Lie algebroid A on M, a transitive Lie subalgebroid B of A and a partially flat connection λ in A, namely, flat over a given involutive distribution F ⊂ T M. The characteristic homomorphism Δ_{#}^{F} : H (ggg, B^{F} ) → H_{F} (M) of the FS-regular Lie algebroid (A^{F} , B^{F} , λ|F ) will be called the tangential characteristic homomorphism of the system (A, B, λ) and the cohomology classes from its image — the tangential characteristic classes of the system (A, B, λ) .

Let now the system (P, P′, ω) be given as in Ex.3.1(3). It determines the FS-regular Lie algebroid (A (P)^{F} , A (P′)^{F} , λ|F), and via this a characteristic homomorphism

Δ_{#}^{F} : H(ggg, A (P′)^{F}) ⟶ H_{F} (M) ,

called the characteristic homomorphism of the system (P, P′, ω) . The cohomology classes from the image of Δ_{#}^{F} should be called the tangential characteristic classes of the system (P, P′, ω) . By construction, they measure the independence of ω and P′ — exactly the same as the exotic characteristic classes of a partially flat principal bundle [K-T].

R e m a r k 3. The problem of the comparing of these two systems of characteristic classes is open.

8. The case of a TC-foliation. This section is a continuation of Example 3.1(4). One can prove the following two propositions.

Proposition 8.1. The Lie algebroid T ℝ × A (M, F) is canonically isomorphic to the Lie algebroid A (ℝ × M, F̃) of the TC-foliation (ℝ × M, F̃) := (ℝ × E_{d}) × (M, F) being the product of the discrete foliation E_{d} of ℝ and the given foliation (M, F) . □

Let B_{0}, B_{1} ⊂ A (M, F) be two Lie subalgebroids of A (M, F) and B̃_{0}, B̃_{1} ⊂ T M — the involutive subbundles corresponding to them (see Ex. 3.1(4)). A simple consequence of the definitions and Proposition 8.1 is the following

Proposition 8.2. B_{0} and B_{1} are homotopic if and only if there exists an involutive subbundle B̄ ⊂ T (ℝ × M) = T ℝ × T M such that

(1) 0 × F ⊂ B̄,
(2) B̄+ (0 × F_{b}) = T ℝ × T M,

(3) v ∈ B̄_{t} ⇔ (θ_{t}, v) ∈ B̄ for t = 0, 1,

(4) the Lie algebra Sec B̄ ∩ L (ℝ × M, F̃) generates at each point (t, x) ∈ ℝ × M the

entire space B̄_{|(t,x)}. □

Here we pass to the Lie algebroid A (G; H) of the TC-foliation H of a connected Lie group G by left cosets of a connected nonclosed Lie subgroup H ⊂ G. Let 𝔥, 𝔥̄, 𝔤 denote, as usual, the Lie algebras of H, H̄ and G, respectively. By the same argument as in Example 7.4.7 from [K4] we assert

Lemma 8.3. Let 𝔟 ⊂ 𝔤 be a Lie subalgebra such that

𝔥 ⊂ 𝔟, 𝔥̄ + 𝔟 = 𝔤;  (8.1)

then the G-left-invariant distribution B̄_{𝔟} ⊂ T G determined by 𝔟 [i.e. the one tangent to the foliation {g · B; g ∈ G} , B being the connected Lie subgroup with the Lie algebra 𝔟] fulfils conditions (1), (2), (3’) from Example 3.1(4), giving at the same time a transitive Lie subalgebroid of A (G; H) . □

It seems to be interesting that 𝔟 can be interpreted as a ”connection”, but in another Lie algebroid. Namely, let H_{1} be the connected Lie subgroup of G whose Lie algebra equals 𝔟 ∩ 𝔥̄. Of course, 𝔥 ⊂ 𝔟 ∩ 𝔥̄ ⊂ 𝔥̄, therefore H ⊂ H_{1} ⊂ H̄, thereby H̄_{1} = H̄. Then, it is clear (see [K4, Ex.8.4.7]) that B̄_{𝔟} is an F-connection where F is the foliation of left cosets of G by H_{1}.

Lemma 8.4. Let 𝔠 ⊂ 𝔤 be a Lie subalgebra such that

𝔠 ∩ 𝔥̄ = 𝔥, 𝔠 + 𝔥̄ = 𝔤;  (8.2)

then the G-left distribution C_{𝔠} determined by 𝔠 is C^{∞} completely integrable and fulfils (a), (b), (c’) from Example 3.1(4), giving at the same time a flat connection in A (G; H) . □

Seeing that the foliation {ℝ × G, H̄} = {ℝ, E_{d}} × {G, H} is equal to the foliation of the Lie group ℝ × G (being the product of the additive Lie group of reals, with G) by left cosets of a Lie subgroup Θ × H, Θ being the null Lie subgroup of ℝ, we assert that

Lemma 8.5. The Lie algebroid T ℝ × A (G; H) is isomorphic to the Lie algebroid A (ℝ × G; Θ × H) . Condition (4) from Theorem 8.2 above is equivalent to

(4’) B̄ is Θ × H̄-right-invariant. □

Now, we calculate the characteristic homomorphism of the FS-transitive Lie algebroid (A (G; H) , B, λ) in which

• B = B_{𝔟} for a Lie subalgebra 𝔟 fulfilling (8.1),

• λ is the flat connection determined by a Lie subalgebra 𝔠 fulfilling (8.2). (According to [K5] for such a Lie subalgebra 𝔠 to exist, π_{1} (G) must be infinite).

Recall that [K4, 8.2.4] φ : G/H̄ × 𝔥̄/𝔥 → ggg, (ḡ, [w]) ↦ [X̄_{w} (g)] , g ∈ π_{b}^{−1} \binom{}{ḡ} , is a

global trivialization of the Lie algebra bundle ggg of A (G; H) , and that the typical fibre 𝔥̄/𝔥 of this bundle is an abelian Lie algebra [X̄_{w} stands for the left-invariant vector field on G generated by a vector w] [K4, 8.1.3]. The equalities h h h = ggg ∩ B and dim (𝔥̄ ∩ 𝔟/𝔥) = rankh h h yield that φ induces a global trivialization φ^{1} : G/H̄ × (𝔥̄ ∩ 𝔟) /𝔥 → h h h. Next, φ and φ^{1} give a global trivialization φ^{2} : G/H̄ × 𝔥̄/(𝔥̄ ∩ 𝔟) → ggg/h h h. Using φ^{2}, we can modify any cross-section ν ∈ Secggg/h h h to the 𝔥̄/ (𝔥̄ ∩ 𝔟)-valued function ν̄ : G/H̄ → 𝔥̄/ (𝔥̄ ∩ 𝔟) , and, analogously, any cross-section Ψ ∈ Sec ⋀^{k} (ggg/h h h)^{⋆} to the function Ψ̄ : G/H̄ → ⋀^{k} (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆} . Analogously to the proof of Prop.7.4.1 from [K4] we assert the following

Proposition 8.6. Ψ is invariant if and only if Ψ̄ is constant. □

In the space ⨁^{k≥0} (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} the differential δ̄ defined by (3.1) works. We

can carry δ̄ over to the space ⋀ (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆} and obtain a differential δ^{=}.

Proposition 8.7. δ^{=} = 0 [hence δ̄ = 0].

P r o o f. Take Ψ̂ ∈ ⋀^{k} (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆} and let Ψ ∈ (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{o}} corresponds to Ψ̂

(i.e. Ψ̂ = Ψ̄(ē) = Ψ̄(ḡ)). For w_{1}, …, w_{k} ∈ 𝔥̄, we have

⟨Ψ̂, [w_{1}] ∧ … ∧ [w_{k}]⟩ = ⟨Ψ \binom{}{ḡ} , [φḡ ([w_{1}])] ∧ … ∧ [φḡ ([w_{k}])]⟩

and then (see [K4], [MO2]) for w_{0}, …, w_{k} ∈ 𝔥̄,

⟨δ^{=}Ψ̂, [w_{0}] ∧ … ∧ [w_{k}]⟩

= ⟨(δ̄Ψ)^{∧} , [w_{0}] ∧ … ∧ [w_{k}]⟩
= ⟨(δ̄Ψ) , [c \binom{}{X̄_{w_{0}}}] ∧ … ∧ [c \binom{}{X̄_{w_{k}}}]⟩ \binom{}{ḡ}
= − ∑_{i<j} (−1)^{i+j} ⟨Ψ, [[[c \binom{}{X̄_{w_{i}}} , c \binom{}{X̄_{w_{j}}}]]] ∧ [c \binom{}{X̄_{w_{0}}}] ∧ … i^{∧} … j^{∧} … ⟩ \binom{}{ḡ}

= − ∑_{i<j} (−1)^{i+j} ⟨Ψ, [c \binom{}{X̄_{[w_{i},w_{j}]}}] ∧ … i^{∧} … j^{∧} … ⟩ \binom{}{ḡ}

= − ∑_{i<j} (−1)^{i+j} ⟨Ψ̂, [[w_{i}, w_{j}]] ∧ … i^{∧} … j^{∧} … ⟩
= 0

because [w_{i}, w_{j}] ∈ 𝔥 ⊂ 𝔟 ∩ 𝔥̄ [𝔥̄/𝔥 is abelian !].

As a corollary we obtain an isomorphism of algebras

H (ggg,B) ≅ H (⋀ (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆} , 0) = ⋀ (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆}

and the fact that the forms from the image of the characteristic homomorphism, Im Δ_{⋆}, are closed.

Take into account the connection λ determined by 𝔠. Let ω be its connection form. Conditions (8.2) determine a decomposition 𝔤/𝔥 = 𝔥̄/𝔥 ⨁ 𝔠/𝔥. Define ω_{o} : 𝔤 → 𝔥̄/𝔥 as the linear mapping being the superposition ω_{o} : 𝔤 → 𝔤/𝔥 = 𝔥̄/𝔥 ⨁ 𝔠/𝔥 ⟶^{pr_{1}} 𝔥̄/𝔥. Take also the canonical linear homomorphism ρ : 𝔥̄/𝔥 → 𝔥̄/ (𝔥̄ ∩ 𝔟) and put ω_{1} = ρ∘ω_{o} : 𝔤 → 𝔥̄/ (𝔥̄ ∩ 𝔟) .

Lemma 8.8. The form Δ_{⋆} (Ψ) for Ψ ∈ (Sec ⋀^{k} (ggg/h h h)^{⋆})_{I^{0}} is a G-left-invariant form

on G/H̄ such that its value Δ_{⋆} (Ψ)ē ∈ ⋀^{k} (𝔤/𝔥̄)^{⋆} at ē is equal to

⟨Δ_{⋆} (Ψ)ē , [w_{1}] ∧ ... ∧ [w_{k}]⟩ = ⟨Ψ̂, ω_{1} \binom{}{w̃_{1}} ∧ ... ∧ ω_{1} \binom{}{w̃_{k}}⟩

where w̃_{i} ∈ 𝔟 are vectors such that \left[{ \atop w̃_{i}}\right] = [w_{i}] ∈ 𝔤/𝔥̄ (notice that 𝔟 ↪ 𝔤 → 𝔤/𝔥̄ is an epimorphism). □

Lemma 8.9. Consider the DG-algebra (⋀ (𝔤/𝔥̄)^{⋆})_{I} of vectors invariant with respect

to the adjoint representation Ad^{⊥} : H̄ → GL (⋀ (𝔤/𝔥̄)^{⋆}) , see [G-H-V, Vol.II, PropXI].

The forms from Im Δ̂_{⋆} are closed, where

Δ̂_{⋆} : ⋀ (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆} ⟶ (⋀ (𝔤/𝔥̄)^{⋆})_{I}

is defined by the equality (w̃_{i} – as above)

⟨Δ̂_{⋆} \binom{}{Ψ̄} , [w_{1}] ∧ ... ∧ [w_{k}]⟩ = ⟨Ψ̄, ω_{1} \binom{}{w̃_{1}} ∧ ... ∧ ω_{1} \binom{}{w̃_{k}}⟩. □

From the above we obtain the fundamental commutative diagram

[figure: H (ggg, B); H_{dR} (G/H̄); Δ_{#}; ⋀ (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆}; H ((⋀ (𝔤/𝔥̄)^{⋆})_{I}) ≅ H_{I} (G/H̄) .; Δ̂_{#}; ≅]

If G is compact, then the right arrow is an isomorphism.

Theorem 8.10. Δ̂_{#} is trivial if and only if 𝔠 ⊂ 𝔟.

P r o o f. Let 𝔠 ⊂ 𝔟. The epimorphy of 𝔟 ↪ 𝔤 → 𝔤/𝔥̄ imply that it is sufficient to show the equality ω_{1} (w) = 0 for w ∈ 𝔟. For this purpose, take an arbitrary point w ∈ 𝔟 and write w = w_{1} + w_{2} for w_{1} ∈ 𝔥̄ and w_{2} ∈ 𝔠. Then w_{1} = w − w_{2} ∈ 𝔟, so ω_{1} (w) = ρ (w_{1}) = 0.

Assume now 𝔠 ⊂/ 𝔟. Take w ∈ 𝔠\𝔟 and let w̃ ∈ 𝔟 be a vector such that [w] = \left[{ \atop w̃}\right] in 𝔤/𝔥̄. Of course, w̃ − w ∈ 𝔥̄\ (𝔥̄ ∩ 𝔟) and w̃ = (w̃ − w) + w ∈ 𝔥̄ + 𝔠. Take a covector Ψ̄ ∈ (𝔥̄/ (𝔥̄ ∩ 𝔟))^{⋆} such that Ψ̂ (w̃ − w) ≠ 0. Then

Δ̂_{⋆} \binom{}{Ψ̂} ([w]) = ⟨Ψ̂, ω_{1} \binom{}{w̃}⟩ = ⟨Ψ̂, [w̃ − w]⟩ ≠ 0.

Since Z ((𝔤/𝔥̄)^{⋆})_{I} → H^{1} ((⋀ (𝔤/𝔥̄)^{⋆})_{I}) is a monomorphism, we obtain Δ̂_{#} \binom{}{Ψ̂} ≠ 0.

Then, for compact G, each case 𝔠 ⊂/ 𝔟 is the source of the nontrivial characteristic homomorphism of a FS-transitive Lie algebroid on the ground of TC-foliations of left cosets.

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