Universitatis Iagellonicae Acta Mathematica, 2002 1

Endomorphisms of the Lie Algebroids T M × 𝔤 up to Homotopy

Bogdan Balcerzak

Institute of Mathematics, Technical University of Łódź

PL-90-924, al. Politechniki 11, Poland email: bogdan@ck-sg.p.lodz.pl

Jan Kubarski

Institute of Mathematics, Technical University of Łódź

PL-90-924, al. Politechniki 11, Poland and Czȩstochowa Technical University, Institute of Mathematics and Informatics,

PL-42-201 Czȩstochowa, ul. Da̧browskiego 69, Poland, email: kubarski@ck-sg.p.lodz.pl

Witold Walas

Institute of Mathematics, Technical University of Łódź

PL-90-924, al. Politechniki 11, Poland email: walwit@ck-sg.p.lodz.pl

November 10, 1999

Abstract

The notion of a homotopy joining two homomorphisms of Lie algebroids comes from J.Kubarski [5] and the definition of the fundamental group π (A) of a Lie algebroid A (which consists of homotopy classes of strong automorphisms of A), from B.Balcerzak [1]. It is given in this paper a homotopic classification of strong homomorphisms of the trivial Lie algebroid T M × 𝔤 and computed the fundamental group of the algebroid T M × ℝ with the aid of H_{dR}^{1} (M) .

1 Introduction

The notion of homotopic homomorphisms between vector bundles or principal bundles and the problem of classification of homomorphisms up to homotopy is important in all classical parts of geometry and differential topology where we have to deal with problems of homotopy invariance (classification of bundles, characteristic classes). Vector bundles and principal bundles have their infinitesimal objects – transitive Lie algebroids – which contain the most of information, interesting from the topological point of view. For example, homotopy of mappings can be expressed by these infinitesimal invariants, moreover, characteristic homomorphisms of principal bundles (primary and secondary [5], [7], [8]) can also be expressed in terms of these invariants. We add that the characteristic homomorphisms can also be defined in the category of Lie algebroids or more generally, in the geometry (commutative) of Lie-Rinehart algebras, see [10], [11], [3], [Lie-Rinehart algebras are called sometimes Lie modules or Lie pseudoalgebras].

A homotopy joining two homomorphisms H_{0}, H_{1} : A′ → A of Lie algebroids is defined in J.Kubarski [6]; it is a homomorphism of Lie algebroids H : T R × A′ → A with H (θ_{0}, ·) = H_{0}, H_{2} (θ_{1}, ·) = H_{1}, where θ_{0} ∈ T_{0}ℝ and θ_{1} ∈ T_{1}ℝ, are null vectors (T ℝ × A′ denotes the product of the trivial Lie algebroid T ℝ and A′). The reader can find the definition of a product of Lie algebroids and a homomorphism (not necessarily strong) in [2], [6], [5]. If A′ and A are algebroids on a manifold M, H_{0} and H_{1} are strong homomorphisms between them, then the homotopy H is called strong if H is defined over projection pr_{2} : ℝ × M → M. This definition is well seen if we consider the following case: let h_{0} and h_{1} : P′ → P be two homomorphisms of G-principal bundles and h_{t} – a familly of homomorphisms (a homotopy) joining h_{0} and h_{1}. This family gives a homomorphism of G-principal bundles h : ℝ × P′ → P. If we consider infinitesimal mappings between the algebroids of these bundles (see [9], [4]), h_{∗} : A (ℝ × P′) → A (P), and indentify ℝ × P′ with the cartesian product of the trivial {e}-principal bundle ℝ and G-bundle P′ we obtain (after identification A (ℝ × P′) ≅ T ℝ × A (P′)) a homomorphism h_{∗} : T ℝ × A (P′) → A (P) which is the homotopy joining h_{0∗} and h_{1∗}, in the terms above. We can interpret a homotopy joining linear homomorphisms of vector bundles (assuming isomorphisms on each fibre) in the category of algebroids (by principal bundles of frames or directly), using (1) the algebroid of a vector bundle (see [9], [5]) which is isomorphic to the algebroid of a frame bundle; (2) the homomophism of algebroids induced by a homomorphism of vector bundles [6].

The problem of classification of endomorphisms for a given Lie algebroid up to homotopy had been initiated by B.Balcerzak in [1], where it was introduced the notion of the fundamental group π (A) of a Lie algebroid A (π (A) is a set of homotopy classes of strong automorphisms in the Lie algebroid A). In [1] the author gave an effective classification of endomorphisms of the trivial Lie algebroid T ℝ^{n} × ℝ (on ℝ^{n} with the 1-dimensional abelian isotropy Lie algebra ℝ) and proved that π (T ℝ^{n} × ℝ) ≅ GL (ℝ) . The aim of this paper is homotopic classification of endomorphisms of the trivial Lie algebroid T M ×𝔤 on a manifold M with an isotropy algebra 𝔤 and computation π (T M × ℝ) using the first de Rham cohomology group H_{dR}^{1} (M). The problem of computation π (T M × 𝔤) and more generally, π (A) for any Lie algebroid A, is still open.

2 Endomorphisms of trivial algebroids

2.1 The case of T M × 𝔤

Let M be a connected smooth manifold and 𝔤 a finite dimensional real Lie algebra. Then (T M × 𝔤, [[·, ·]], pr_{1}) is a transitive Lie algebroid with the anchor pr_{1} : T M × 𝔤 → T M and the Lie algebra structure [[·, ·]] in the module of cross-sections Sec (T M × 𝔤) = {(X, σ) ; X ∈ 𝔛(M), σ ∈ C^{∞} (M; 𝔤)} defined by

[[(X, σ), (Y, τ)]] = ([X, Y], 𝔏_{X}(η) − 𝔏_{Y} (σ) + [σ, τ]),

where X, Y ∈ 𝔛(M) and σ, τ ∈ C^{∞} (M; 𝔤). An endomorphism of vector bundles H : T M × 𝔤 → T M × 𝔤 is a homomorphism of Lie algebroids if pr_{1} ∘ H = pr_{1} and Sec H : Sec (T M × 𝔤) → Sec (T M × 𝔤) is a homomorphism of Lie algebras.

Let

π_{ℝ} : ℝ × M ⟶ ℝ, π_{M} : ℝ × M ⟶ M, π : T M ⟶ M

be (in this paper) canonical projections and

j_{t} : M ⟶ ℝ × M, x ⟼ (t, x) , j_{x} : ℝ ⟶ ℝ × M, t ⟼ (t, x) ,

injections.

Theorem 2.1 Each endomorphism H : T M ×𝔤 → T M ×𝔤 of the Lie algebroid T M × 𝔤 is given by

H (u, r) = (u, ω_{π(u)} (u) + ϕ_{π(u)} (r))

for some smooth function ϕ : M → End (𝔤) and 1-form ω ∈ Ω^{1} (M; 𝔤) such that

W1 δω (X, Y) + [ω (X) , ω (Y)] = 0,

W2 ((dϕ) X) (η) = [ϕ (η) , ω (X)] , X, Y ∈ 𝔛 (M) , η ∈ C^{∞} (M; 𝔤) , ϕ (η) : M → 𝔤, x ↦ ϕ_{x} (η_{x}).

ϕ and ω are uniquely determined. H is an automorphism if and only if ϕ_{x} is a linear automorphism of 𝔤 for all x ∈ M. Under this condition there is H^{−1} given by H^{−1} (u, r) = (u, ϕ^{−1} (ω (u)) + ϕ^{−1} (r)) .

Proof. Let H be an endomorphism of the Lie algebroid T M × 𝔤. Since pr_{1} ∘ H = pr_{1}, we obtain H (u, r) = (u, λ (u, r)) , u ∈ T M, r ∈ 𝔤 for a smooth function λ : T M × 𝔤 → 𝔤. Each function λ_{x} : T_{x}M × 𝔤 → 𝔤 is linear and hence λ_{x} (u, r) = ω_{x} (u) + ϕ_{x} (r) for ω_{x} ∈ Hom (T_{x}M; 𝔤) and ϕ_{x} ∈ Hom (𝔤; 𝔤). Thus

H (u, r) = (u, ω_{π(u)} (u) + ϕ_{π(u)} (r))

for some 1-form ω ∈ Ω^{1} (M; 𝔤) and smooth function ϕ : M → Hom (𝔤; 𝔤).

Sec H : Sec (T M × 𝔤) → Sec (T M × 𝔤), ξ ↦ H ∘ ξ is a homomorphism of Lie algebras, so it satisfies H ∘ [[(X, σ) , (Y, η)]] = [[H ∘ (X, σ) , H ∘ (Y, η)]] for all X, Y ∈ 𝔛 (M), σ, η ∈ C^{∞} (M; 𝔤). If σ = η = 0, then we easily obtain ω ([X, Y]) = X (ω (Y)) − Y (ω (X)) + [ω (X) , ω (Y)] , i.e. W1. If X = Y = 0, we have [ϕ (σ) , ϕ (η)] = ϕ ([σ, η]) , so linear endomorphisms ϕ_{x} are endomorphisms of the Lie algebra 𝔤 for all x ∈ M. Finally, for any X ∈ 𝔛 (M), η ∈ C^{∞} (M, 𝔤) and Y = 0, σ = 0 we obtain X (ϕ (η)) + [ω (X) , ϕ (η)] = ϕ (X (η)), hence W2 holds.

Conversely, define a linear endomorphism H : T M × 𝔤 → T M × 𝔤 by

H (u, r) = (u, ω_{π(u)} (u) + ϕ_{π(u)} (r))

for given ϕ : M → End (𝔤) and ω ∈ Ω^{1} (M; 𝔤) which satisfy W1 and W2. This endomorphism is a homomorphism of Lie algebroids because (a) condition pr_{1} ∘ H = pr_{1} holds evidently, (b) Sec H is a homomorphism of Lie algebras. In fact, for X, Y ∈ 𝔛 (M) , σ, η ∈ C^{∞} (M; 𝔤) we compute

[[H ∘ (X, σ) , H ∘ (Y, η)]]
= [[(X, ω (X) + ϕ (σ)) , (Y, ω (Y) + ϕ (η))]]
= ([X, Y] , δω (X, Y) + [ω (X) , ω (Y)]
+ ((dϕ) X) (η) − [ϕ (η) , ω (X)]
− ((dϕ) Y) (σ) + [ϕ (σ) , ω (Y)])
+ϕ ((X (σ)) − ϕ (Y (η)) + ϕ [σ, η] + ω ([X, Y]))
= ([X, Y] , ω ([X, Y]) + ϕ (X (η)) − ϕ (Y (σ)) + ϕ ([σ, η]))
= H ∘ ([X, Y] , X (η) − Y (σ) + [σ, η])
= H ∘ [[(X, σ) , (Y, η)]].

The rest is obvious.

We can easily generalize the theorem above in the case of strong homomorphisms of trivial Lie algebroids with the different isotropy algebras.

Theorem 2.2 Each homomorphism H : T M × 𝔤 → T M × 𝔤′ of Lie algebroids is given by

H (u, r) = (u, ω_{π(u)} (u) + ϕ_{π(u)} (r))

for some smooth function ϕ : M → Hom (𝔤; 𝔤′) and 1-form ω ∈ Ω^{1} (M; 𝔤′) such that

W1’ δω (X, Y) + [ω (X) , ω (Y)] = 0,

W2’ ((dϕ) X) (η) = [ϕ (η) , ω (X)] , X, Y ∈ 𝔛 (M) , η ∈ C^{∞} (M; 𝔤) , ϕ (η) : M → 𝔤, ϕ (η) (x) = ϕ_{x} (η_{x}).

2.2 The case of T M × ℝ

Consider the 1-dimensional abelian Lie algebra 𝔤 = ℝ. It follows from Theorem 2.1 that each strong endomorphism H of the Lie algebroid T M × ℝ is given by

H (u, r) = (u, ω_{π(u)} (u) + ϕ_{π(u)} (r)) , (u, r) ∈ T M × ℝ

for an uniquely determined function ϕ : M → End (ℝ) and closed 1-form ω ∈ Ω^{1} (M) such that ((dϕ) X) (η) = 0 for X ∈ 𝔛 (M) , η ∈ C^{∞} (M). Since ϕ_{x} (r) = B (x) · r, r ∈ ℝ, B ∈ C^{∞} (M), we see that ((dϕ) X) (η) = X (B) · η = 0. In particular, if we take a non-zero η, then X (B) = 0,which means that B is constant. Hence we have

Corollary 2.3 Each strong endomorphism H of the Lie algebroid T M × ℝ is given by

H (u, r) = (u, ω_{π(u)} (u) + B · r) ,

where B is a real number and ω is a closed 1-form on M. H is an automorphism if and only if B ≠ 0; then H^{−1} (u, r) = (u, \frac{1}{B} ω (u) + \frac{1}{B} r) .

Example 2.4 [1, Theorem. 2.1]Consider the special case M = ℝ^{n}. Since the

fact that the 1-form ω = n ; ∑_{i=1}A^{i}dx^{i} ∈ Ω^{1} (ℝ^{n}), A^{i} ∈ C^{∞} (ℝ^{n}) , is closed is

equivalent to

\frac{∂ A^{i}}{∂ x^{j}} = \frac{∂ A^{j}}{∂ x^{i}} , i, j = 1, ..., n,  (2.1)

an endomorphism H of the trivial Lie algebroid T ℝ^{n} ×ℝ has the following form

H (u, r) = (u, ∑^{n} ; i=1A^{i} (x) · u^{i} + B · r)

(u^{i} are coordinates of the vector u with respect to the basis \frac{∂}{∂ x^{i}}), where B ∈ ℝ, and A^{i} ∈ C^{∞} (ℝ^{n}) satisfy condition (2.1).

3 Homotopic classification of endomorphisms of trivial algebroids

3.1 The case of T M × 𝔤

Consider the trivial Lie algebroids T M × 𝔤, T (ℝ × M) × 𝔤 and the product of algebroids T ℝ × (T M × 𝔤) . Mappings

K : T (ℝ × M) × 𝔤 ⟶ T ℝ × (T M × 𝔤) , (u, r) ⟼ (dπ_{ℝ} (u) , (dπ_{M} (u) , r)) ,
L : T (ℝ × M) × 𝔤 ⟶ π_{M}^{∧} (T M × 𝔤) , (u, r) ⟼ (u, (dπ_{M} (u) , r)) ,

are isomorphisms of Lie algebroids, where π_{M}^{∧} (T M × 𝔤) denotes the inverseimage of T M × 𝔤 by π_{M}, see.[5].

Let H_{0}, H_{1} be strong endomorpisms of the Lie algebroid T M × 𝔤. They are given by

H_{i} (u, r) = (u, ω_{π(u)}^{i} (u) + ϕ_{π(u)}^{i} (r)) , i = 0, 1,  (3.1)

where ϕ^{i} : M → End (𝔤), ω^{i} ∈ Ω^{1} (M; 𝔤) and (a) δω^{i} (X, Y) + [ω^{i} (X) , ω^{i} (Y)] = 0,

(b) ((dϕ^{i}) X) (η) = [ϕ^{i} (η) , ω^{i} (X)] , X, Y ∈ 𝔛 (M), η ∈ C^{∞} (M; 𝔤).

The theorem below gives characterization of the homotpy joining these two endomorphisms.

Theorem 3.1 Endomorphisms H_{0} and H_{1} of the trivial Lie algebroid T M × 𝔤 given by (3.1), are homotopic if and only if there exists a mapping ϕ : ℝ × M → End (𝔤) and a 1-form ω ∈ Ω^{1} (ℝ × M; 𝔤) such that

ϕ (0, ·) = ϕ^{0}, ϕ (1, ·) = ϕ^{1}, ω^{0} = j_{0}^{∗}ω , ω^{1} = j_{1}^{∗}ω,

and

δω (X, Y) + [ω (X) , ω (Y)] = 0, ((dϕ) X) (η) = [ϕ (η) , ω (X)] ,  (3.2)

for X, Y ∈ 𝔛 (ℝ × M) , η ∈ C^{∞} (ℝ × M; 𝔤).

Proof. Suppose that H : T ℝ × (T M × 𝔤) ⟶ T M × 𝔤 is a homotopy (over π_{M}) joining endomorphisms H_{0} i H_{1}, that is

H (θ_{0}, ·) = H_{0}, H (θ_{1}, ·) = H_{1},  (3.3)

(θ_{0} ∈ T_{0}ℝ and θ_{1} ∈ T_{1}ℝ are null vectors). H is canonically represented as the superposition H = χ ∘ \overline{H} of the homomorphisms \overline{H} : T ℝ × (T M × 𝔤) → π_{M}^{∧} (T M × 𝔤) and χ : π_{M}^{∧} (T M × 𝔤) → T M ×𝔤, [5], see [6]. Hence the mapping L^{−1} ∘ \overline{H} ∘ K is an endomorphism of the Lie algebroid T (ℝ × M) × 𝔤. It follows from Theorem 2.1 that there exists a mapping ϕ : ℝ × M → End (𝔤) and a 1-form ω ∈ Ω^{1} (ℝ × M; 𝔤) which satisfy (3.2) and

(L^{−1} ∘ \overline{H} ∘ K) (w, r) = (w, ω_{π(w)} (w) + ϕ_{π(w)} (r)) , w ∈ T (ℝ × M) , r ∈ ℝ.

Therefore (\overline{H} ∘ K) (w, r) = (w, (dπ_{M} (w) , ω_{π(w)} (w) + ϕ_{π(w)} (r))) , whence we have \overline{H}_{(t,x)} (v, (u, r)) = (\overline{H} ∘ K) (w, r) for v ∈ T_{t}ℝ, u ∈ T_{x}M and w = dj_{x} (v) + dj_{t} (u) . Since π_{M}^{∧} (T M × 𝔤) ⊂ T (ℝ × M) ⨁ π_{M}^{∗} (T M × 𝔤) and χ is defined as the projection on the second term,

H_{(t,x)} (v, (u, r)) = (u, ω_{(t,x)} (dj_{x} (v) + dj_{t} (u)) + ϕ_{(t,x)} (r)) ,  (3.4)

because \overline{H} (v, (u, r)) = (u, H (v, (u, r))). By (3.3),

(u, ω_{(0,x)} (dj_{0} (u)) + ϕ_{(0,x)} (r)) = (u, ω_{x}^{0} (u) + ϕ_{x}^{0} (r)) ,
(u, ω_{(1,x)} (dj_{1} (u)) + ϕ_{(1,x)} (r)) = (u, ω_{x}^{1} (u) + ϕ_{x}^{1} (r)) ,

hence we have

ϕ_{x}^{0} (r) = ϕ_{(0,x)} (r) ,
ϕ_{x}^{1} (r) = ϕ_{(1,x)} (r) ,
ω_{x}^{0} (u) = ω_{(0,x)} (dj_{0} (u)) = (j_{0}^{∗}ω)_{x} (u) ,
ω_{x}^{1} (u) = ω_{(1,x)} (dj_{1} (u)) = (j_{1}^{∗}ω)_{x} (u) .

Conversely, let ϕ : ℝ × M → End (𝔤) and ω ∈ Ω^{1} (ℝ × M; 𝔤) satisfy (3.2). Then we see that H : T ℝ×(T M × 𝔤) ⟶ T M ×𝔤 defined by (3.4) is a homotopy joining endomorphisms H_{0} and H_{1}.

3.2 The case of T M × ℝ

The general form of strong endomorphisms of the trivial Lie algebroid T M × ℝ gives Corollary (2.3).

Theorem 3.2 Let H_{0}, H_{1} be two endomorphisms of the Lie algebroid T M × ℝ given by

H_{i} (u, r) = (u, ω_{π(u)}^{i} (u) + B_{i} · r) ,  (3.5)

where B_{i} are real numbers and ω^{i} are closed 1-forms on M. The endomorphisms H_{0} and H_{1} are homotopic if and only if B_{0} = B_{1} and the forms ω^{0} and ω^{1} are homologous, [ω^{0}] = [ω^{1}] ∈ H_{dR}^{1} (M).

Proof. Suppose that endomorphisms H_{0} and H_{1} are homotopic. By Theorem 3.1, there is some closed 1-form ω ∈ Ω^{1} (ℝ × M) and mapping ϕ : ℝ×M → End (ℝ) such that ω^{0} = j_{0}^{∗}ω, ω^{1} = j_{1}^{∗}ω and dϕ = 0 with ϕ_{(0,·)} = ϕ^{0} and ϕ_{(1,·)} = ϕ^{1}, where ϕ_{x}^{0} (r) = B_{0} · r, ϕ_{x}^{1} (r) = B_{1} · r for all x ∈ M. Whence ϕ_{(t,x)} (r) = B · r for some B ∈ ℝ and thus B_{0} = B = B_{1}. Since j_{0} and j_{1} are homotopic, [ω^{0}] = j_{0}^{#} [ω] = j_{1}^{#} [ω] = [ω^{1}] .

Conversely, suppose that B_{0} = B_{1} and there is a function f ∈ C^{∞} (M), with ω^{1} −ω^{0} = δf. Define a closed 1-form on ℝ×M by ω = π_{M}^{∗}ω_{0} +δ (π_{ℝ} · (f ∘ π_{M})) . Since (π_{ℝ} · (f ∘ π_{M})) ∘ j_{0} ≡ 0 and (π_{ℝ} · (f ∘ π_{M})) ∘ j_{1} = f, we obtain j_{0}^{∗}ω = ω_{0} and j_{1}^{∗}ω = ω^{1}.

Next, let ϕ : ℝ × M → End (ℝ) be defined by ϕ_{(t,x)} (r) = B · r, where B = B_{0} = B_{1}. Then ϕ_{(0,·)} = ϕ^{0} = ϕ^{1} = ϕ_{(1,·)} and dϕ = 0 which means that H_{0} and H_{1} are homotopic (by Theorem 3.1).

Corollary 3.3 If H_{dR}^{1} (M) = 0 (e.g. M is contractible, in particular if M = ℝ^{n} – see [1]), any two endomorphisms H^{0} and H^{1} of the Lie algebroid T M × ℝ given by (3.5) are homotopic if and only if B_{0} = B_{1}.

4 The fundamental group of T M × ℝ

Let A be a transitive Lie algebroid on a manifold M. Consider the set π (A) of homotopy classes [H], where H : A → A is a strong automorphism of A. π (A) under the binary operation [H] · [G] = [H ∘ G] forms a group (the unity element is the class of the identity on A and the inverse of [H] is [H^{−1}], see [1]). The group π (A) is called the fundamental group of A.

Theorem 4.1 The fundamental group π (T M × ℝ) is isomorphic to the group

(H_{dR}^{1} (M) × (ℝ \ {0}) , ∘) ,

where

([Φ] , a) ∘ ([Ψ] , b) = ([Φ] + a [Ψ] , ab) ,

Φ, Ψ ∈ Ω^{1} (M) , a, b ∈ ℝ \ {0}.

Proof. Let G and H be automorphisms of the algebroid T M × ℝ,

G (u, r) = (u, ω_{π(u)}^{G} (u) + B_{G} · r) , H (u, r) = (u, ω_{π(u)}^{H} (u) + B_{H} · r) ,

where B_{G}, B_{H} are non-zero real numbers and ω^{G}, ω^{H} are closed 1-forms on M. Then

(G ∘ H) (u, r) = (u, ω_{π(u)}^{G} (u) + B_{G} · ω_{π(u)}^{H} (u) , B_{G} · B_{H} · r) ,

(u, r) ∈ T M × ℝ. Hence we see that the mapping

Δ : π (T M × ℝ) ⟶ H_{dR}^{1} (M) × (ℝ \ {0}) , [H] ⟼ ([ω^{H}] , B_{H}) ,

is an isomorphism of the groups π (T M × ℝ) and (H_{dR}^{1} (M) × (ℝ \ {0}) , ∘). Elements of the group H_{dR}^{1} (M) × (ℝ \ {0}) can be represented by the ma-

trices | \begin{matrix}a & φ \\ 0 & 1 \end{matrix} | , a ∈ ℝ \ \{0\} , φ ∈ H_{dR}^{1} (M) , with the standard multiplication

of matrices.

Corollary 4.2 If H_{dR}^{1} (M) = 0 then

π (T M × ℝ) ≅ (ℝ \ {0} , ·) ≅ GL (ℝ) .

It is a generalization of the result from [1].

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