Modular classes of Lie algebroids homomorphisms as Chern–Simons ...

UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLVII 2009

MODULAR CLASSES OF LIE ALGEBROIDS HOMOMORPHISMS AS CHERN–SIMONS FORMS

by Bogdan Balcerzak Abstract. We study the Chern–Simons forms for a pair of some R-linear connections. We show that the modular class of a base-preserving homomorphism of Lie algebroids is such a Chern–Simons form induced by a pair of some R-linear closed 1-forms. We prove that two arbitrary homotopic homomorphism of Lie algebroids have the same modular classes.

1. Preface. In this paper we examine R-linear connections and induced by them Chern–Simons forms on Lie algebroids. In particular we obtain modular classes of homomorphisms of Lie algebroids as a Chern–Simons form for a pair of some R-linear connections. Such connections in the context of Lie algebroids were considered earlier by Evens, Lu and Weinstein in [7] and by Crainic and Fernandes [4, 6]. They discuss so-called representations up to homotopy and non-linear connections. The notion of a representation up to homotopy of a Lie algebroid on a super-complex of vector bundles comes from [7]. Recall that a representation up to homotopy of a Lie algebroid A (over a manifold M ) on a super-complex of vector bundles (E = E0 ⊕ E1 , ∂) is a homomorphism D : Γ (A) → CDO (E) of R-Lie algebras acting to the module of sections of the Lie algebroid of E such that for all a ∈ Γ (A) and f ∈ C ∞ (M ), Da preserves the grading and commutes with the action of ∂ and Df a = f Da + [I (a, f ) , ∂]s for some bundle map I (a, f ) : E → E of degree 1, and where [·, ·]s denotes the super-commutator in the super-Lie algebra End Γ (E). Studying exotic characteristic classes of Lie algebroids, Crainic and Fernandes introduced non-linear connections on a super-vector bundle as local R-linear connections on the Lie algebroid of the vector bundle, which preserve the grading. Moreover, they consider non-linear forms on Lie algebroids as R-multilinear antisymmetric local maps (see [4, 6]).

12

In the paper we extend these notions to R-linear connections understood as R-linear operators ∇ : Γ (A) → Γ (B) preserving the anchors of Lie algebroids A and B (both over the same base manifold) and (not necessarily local) Rlinear forms. If A and B are Lie algebroids on the same manifold M , then an A-connection in B is a homomorphism of vector bundles D : A → B commuting with the anchors (see [1]). Therefore, D induces on sections the C ∞ (M )-linear operator Sec D : Γ (A) → Γ (B) preserving anchors. For this reason the notion of an R-linear connection covers (linear) connections of Lie algebroids. The modular class of a Lie algebroid first appeared in [18] and was studied by Evens, Lu and Weinstein in [7]. There is an adjoint representation up to homotopy for every Lie algebroid (see [7]). The modular class of a Lie algebroid is a cohomology class of a Chern–Simons forms of the adjoint representation up to homotopy and its metric connection (see [5, 6, 8]). The idea of the modular class of a base-preserving homomorphisms of Lie algebroids was proposed by Grabowski, Marmo and Michor in [9] and was discussed by Kosmann-Schwarzbach and Weinstein in [12]. Using the Grabowski–Marmo– Michor approach, we show that the modular class of a homomorphism of Lie algebroids is equal to the Chern–Simons form for a pair of connections determined by some distinguished divergences. The notion of the modular class of arbitrary homomorphism of Lie algebroids is proposed by Kosmann-Schwarzbach, Laurent-Gengoux and Weinstein in [13] (and where it is called a relative modular class). In the paper we study these generalized modular classes of homotopic homomorphism of Lie algebroids and show their homotopic invariance. The notion of a homotopy joining two homomorphisms of Lie algebroids was first introduced by Kubarski in [15] (see also [2]). Using the existence of a homotopy operator joining pullbacks induced by homotopic homomorphisms, we prove that two homomorphisms of Lie algebroids (not necessarily over the identity) have the same modular classes. 2. R-linear connections and R-linear forms on Lie algebroids. A Lie algebroid is a triple (A, ρA , [[·, ·]]A ), where A is a vector bundle over a manifold M , ρA : A → T M is a homomorphism of vector bundles called an anchor, (Γ (A) , [[·, ·]]A ) is an R-Lie algebra and the following Leibniz identity holds [[a, f · b]]A = f · [[a, b]]A + ρA (a) (f ) · b for all a, b ∈ Γ (A), f ∈ C ∞ (M ). The representation % : C ∞ (M ) −→ EndC ∞ (M ) (Γ (A)) , % (ν) (a) = ν · a,

13

ν ∈ C ∞ (M ) , a ∈ Γ (A), is faithful (ker % = 0), which implies (see [10]) that Sec ρA : Γ (A) → X (M ), a 7→ ρA ◦ a, is a homomorphism of Lie algebras. Let (A, ρA , [[·, ·]]A ) and (B, ρB , [[·, ·]]B ) be Lie algebroids over the same manifold M . An R-linear connection of A in B is an R-linear operator ∇ : Γ (A) → Γ (B) such that Sec ρB ◦ ∇ = Sec ρA , where Sec ρA : Γ (A) → X (M ) and Sec ρB : Γ (B) → X (M ) are induced by anchors in such way that Sec ρA (a) = ρA ◦ a, Sec ρB (b) = ρB ◦ b for all a ∈ Γ (A), b ∈ Γ (B). An R-linear connection ∇ : Γ (A) → Γ (B) in B is called an A-connection if it is also C ∞ (M )-linear operator. In particular, if E is a vector bundle over M and B = A (E) is a Lie algebroid of the vector bundle E we obtain connections of A on E. The module CDO (E) of sections of the Lie algebroid A (E) is the space of all covariant differential operators in E, i.e. R-linear operators L : Γ (E) → Γ (E) such e ∈ X (M ) with L (f s) = f L (s) + L e (f ) s for all that there exists exactly one L ∞ f ∈ C (M ) and s ∈ Γ (E); see for example [14, 16, 17]. The notion of an A-connection in B coincides with the notion of a connection introduced in [1]. Moreover, the definition of an R-linear connection covers so-called non-linear connections (see [4, 6]). Let (A, ρA , [[·, ·]]A ) be a Lie algebroid on a manifold M . An R-multilinear antisymmetric map ω : Γ (A)×· · ·×Γ (A) → C ∞ (M ) is called an R-linear form on A. The space of all R-linear n-forms onLA we will denote by ΩnR (A) and the ΩnR (A), where Ω0R (A) = C ∞ (M ). space of R-linear forms on A by ΩR (A) = n≥0

Let E be a vector bundle over M . The space of all R-n-multilinear antisymmetric maps θ : Γ (A) × · · · × Γ (A) → Γ (E) will be denoted by ΩnR (A; E), L k ΩR (A; E). Consider an Ω0R (A; E) = C ∞ (M ), and by ΩR (A; E) the space k≥0

R-linear connection ∇ : Γ (A) → CDO (E) of A in the algebroid of a vector bundle E. For ∇ we define a 2-form R∇ ∈ Ω2R (A; End E) by R∇ (α, β) = ∇α ◦ ∇β − ∇β ◦ ∇α − ∇[[α,β]]A for all α, β ∈ Γ (A), which is called a curvature of ∇. If the curvature of a connection ∇ is equal to zero, we say that ∇ is flat. Every flat R-connection ∇ : Γ (A) → CDO (E) on E induces an differential operator •+1 • d∇ R : ΩR (A; E) −→ ΩR (A; E)

14

defined by the classical Chevalley–Koszul–Eilenberg formula n+1 X
d∇ ω (a , . . . , a ) = (−1)i+1 ∇ai (ω (a1 , . . . ˆı . . . , an+1 )) 1 n+1 R i=1

+

X

(−1)i ω (a1 , . . . ˆı . . . , aj−1 , [[ai , aj ]]A , . . . , an+1 )

i