Locally conformal Dirac structures and infinitesimal automorphisms

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1 Locally conformal Dirac structures and infinitesimal automorphisms Aïssa Wade Department of Mathematics, The Pennsylvania State University University Park, PA Abstract We study the main properties of locally conformal Dirac bundles, which include Dirac structures on a manifold and locally conformal symplectic manifolds. It is proven that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Furthermore we show that, given a locally conformal Dirac bundle over a smooth manifold M, there is a Lie homomorphism between a subalgebra of the Lie algebra of infinitesimal automorphisms and the Lie algebra of admissible functions. We also show that Dirac manifolds can be obtained from locally conformal Dirac bundles by using an appropriate covering map. Finally, we extend locally conformal Dirac bundles to the context of Lie algebroids. 1 Introduction Dirac structures generalize foliations, Poisson structures, and pre-symplectic 2-forms. They were originally studied by Courant and Weinstein (see [C] and [CW]), who showed their connection with constraint Hamiltonian systems. On the other hand, locally conformal symplectic manifolds play an important role in the theory of Jacobi manifolds and in physics. Indeed, locally conformal symplectic manifolds may be considered as phase spaces of Hamiltonian systems (see [V]). They occur, for instance, in the study of Gaussian isokinetic dynamics (see [WL]). Motivated by these facts, we introduce the notion of a locally conformal Dirac bundle which extends that of a Dirac structure on a manifold. The purpose of this paper is to investigate the basic properties of locally conformal Dirac bundles. 1

2 A locally conformal Dirac bundle over M is a sub-bundle of the vector bundle T M T M M which is maximally isotropic with respect to the canonical symmetric bilinear operation on T M T M and such that there exist an open cover U = (U i ) i I of M and smooth positive functions f i : U i R for which the sub-bundle L i = {(X Ui, f i α Ui ) (X, α) L} T U i T U i is integrable in the sense that the space Γ(L i ) of smooth sections of L i is closed under the Courant bracket for all i I (see Section 2 for the definition of the Courant bracket). To any locally conformal Dirac bundle L T M T M, there corresponds a skew-symmetric bilinear map Ω L on L which induces a 2-form on the distribution ρ(γ(l)) = {X χ(m) (X, α) Γ(L)}: Ω L (X, Y ) = α(y ), for any (X, α), (Y, β) Γ(L). Any locally conformal Dirac bundle L over M induces a singular foliation whose leaves are locally conformal pre-symplectic manifolds. Precisely, the foliation is determined by the distribution ρ(γ(l)). If L is a locally conformal Dirac bundle L such that the rank of Ω L is greater than 2 at every point x M then there exists a unique 1-cocycle ω L satisfying dω L = ω L Ω L, where d denotes the derivative along the foliation defined by ρ(γ(l)). The 1-cocycle ω L is called the Lee 1-form of L. There are two fundamental Lie algebras associated with a locally conformal Dirac bundle L whose 2-form Ω L has a rank r(x) > 2 at every point: (i) The Lie algebra of L-admissible functions. (ii) The Lie algebra A(L) of infinitesimal automorphisms. Given a locally conformal Dirac bundle L with Lee 1-form ω L, we show that if there exists a vector field Z such that (Z, ω L ) Γ(L) then there is a Lie homomorphism between a subalgebra of A(L) and the Lie algebra of L-admissible functions. We then recover results given in [B1], [GL], and [V]. Theorem 5.5 shows that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Two locally conformal Dirac bundles L 1, L 2 T M T M are conformally equivalent if ρ(γ(l 1 )) = ρ(γ(l 2 )) and there exists a nowhere vanishing function f such that Ω L1 = fω L2, 2

3 where ρ : Γ(L i ) χ(m) is the canonical projection for i = 1, 2. An equivalence class of such locally conformal Dirac bundles is called a locally conformal Dirac structure on M. We show that, for a large family of locally conformal Dirac structures D, the extended Lee homomorphism does not depend on the locally conformal Dirac bundle representing D. One can extend the concept of a locally conformal Dirac bundle to more general Lie algebroids. This idea was already suggested in [Wa], where we study conformal E 1 (M)-Dirac structures in connection with Jacobi manifolds. The last section of this paper is devoted to the generalization of locally conformal Dirac bundles to the context of Lie algebroids. The paper is organized as follows. In Section 2, we recall some known results concerning Dirac structures on manifolds. We give, in Section 3, the characterizations of locally conformal Dirac bundles over a smooth manifold M (see Theorem 3.5). We present some examples of locally conformal Dirac bundles in Section 4. In Section 5, we introduce the concept of an admissible function relative to a given locally conformal Dirac bundle L. Then, we show that the space of admissible functions is a Lie algebra. Moreover, we prove that a reducible locally conformal Dirac bundle induces a Jacobi bracket on a quotient manifold. In Section 6, we develop the properties of the infinitesimal automorphisms of a locally conformal Dirac bundle L. We also establish Theorems 6.5 and 6.6. Finally, in Section 7, we consider the generalized locally conformal Dirac structures. 2 Dirac structures Let M be a smooth manifold. We denote by.,. + the canonical symmetric bilinear operation on the vector bundle T M T M M. This induces a symmetric C -bilinear operation on the space of sections of T M T M given by: (X 1, α 1 ), (X 2, α 2 ) + = 1 2 (i X 2 α 1 + i X1 α 2 ), for any (X 1, α 1 ), (X 2, α 2 ) Γ(T M T M). Definition 2.1 ( [C]) An almost Dirac structure on M is a vector subbundle of T M T M M which is maximally isotropic with respect to.,. +. Every almost Dirac structure L on M induces a skew-symmetric C -bilinear map Ω L : ρ(γ(l)) ρ(γ(l)) C (M), where ρ is the canonical projection 3

4 of Γ(L) onto the Lie algebra of vector fields χ(m). Precisely, Ω L is given by (1) Ω L (X, Y ) = α(y ), for any (X, α), (Y, β) Γ(L). In fact Ω L is well-defined since L is isotropic. Consider the Courant bracket on Γ(T M T M), that is, the skew-symmetric R-bilinear operation defined by: [(X 1, α 1 ), (X 2, α 2 )] C = ([X 1, X 2 ], L X1 α 2 L X2 α d(i X 2 α 1 i X1 α 2 )), where L X = d i X + i X d is the Lie derivation by X. This bracket does not satisfy the Jacobi identity. Definition 2.2 ( [C]) A Dirac structure L on M is an almost Dirac structure which is integrable in the sense that the space of all sections of L is closed under the Courant bracket. In this case, the pair (M, L) is called a Dirac manifold. Basic examples of Dirac manifolds are foliations, pre-symplectic and Poisson manifolds. Now, we review some properties of Dirac manifolds that will be used later. Proposition 2.3 ( [C]) Let L be an almost Dirac structure on M. If L is integrable then dω L = 0, where Ω L is the skew-symmetric C -bilinear map defined as in (1). This proposition is an immediate consequence of the following lemma. Lemma 2.4 ( [C]) Let L be a Dirac structure on M. For any e 1 = (X, α), e 2 = (Y, β), e 3 = (Z, γ) in Γ(L), we have 2 [e 1, e 2 ] C, e 3 + = dω L (X, Y, Z). The proof of 2.4 is straightforward. It can be found in [C]. Remark 1. This lemma is still valid if one weakens the hypothesis of Lemma 2.4 by assuming that L is an almost Dirac structure on M such that the canonical projection of Γ(L) onto χ(m) defines an involutive distribution. Corollary 2.5 ( [C]) A Dirac structure on M gives rise to a singular foliation by pre-symplectic leaves. 4

5 Proof: Clearly, if L is a Dirac structure on M then the canonical projection of Γ(L) onto χ(m) defines an involutive distribution. In other words, we have a singular foliation in the sense of Stefan and Sussmann. Moreover, Ω L is closed on each leaf by Proposition 2.3. Hence, each leaf is a pre-symplectic manifold. 3 Locally conformal Dirac bundles Definition 3.1 An almost Dirac structure L on M is a locally conformal Dirac bundle over M if there exist an open cover (U i ) i I of M and smooth positive functions f i : U i R such that L i = {(X Ui, f i α Ui ) (X, α) L} is a Dirac structure on U i. We say that L is a globally conformal Dirac bundle if there exists a smooth positive function f : M R such that L = {(X, f α) (X, α) L} is a Dirac structure on M. Remark 2. a. An immediate consequence of Definition 3.1 is that if L is a locally conformal Dirac bundle then the canonical projection of Γ(L) onto χ(m) defines an involutive distribution. Thus, a locally conformal Dirac bundle over M induces a singular foliation. However, the leaves are not, in general, pre-symplectic manifolds. b. If L is an almost Dirac structure whose 2-form Ω L vanishes then (X, fα) L, for any (X, α) L and for any f C (M). In other words, one obtains again L by scaling the second component. It follows from Remark 1 that L is a Dirac structure if and only if the canonical projection of Γ(L) onto χ(m) defines an involutive distribution. This kind of Dirac structure is called a null Dirac structure. We will not consider null Dirac structures here. Proposition 3.2 Suppose that L together with the open cover U = (U i ) i I of M and the smooth positive functions f i : U i R defines a locally conformal Dirac bundle over M. Denote by ρ the canonical projection of Γ(L) onto χ(m) and Ω L the skew-symmetric 2-form on the C (M)-module ρ(γ(l)) defined as in (1). If the rank of Ω L is greater than 2 at every point x M then there exits a unique C -linear map ω L : ρ(γ(l)) C (M) such that dω L = ω L Ω L and dω L = 0. The proof of this proposition is based on the following lemma: 5

6 Lemma 3.3 Consider an almost Dirac structure L on M. Assume the rank of the corresponding 2-form Ω L is greater than 2 at every point. If ξ : ρ(γ(l)) C (M) is a C -linear map such that then ξ = 0. (ξ Ω L )(X, Y, Z) = 0, X, Y, Z ρ(γ(l)) Proof: Pick a point x in M. If ξ does not vanish at x, then we can choose u 1,, u k such that {ξ(x), u 1,, u k } is a basis for the dual of the vector space ρ(l x ) T x M. Thus, we have Ω L (x) = ξ(x) n d i u i + c ij u i u j = ξ(x) θ + c ij u i u j. i<j i<j i=1 Using the fact that ξ Ω L = 0, we get c ij = 0, for all i, j. This implies Ω L (x) = ξ(x) θ, which is not possible since the rank of Ω L (x) is greater than 2. Therefore, ξ = 0 on M. Proof of Proposition 3.2: Existence of ω L : Set ω L = (d ln f i ) ρ(γ(l)) on each U i. Here (d ln f i ) ρ(γ(l)) denotes the restriction of d ln f i to the leaves of the foliation associated with L. We have to show that d(ln f i ln f j ) ρ(γ(l)) = 0 on U i U j when this intersection is not empty. By definition L i = {(X Ui, f i α Ui ) (X, α) L} is a Dirac structure on U i. Therefore Proposition 2.3 ensures that dω Li = 0. But, Ω Li = f i Ω L on U i. This gives ( ) 0 = d(ω Li ) = f i (d ln f i ) ρ(γ(l)) Ω L + dω L on U i, i I. It follows (d ln f i ) ρ(γ(l)) Ω L + dω L = 0 on U i, i I. Consequently, for two different open sets U i and U j such that U i U j is not empty, one has (d ln f i d ln f j ) ρ(γ(l)) Ω L = 0, on U i U j. Using Lemma 3.3, we obtain (d ln f i d ln f j ) ρ(γ(l)) = 0 on U i U j. This shows that ω L is well-defined. Uniqueness of ω L : Assume that dω L = ω 1 Ω L = ω 2 Ω L Then (ω 1 ω 2 ) Ω L = 0. 6

7 By Lemma 3.3, we obtain ω 1 = ω 2 at every point x M. Definition 3.4 If L is a locally conformal Dirac bundle over M such that the rank of Ω L is greater than 2 at every point x M then its corresponding 1-form ω L : ρ(γ(l)) C (M) is called the Lee 1-form of L. Let L be an almost Dirac structure on M and let ω : ρ(γ(l)) C (M) be a C -linear map. On Γ(L), we define: [(X 1, α 1 ), (X 2, α 2 )] ω C = ([X 1, X 2 ], L X1 α 2 L X2 α d(i X 2 α 1 i X1 α 2 ) Define the 3-tensor + (i X1 ω)α 2 (i X2 ω)α (i X 2 α 1 i X1 α 2 )ω). T (e 1, e 2, e 3 ) = 2 [e 1, e 2 ] ω C, e 3 +, Now, we can state our first main result: e 1, e 2, e 3 Γ(L) Theorem 3.5 Let η be a closed 1-form on a smooth manifold M. Let L be an almost Dirac structure on M such that the distribution ρ(γ(l)) is involutive. Assume that the rank of Ω L is greater than 2 at every point. Then the following statements are equivalent: 1. L is a locally conformal Dirac bundle with Lee 1-form ω = η ρ(γ(l)) 2. (dω L + ω Ω L )(X, Y, Z) = 0 for any vector fields X, Y, Z ρ(γ(l)). 3. The set of all sections of L is closed under [, ] ω C. 4. T (e 1, e 2, e 3 ) = 0 for any e 1, e 2, e 3 Γ(L). Proof: Proposition 3.2 shows 1 2. Now assume that Statement 2 is true. By Poincaré Lemma, there exist an open cover (U i ) i I of M and smooth positive functions f i : U i R such that η = d ln f i on the open set U i. Let ω = (d ln f i ) ρ(γ(l)) on U i. Define L i = {(X Ui, f i α Ui ) (X, α) L}. It is clear that L i is an almost Dirac bundle. Moreover, the 2-form corresponding to L i is Ω Li = f i Ω L Ui on U i. We have: dω Li = (df i ) ρ(γ(l)) Ω L Ui + f i dω L Ui = f i (ω Ω L Ui + dω L Ui ) = 0. 7

8 Using Remark 1, one deduce that L i is integrable. This proves that 2 1. Clearly, Statements 3 and 4 are equivalent. We are going to show that 2 4. Consider three smooth sections e i = (X i, α i ), i = 1, 2, 3 Γ(L). Then T (e 1, e 2, e 3 ) = 2 [e 1, e 2 ] ω C, e 3 + = [X 1, X 2 ], α 3 + X 3, L X1 α 2 L X2 α d(i X 2 α 1 i X1 α 2 ) + X 3, (i X1 ω)α 2 (i X2 ω)α (i X 2 α 1 i X1 α 2 )ω. This can be written in the form ( ) T (e 1, e 2, e 3 ) = Ω L (X 3, [X 1, X 2 ])+X 1 Ω L (X 2, X 3 )+c.p +(ω Ω L )(X 1, X 2, X 3 ). Thus, this is equivalent to the following equation We deduce that 2 4. T (e 1, e 2, e 3 ) = (dω L + ω Ω L )(X 1, X 2, X 3 ) Remark 3. If one weakens the hypothesis of Theorem 3.5 by removing the fact that the Lee 1-form ω comes from a differential 1-form η Γ(T M) then one gets In fact, Poincaré Lemma applies to regular foliations but it is not always valid for singular foliations. 4 Examples and Applications 4.1 Locally conformal pre-symplectic manifolds A locally conformal pre-symplectic manifold is a manifold M endowed with a pair (Ω, ω), where Ω is a differential 2-form and ω is a closed 1-form such that dω = ω Ω. When Ω is non-degenerate, this definition is equivalent to that of a locally conformal symplectic manifold (see [B1], [Li1], [V]). It is easy to check that (M, Ω, ω) is a locally conformal pre-symplectic manifold if and only if the graph of Ω is a locally conformal Dirac bundle over M having ω as a Lee 1-form. As an immediate consequence of Proposition 3.2, one has: Corollary 4.1 Consider a locally conformal Dirac bundle L T M T M such that the rank of Ω L is greater than 2 at every point. Then L induces a foliation by locally conformal pre-symplectic manifolds. 8

9 4.2 Examples related to Jacobi manifolds A Jacobi structure on a manifold M is given by a pair (π, E) formed by a bivector field π and a vector field E such that ( [Li1]) [E, π] SN = 0, [π, π] SN = 2E π, where [, ] SN is the Schouten-Nijenhuis bracket on the space of multi-vector fields. A manifold endowed with a Jacobi structure is said to be a Jacobi manifold. When E is zero, we get a Poisson structure. After recalling the definition of a Jacobi manifold, we are going to give examples of locally conformal Dirac bundles which are related to Jacobi manifolds. Proposition 4.2 Let π be a bivector field on M whose rank is greater than 2 at every point. Let ω be a closed 1-form on M. Define the vector field E = π(ω) and L π = {(πα, α), α T M}. Then L is a locally conformal Dirac bundle over M with Lee 1-form ω ρ(γ(l)) if and only if (π, E) is a Jacobi structure on M. To prove this proposition, we will use the following lemmas: the skew- Lemma 4.3 Let π be a bivector field on M. We denote by {, } π symmetric operation 1-forms defined by: {α, β} π = L πα β L πβ α d(π(α, β)), for any 1-forms α and β on M. Then 1 2 [π, π] SN (α, β) = π{α, β} π [πα, πβ]. This lemma is proven for instance in [KM]. Lemma 4.4 If the distribution generated by the vector fields of the form πα is involutive then we have (dω L + ω ρ(γ(l)) Ω L )(πα, πβ, πγ) = ( 1 2 [π, π] SN + E π)(α, β, γ), where Ω Lπ is the skew-symmetric 2-form associated with L π and ρ : Γ(L π ) χ(m) is the canonical projection. 9

10 Proof: Let e α = (πα, α), e β = (πβ, β), and e γ = (πγ, γ) be in Γ(L). On the one hand, by Remark 1, one gets On the other hand, Therefore, dω L (πα, πβ, πγ) = 2 [e α, e β, ], e γ + = [πα, πβ], γ + πγ, {α, β} π = [πα, πβ] π{α, β} π, γ = 1 [π, π](α, β, γ). 2 ω ρ(γ(l)) Ω L ((πα, πβ, πγ) = πα, ω Ω L (πβ, πγ) + c.p = i E α π(α, β) + c.p. = (E π)(α, β, γ). (dω L + ω ρ(γ(l)) Ω L )(πα, πβ, πγ) = ( 1 2 [π, π] SN + E π)(α, β, γ). Proposition 4.2 is an immediate consequence of Lemma 4.4 and Theorem 3.5. From now on, we will only consider locally conformal Dirac bundles L such that the rank of Ω L (x) is greater than 2 at every point x M. 5 Admissible functions Definition 5.1 Let L be a locally conformal Dirac bundle over M with Lee 1-form ω. We say that a smooth function f on M is L-admissible if there exists a vector field X f on M such that (X f, d ω f) is a smooth section of L, where d ω f = df + fω. On the space of all L-admissible functions, we define the bracket {, } : (2) {f, g} = X f g + gω(x f ), for any (X f, d ω f), (X g, d ω g) Γ(L). This is well-defined and skew-symmetric since L is isotropic. We have the following result: Lemma 5.2 Let L be a locally conformal Dirac bundle over M with Lee 1-form ω. If e f = (X f, d ω f) and e g = (X g, d ω g) are in Γ(L) then [e f, e g ] ω C = ([X f, X g ], d ω {f, g}). 10

11 Proof: To simplify the proof, we use the deformed Lie differential L ω X = i X d ω + d ω i X, where d ω µ = dµ + ω µ. We get a simpler expression for the bracket [, ] ω C : [(X 1, α 1 ), (X 2, α 2 )] ω C = ([X 1, X 2 ], L ω X 1 α 2 L ω X 2 α dω (i X2 α 1 i X1 α 2 )), for any (X 1, α 1 ), (X 2, α 2 ) Γ(L). Assume that e f = (X f, d ω f) and e g = (X g, d ω g) are in Γ(L). Using the fact that L ω X dω = d ω L ω X, we get [e f, e g ] ω C = ([X f, X g ], L ω X f d ω g L ω X g d ω f + d ω ({f, g})) = ([X f, X g ], d ω L ω X f g) = ([X f, X g ], d ω {f, g}). Theorem 5.3 Let L be a locally conformal Dirac bundle over M with Lee 1-form ω. Then, the space of all L-admissible functions is a Lie algebra. Proof: For any e f = (X f, d ω f), e g = (X g, d ω g), and e h = (X h, d ω h) in Γ(L), we have 2 [e f, e g ] ω, e C h + = [X f, X g ], d ω h + X h, d ω {f, g} = L ω [X f,x g ] h + {h, {f, g}}. Applying the relation L ω [X f,x g] = Lω X f L ω X g L ω X g L ω X f, we obtain 2 [e f, e g ] ω, e C h + = L ω X f L ω X g h L ω X g L ω X f h + {h, {f, g}} = {f, {g, h}} {g, {f, h}} + {h, {f, g}}. By Theorem 3.5, we have 0 = [e f, e g ] ω C, e h + = 1 2 ( ) {f, {g, h}} {g, {f, h}} + {h, {f, g}}. Thus, the Jacobi identity is satisfied. Furthermore, the bracket {, } is R- bilinear and skew-symmetric. Hence, the set of all L-admissible functions is a Lie algebra. 11

12 Lemma 5.4 Let L be a locally conformal Dirac bundle over M with Lee 1-form ω. Suppose there exists a vector field Z such that (Z, ω) Γ(L) then the product f g of two L-admissible functions is L-admissible. Moreover the bracket (2) gives {fg, h} = f{g, h} + g{f, h} + fgω(x h ), for any e f = (X f, d ω f), e g = (X g, d ω g), and e h = (X h, d ω h) in Γ(L). Proof: The condition (Z, ω) Γ(L) means that the constant functions are L-admissible. Furthermore, fe g + ge f (fgz, fgω) = (fx g + gx f fgz, d ω (fg)) Γ(L). Hence fg is L-admissible. We have: There follows the lemma. {fg, h} = d ω h(fx g + gx f fgz) = f{g, h} + g{f, h} fg(d ω h(z)) = f{g, h} + g{f, h} + fgω(x h ). Remark 4. From Lemma 5.4, on gets the following relation: {fg, 1} = f{g, 1} + g{f, 1}, e f, e g Γ(L) This shows that there exists a unique vector field E such that {, 1} = E. Let L be a locally conformal Dirac bundle over M. The distribution ϱ(l T M) is called the characteristic distribution of L. It is clear that ϱ(l T M) is involutive. Therefore, it determines a singular foliation F L on M. Following [LWX], we say that L is reducible if F L is simple, i.e. M/F L is a smooth manifold and the projection π : M M/F L is a submersion. Here is our second main result: Theorem 5.5 Let L be a reducible locally conformal Dirac bundle over M with Lee 1-form ω. Assume that there exists a vector field Z such that (Z, ω) Γ(L). Then L induces a Jacobi structure on M/F L. 12

13 Proof: Since (Z, ω) Γ(L), one has d ω f(x) = df(x) + fω(x) = df(x), X L T M Hence f is L-admissible if and only if f is constant along F L. So, we can identify the set of all L-admissible functions with C (M/F L ). Lemma 5.4 ensures that the bracket is local. Applying Theorem 5.3, we conclude that (2) defines a Jacobi bracket on C (M/F L ). Let pr : T M T M/L T M be the canonical projection. We set E = pr(z). Define a bivector field Π M/F L as follows: Π(df, ) = pr(x f ) fe for all (X f, d ω f) Γ(L). Then (Π, E) defines the Jacobi structure on M/F L. Remark 5. Theorem 5.5 is a generalization of a result given in [C] (see also [LWX]). Precisely, Courant proved that a Dirac structure L on a smooth manifold M induces a Poisson bracket on a quotient space of the type M/F, where F is a foliation associated with L. It may be valuable to recall the standard description of the Dirac reduced phase space in relation with Courant s result. Let M be a finite-dimensional smooth manifold. Its cotangent bundle P = T M together with the canonical symplectic form Ω = dη is the phase space of a mechanical system. Consider k functions H 1,..., H k C (P ). The constraint hypersurface is the subset N of P defined by the equations H u = 0, u = 1, 2,..., k. We assume that N is a smooth submanifold of P of codimension k. The symplectic form Ω induces a closed 2-form Ω N = i Ω, where i : N P is the inclusion. If, in addition, Ω N has a constant rank then the characteristic distribution C = ker(ω N ) defines a regular foliation on N. Suppose that the foliation is simple then there is a symplectic form on N/C induced by Ω N. The quotient N/C is called the space of physical states. One may replace the symplectic 2-form by a locally conformal symplectic 2-form on the phase space. In this case, the induced 2-form on N/C is a locally conformal symplectic 2-form (see for instance [MR] for more details on the reduction of Poisson manifolds). Here the quotient space obtained in Theorem 5.5 plays the role of the space of physical states. The converse of Theorem 5.5 will be studied elsewhere. We close this section by reminding that conformal symplectic 2-forms are also relevant to statistical mechanics (see [BFLS] and [Li2]). 13

14 6 The extended Lee homomorphism We start this section by introducing the notion of a pull-back of an almost Dirac structure by a surjective submersion (see [LWX]). This notion allows to better understand the Lie algebra of infinitesimal automorphisms of a locally conformal Dirac structure. Let V and W be two vector spaces over R. Given a surjective linear map Φ : V W, its dual Φ : W V is injective. Let L W be a maximally isotropic subspace of W W. Then L V = {(x, Φ ξ) x V, ξ W, (Φ(x), ξ) L W } is a maximally isotropic subspace of V V, which is called the pull-back of L W by the map Φ. This definition can be extended to the context of vector bundles. Namely, if E M and F N are two vector bundles, and Φ : E F a surjective bundle map over Ψ : M N. Then, one defines the pull-back L E of an almost Dirac structure L F by Φ as its fiberwise pull-back. Lemma 6.1 Let ϕ : M N be a surjective submersion and let L be a locally conformal Dirac bundle over N. Denote by L ϕ the pull-back of L by ϕ. Then Ω Lϕ = ϕ Ω L. Proof: For any x M, (X, ϕ α) and (Y, ϕ β) Γ(L ϕ ), we have This shows that Ω Lϕ = ϕ Ω L. Ω (X, Y )(x) L ϕ = X, ϕ β (x) = ϕ X, β (ϕ(x)) = Ω L (ϕ X, ϕ Y )(ϕ(x)) = (ϕ Ω L )(X, Y )(x). Definition 6.2 Two locally conformal Dirac bundles L 1 and L 2 over M with respective Lee 1-forms ω 1 and ω 2 are conformally equivalent if there exits a smooth function a on M such that a(x) 0 at every point x M and L 2 = {(X, aα) (X, α) L 1 }. This gives Ω L2 = aω L1 and ω 2 = ω 1 d(ln a ) ρ(γ(l)). The equivalence class of a locally conformal Dirac bundle L is called a locally conformal Dirac structure on M and denoted by D = [L]. The pair (M, D) is called a locally conformal Dirac manifold. 14

15 One can notice that if two locally conformal Dirac bundles are conformally equivalent then they have the same foliation. Let D = [L] be a locally conformal Dirac structure on M. A diffeomorphism ϕ is in Diff (M, D) if the pull-back of L by ϕ is conformally equivalent to L. Therefore, there exists a smooth function a L,ϕ : M R such that ϕ Ω L = a L,ϕ Ω L. Clearly, this definition does not depend on the locally conformal Dirac bundle L representing D. Moreover, any ϕ Diff (M, [L]) preserves the foliation associated with L. We have the following definition: Definition 6.3 An infinitesimal automorphism of D = [L] is a smooth vector field X χ(m) satisfying the conditions 1. [X, ρ(γ(l))] ρ(γ(l)); 2. L X Ω L = u X Ω L, where u X is a smooth function on M. Here we use the formula (L X Ω L )(Y 1, Y 2 ) = X Ω L (Y 1, Y 2 ) Ω L ([X, Y 1 ], Y 2 ) Ω L (Y 1, [X, Y 2 ]), for (Y i, α i ) Γ(L). We denote by χ(m, D) the space of all infinitesimal automorphisms of D. Let T (M, D) be the subspace formed by elements of χ(m, D) which are contained in ρ(γ(l)). It is easy to check that χ(m, D) is a Lie algebra. Moreover, T (M, D) is a subalgebra of χ(m, D). One can observe that, when M is compact, the flow φ X t of any infinitesimal automorphism X preserves D, i.e. φ X t Diff (M, D) for all t R. Proposition 6.4 Let L be a locally conformal Dirac bundle with Lee 1-form ω. Denote χ 0 (M, L) = {X χ(m) L X Ω L = 0 and [X, ρ(γ(l))] ρ(γ(l))}. Then X is in χ 0 (M, L) if and only if ([X, Y ], L X β) is in Γ(L) for any (Y, β) Γ(L). Proof: Suppose [X, ρ(γ(l))] ρ(γ(l)). Pick (Y, β), (Z, γ) Γ(L) then 2 ([X, Y ], L X β), (Z, γ) + = γ([x, Y ]) + L X β(z) = γ([x, Y ]) + X β(z) β([x, Z]) 15

16 There follows the proposition. = Ω(Z, [X, Y ]) + X Ω(Y, Z) Ω(Y, [X, Z]) = (L X Ω)(Y, Z). Now, we can state our third main theorem, which is a generalization of results proven in [GL], [V], and [B1]. Theorem 6.5 Let L be a locally conformal Dirac bundle with Lee 1-form ω. Assume that there exists a vector field Z such that (Z, ω) Γ(L). If X is an infinitesimal automorphism of D = [L] which belongs to ρ(γ(l)), then there exists a smooth function c X on M which is constant along the leaves of the foliation determined by ρ(γ(l)) and satisfies d ω θ = c X Ω L, where θ = i X Ω L. Furthermore the mapping X c X is a Lie homomorphism C : T (M, D) A (M), where A (M) is the Lie algebra of L-admissible L L functions. This mapping C is called the extended Lie homomorphism of L. When A (M) = C (M) the extended Lie homomorphism is an invariant of D = [L], i.e. C does not depend on the vector bundle L representing L D. Proof: We have d ω θ = di X Ω L + ω i X Ω L = L X Ω L i X dω L + ω i X Ω L = u X Ω L + i X (ω Ω L ) + ω i X Ω L = (u X + ω(x))ω L. Set Then d ω θ = c X Ω L. Moreover c X = u X + ω(x). 0 = d ω d ω θ = d ω (c X Ω L ) = dc X ρ(γ(l)) Ω L + c X (dω L + ω Ω L ) = dc X ρ(γ(l)) Ω L. Applying Lemma 3.3, we get dc X ρ(γ(l)) = 0. This shows that c X along the leaves of the foliation associated with L. is constant 16

17 Now we are going to prove that X c X is a Lie homomorphism. By hypothesis, there exists a vector field Z such that (Z, ω) Γ(L). Therefore, if X, Y ρ(γ(l)) are two infinitesimal automorphisms of D = [L] then (c X Z, d ω c X ) and (c Y Z, d ω c Y ) are sections of L. In other words, c X and c Y are L-admissible functions. Furthermore, {c X, c Y } = 0. We also have: Consequently L [X,Y ] Ω L = L X (u Y Ω L ) L Y (u X Ω L ) = (L X u Y L Y u X )Ω L Therefore, u [X,Y ] = L X u Y L Y u X = L X (c Y ω(y )) L Y (c X ω(x)) = X (ω(y )) + Y (ω(x)) c [X,Y ] = u [X,Y ] + ω([x, Y ]) = X (ω(y )) + Y (ω(x)) + ω([x, Y ]) = dω(x, Y ) = 0. Hence C is a Lie homomorphism. Assume that A (M) = C (M). On can easily show that if L is conformally equivalent to L then A (M) = L C (M). Moreover, we have L Ω L = fω L and ω = ω d ln f, for some smooth function f. Since (Z, ω) Γ(L), we obtain that there exists of a vector field Z such that (Z, ω d ln f ) Γ(L ). Let X ρ(γ(l)) be an infinitesimal automorphism of D. Then L X Ω L = L X (fω L ) = (u X + d ln f (X))Ω L. It follows that Hence, u = u X X + d ln f (X). c X = u X + ω (X) = u X + d ln f (X) + ω(x) d ln f (X) = u X + ω(x) = c X. This completes the proof of Theorem

18 Remark 6. Examples where the situation A L (M) = C (M) occurs are given in Proposition 4.2. The following theorem slightly generalizes a result obtained in [B1] and [B2]: Theorem 6.6 Let ω be a closed 1-form on a smooth manifold M and let L be a locally conformal Dirac bundle with Lee 1-form ω L = ω ρ(γ(l)). Let p : M M be the Galois covering which resolves ω, that is, there exists a positive smooth function f : M R satisfying p ω = d ln f. Then, M is endowed with a Dirac structure L. Precisely, if L is the pull-back of L by the derivative Φ = T p : T M T M then L = {(X, fη) (X, η) L} is a Dirac structure. Furthermore, the conformal class of L is independent of f and the choice of L in D = [L]. Proof: Let ΩˆL be the skew-symmetric 2-form associated with L, that is, ΩˆL = p Ω L. Then It follows that Ω L(X, Y ) = fω ˆL(X, Y ) = f(p Ω L )(X, Y ). (3) Ω L = fp Ω L. Therefore dω L = f(d ln f ρ(γ( L)) p Ω L + p dω L ) = fp (ω L Ω L + dω L ) = 0. Moreover, it follows immediately from Equation (3) that the conformal class of L is independent of f and the choice of L in D = [L]. This completes the proof of Theorem Generalized locally conformal Dirac structures A Lie algebroid over a smooth manifold M is a vector bundle A M together with a Lie algebra structure on the space Γ(A) of smooth sections of A, and a bundle map ϱ : A T M such that 18

19 ϱ induces a Lie homomorphism from Γ(A) to χ(m). [X 1, fx 2 ] = f[x 1, X 2 ] + (ϱ(x 1 )f)x 2. The map ϱ is called the anchor of the Lie algebroid. There is a coboundary operator d A associated with A. It is defined as follows: (d A ξ)(x 1,..., X k ) = k ( 1) i+1 ϱ(x i )(ξ(x 1,, ˆX i,..., X k )) i=1 + i<j ( 1) i+j ξ([x i, X j ],, ˆX i,..., ˆX j,..., X k ), for ξ Γ(Λ k A ), X 1,..., X k Γ(A). The corresponding cohomology is called the Lie algebroid cohomology. When A = T M, one gets the de Rham cohomology. Let A be a Lie algebroid with anchor ϱ and let η Γ(A ) be a 1-cocycle. On Γ(A A ), we define [(X 1, ξ 1 ), (X 2, ξ 2 )] η C = ([X 1, X 2 ], L η X 1 ξ 2 L η X 2 ξ dη A(i X2 ξ 1 i X1 ξ 2 )), where d η Aξ = d A + η ξ and L η X 1 = i X1 d η A + d η Ai X1. Consider the bilinear 2-form defined by for (X 1, ξ 1 ), (X 2, ξ 2 ) A A. (X 1, ξ 1 ), (X 2, ξ 2 ) + = 1 2 (i X 2 ξ 1 + i X1 ξ 2 ), Definition 7.1 A sub-bundle L of the vector bundle A A M is said to be an (A, η)-dirac bundle over M if it is maximally isotropic with respect to, +, and Γ(L) is closed under the bracket [, ] η. In particular, Dirac C structures on M are just (T M, 0)-Dirac bundles. A maximally isotropic sub-bundle L A A is called a locally conformal (A, η)-dirac bundle if there exist an open cover (U i ) i I of M and smooth positive functions f i : U i R such that L i = {(X Ui, f i ξ Ui ) (X, ξ) L} is an (A, η)-dirac bundle over U i for all i I. Thus, locally conformal Dirac bundles coincide with locally conformal (T M, 0)-Dirac bundles. 19

20 Of course, any locally conformal (A, η)-dirac bundle induces a foliation on M which is given by the distribution ϱ(pr 1 (L))), where ϱ is the anchor of the Lie algebroid A and pr 1 : L A is the projection of the first component. Furthermore, there is a skew-symmetric C -bilinear map Ω L : Λ 2 P C (M) associated with L and defined as in (1), where P = pr 1 (Γ(L)). By an argument similar to the one used in Proposition 3.3, one may show that if the rank of Ω L is greater than 2 at every point x M then there exists a unique map ω L : P C (M) such that d A Ω L = ω L Ω L. Here we restrict d A to the space of all the n-forms θ : ω L : Λ n (P ) C (M). We also can extend Definition 6.2. Specifically, Definition 7.2 Two locally conformal (A, η)-dirac bundles L 1 and L 2 over M with respective Lee 1-forms ω 1 and ω 2 are conformally equivalent if there exits a smooth function f on M such that f(x) 0 at every point x M and L 2 = {(X, fα) (X, α) L 1 }. One may state results analogous to Theorem 3.5, 5.3, and 6.6. We will end this paper by giving a example of generalized locally conformal Dirac structure. Example. Let A = T M R, with the Lie algebroid bracket and anchor given by: [(X, f), (Y, g)] = ([X, Y ], X g Y f), and ϱ(x, f) = X, Consider the 1-cocycle η = (0, 1) Γ(T M R). Denote E 1 (M) = A A. It was observed in [GM] that [, ] η coincides with the bracket we used in C [Wa] to define the E 1 (M)-Dirac structures. Namely, ( ) [(X 1, f 1 ) + (α 1, g 1 ), (X 2, f 2 ) + (α 2, g 2 )] η = [X C 1, X 2 ], X 1 f 2 X 2 f 1 ( + L X1 α 2 L X2 α d(i X 2 α 1 i X1 α 2 ) +f 1 α 2 f 2 α (g 2df 1 g 1 df 2 f 1 dg 2 + f 2 dg 1 ), ) X 1 g 2 X 2 g (i X 2 α 1 i X1 α 2 f 1 g 2 + f 2 g 1 ), for any (X i, f i ) + (α i, g i ) Γ(E 1 (M)). More precisely, E 1 (M)-Dirac structures are exactly (T M R, (0, 1))- Dirac bundles over M. Furthermore, the concept of a conformal equivalence 20

21 class considered in this paper is slightly different from the one introduced in [Wa]. We defined an equivalence relation among (T M R, (0, 1))-Dirac bundles in [Wa], while the locally conformal Dirac bundles are considered in Definition 7.2. Acknowledgment. I would like to thank A. Banyaga, P. Brassler, P. Foth, Y. Kosmann-Schwarzbach, C.-M. Marle, A. Weinstein, P. Xu, and N-T. Zung for discussions and for their interest in this work. Thanks go also to the Institut de Mathématiques de Jussieu for the hospitality while I was working on the revised version of this paper. Many thanks to the referee for helpful remarks. References [B1] [B2] [BFLS] A. Banyaga, On the geometry of locally conformal symplectic manifolds, Infinite dimensional Lie groups in geometry and representation theory (Washington, DC, 2000), 79-91, World Sci. Publishing, River Edge, NJ, A. Banyaga, A geometric integration of the extended Lee homomorphism, J. Geom. Phys. 39 (2001), H. Basart, M. Flato, A. Lichnerowicz, D. Sternheimer, Deformation theory applied to quantization and statistical mechanics, Lett. Math. Phys. 8 (1984), [C] T. Courant, Dirac structures, Trans. A.M.S. 319 (1990), [CW] T. Courant, A. Weinstein, Beyond Poisson structures, Séminaire Sud-Rhodanien de Geométrie, Travaux en cours 27, (1988), 39-49, Hermann, Paris. [DLM] [GL] [GM] P. Dazord, A. Lichnerowicz, C.-M. Marle, Structure locale des variétés de Jacobi, J. Math. Pures Appl. 70 (1991), F. Guédira, A. Lichnerowicz, Géométrie des algèbres de Lie locales de Kirillov, J. Math. pures et appl. 63 (1984), J. Grabowski, G. Marmo, The graded Jacobi algebras and (co)homology, Preprint arxiv:math.dg/

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