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An Approach to Omni-Lie Algebroids Using Quasi-Derivations | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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An Approach to Omni-Lie Algebroids Using Quasi-Derivations

Dennise Garc´ıa-Beltr´an and Jos´e A. Vallejo*

Faculty of Sciences, State University of San Luis Potosi, Lat. Av. Salvador Nava s/n, 78290 San Luis Potos´ı, Mexico

Corresponding Author:
Jos´e A. Vallejo
Faculty of Sciences
State University of San Luis Potosi
Lat. Av. Salvador Nava s/n
78290 San Luis Potos´ı, Mexico
Email: [email protected]

Received date: 04 August 2010; Accepted date: 10 January 2011

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

We introduce the notion of left (and right) quasi-Loday algebroids and a “universal space” for them, called a left (right) omni-Loday algebroid, in such a way that Lie algebroids, omni-Lie algebras and omni-Loday algebroids are particular substructures.

MSC 2010: 17A32, 53D17, 58H99

Introduction

There are several ways in which Lie algebras can be generalized. Recall that if M is an R-module1 endowed with an R-bilinear bracket [ , ] : M ×M → M such that for all u, v,w ∈ M

(i) [u, v] = −[v, u] (antisymmetry),

(ii) [u, [v,w]] + [w, [u, v]] + [v, [w, u]] = 0 (Jacobi identity),

then (M,[ , ]) is a Lie algebra structure over M. When R = R (resp. C) we speak about a real Lie algebra (resp. a complex Lie algebra).

Let us briefly mention some of these generalizations.

(a) We can lift the restriction of antisymmetry. We then get the notion of Loday (or Leibniz2) algebra [6,8]. More precisely, the pair (M,[ , ]) is a left Loday algebra if, instead of conditions (i), (ii) above, it satisfies the left

(iii) [u, [v,w]] = [[u, v], w] + [v, [u,w]].

Note that this condition can be expressed by saying that [u, ] is a derivation with respect to the product [ , ]. Analogously, we can define right Loday algebras over M, by imposing that [ , w] is a derivation with respect to [ , ]:

(iii’) [[u, v], w] = [[u,w], v] + [u, [v,w]]. Leibniz identity:

Note that any one of (iii) or (iii’) is equivalent to the Jacobi identity when [ , ] is antisymmetric.

(b) Also, it is possible to consider a family of Lie algebras parameterized by points on a manifold M which, with some natural geometric assumptions, leads to the idea of Lie algebroid introduced by J. Pradines (see [5,9]). To be precise, a Lie algebroid over a manifold M (assume it real for simplicity) is given by a vector bundle π : E → M, an R-bilinear bracket [ , ] : ΓE × ΓE → ΓE defined on the C (M)-module of sections of E, and a mapping qE : ΓE → X(M) (called the anchor map) such that, for all X, Y ∈ ΓE, f ∈ C (M):

(1) (ΓE, [ , ]) is a real Lie algebra,

(2) [X, fY ] = f[X, Y ] + qE(X)(f) · Y .

Note that, in the case when M reduces to a single point, a Lie algebroid over M = {∗} is just a Lie algebra. A basic property of Lie algebroids is that the anchor map qE is a Lie algebra morphism when the bracket on X(M) is taken as the Lie bracket of vector fields [2,7].

(c) Finally, A. Weinstein introduced in [15] the concept of omni-Lie algebra, a structure that can be thought of as a kind of “universal space” for Lie algebras: take any natural number n ≥ 2 and consider the space product εn = gln × Rn endowed with the R-bilinear form { , } : εn × εn → εn given by

Equation

where [A,B] = A◦B−B◦A is the Lie bracket of gln. Then, (εn, { , }) is the n-dimensional omni-Lie algebra. The reason behind this denomination is that any n-dimensional real Lie algebra g is a closed maximal subspace of (εn, { , }).

Our goal is to define a structure for which the constructions mentioned in (a), (b), (c) appear as particular cases. In an absolutely unimaginative way, we will call it a left omni-Loday algebroid (of course, there exists the corresponding “right” definition). As we will see, this also includes as a particular case the notion of omni-Lie algebroid. Actually, the object we will construct will carry on a bracket that has already appeared in the literature, although under a different approach. In [4], M. K. Kinyon and A. Weinstein attacked the problem of integrating (in the sense of S. Lie’s “Third Theorem”) a Loday algebra3, and they gave the following example: take (h, [ , ]) a Lie algebra, and let V be an h-module with left action on V given by (ζ, x) → ζx. Then, we have the induced left action of h on h × V :

Equation

A binary operation · can be defined on ε = h × V through

Equation

It turns out that (ε, ·) is a Loday algebra, and if h acts nontrivially on V , then (ε, ·) is not a Lie algebra. Kinyon and Weinstein called ε with this Loday algebra structure the hemisemidirect product of h with V . Our omni-Loday algebroid will be a particular case of this construction, taking gl(V ) as h (see Definition 17).

To achieve our goal, let us note that it is necessary to recast the definition of a Lie algebroid in a form more suitable to an algebraic treatment, as in (a), (c). This can be easily done, just note that C (M) can be replaced by any R-algebra A, with unit element 1A, and commutative ΓE by a faithful A-module F, and X(M) by the module of derivations DerR(A).

This idea was cleverly exploited by J. Grabowski [2] who used it to prove that the property of the anchor map is a Lie algebra morphism. In the same paper, it is proved that there exist obstructions to the existence of Loday algebroid structures on vector bundles over a manifold M, stated in terms of the rank of these bundles (see Theorems 11, 12 below). As we will see, we can bypass these obstructions by considering left and right structures separately.

Quasi-derivations

The basic properties of a Lie algebroid are encoded in its anchor map, which in this context is a mapping ρ : F → DerR(A). We will assume that F is endowed with an R-bilinear bracket [[ , ]], then ρ is determined by two adjoint maps Equation.

Under certain mild conditions, these mappings are quasi-derivations of F, a property which is basic in the study of ρ. For instance, the fact that Equation are quasi-derivations allows us to prove that ρ is a morphism of Lie algebras (when (F, [[ , ]]) is Lie and we take the commutator of endomorphisms as the bracket on EndR(F)); see [2] (we refer the reader to that paper for the proof of the results stated in this section).

We recall that an operator D ∈ EndR(F) is a quasi-derivation if for a given f ∈ A there exists g ∈ A such that

Equation

where [ , ] is the commutator of endomorphisms of F, and μh(X) = h·X, for any h ∈ A, X ∈ F. A quasi-derivation is called a tensor operator when

Equation

Note that this is equivalent to D being A-linear (and not just R-linear). Some other straightforward properties of quasi-derivations are as follows:

(1) the set of all the quasi-derivations of F, QDerR(F) is an R-module;

(2) the commutator of endomorphisms on DerR(F) restricts to a closed bracket on QDerR(F) (i.e. if D1,D2 ∈ QDerR(F), then [D1,D2] ∈ QDerR(F)). Thus (QDerR(F), [ , ]) inherits the Lie algebra structure of (EndR(F), [ , ]);

(3) QDerR(F) is not just an R-module. Defining, for any f ∈ A and D ∈ QDerR(F)

Equation

it results that QDerR(F) is an A-module too;

(4) (QDerR(F), [ , ]) is not just a Lie algebra, but also a Poisson algebra (with the product given by the composition of endomorphisms).

The following results will be crucial in the sequel.

Theorem 1. There exists an R-linear mapping: QDerR(F) → DerR(A) such that

Equation

Corollary 2. The R-linear mapping extends to a Lie algebra morphism:

Equation

for all D1,D2 ∈ QDerR(F), f ∈ A.

Left Loday quasi-algebroids

The formula obtained in Corollary 3 looks very similar to condition (b2) in the definition of Lie algebroid. We can formalize this observation generalizing at once the definition, simply by replacing the Lie structure on ΓE (our F in the algebraic setting) by a Loday one. Thus, let (F, [[ , ]]) be a left Loday algebra. Given an X ∈ F, denote by Equation the endomorphisms Equation.

Note that if [[ , ]] is antisymmetric, then Equation.

Definition 4. The pair (F, [[ , ]]) is called a left Loday quasi-algebroid if Equation ∈ QDerR(F), for all X ∈ F. This amounts to the condition that, given X ∈ F, f ∈ A,

Equation

and motivates the following definition.

Definition 5. The mapping

Equation

is called the anchor of the left Loday quasi-algebroid. If Equation is tensorial, it is said that Equation is a left Loday algebroid on F.

The condition in Definition 4 now reads

Equation

with this justifying the terminology with the “left” prefix.

Remark 6. There is the corresponding notion of right Loday quasi-algebroid, when Equation. In this case, the formula reads

Equation

Theorem 7. Let Equation be a left Loday quasi-algebroid. Then, Equation is a morphism of left Loday R-algebras.

Proof. First, let us note that the condition of [[ , ]] being a Loday bracket on F means that

Equation

To check this, let Z ∈ F and compute

Equation

As this is valid for all Z ∈ F, we get the stated equivalence.

Now, Corollary 2 says that for all X, Y ∈ F,

Equation

Remark 8. This result partly answers a question raised in [14, Remark 3.3 (1)].

The definitions just given can be particularized to the case of Lie algebras (i.e. [[ , ]] antisymmetric).

Definition 9. Let (F, [[ , ]]) be a Lie algebra. If Equation, we say that (F, [[ , ]], qF), is a Lie quasi-algebroid, where

Equation

is the anchor map. If qF is tensorial (A-linear), then we say that (F, [[ , ]], qF) is a Lie algebroid.

Remark 10. Note that in this case the distinction between the left and right cases is irrelevant: each left Lie quasialgebroid with anchor qF is also a right Lie quasi-algebroid with anchor −qF.

How different are left (and right) Loday quasi-algebroids, Lie quasi-algebroids and Loday algebroids? In some cases, there is no such distinction: if we take Equation a vector bundle over a manifold M, Grabowski calls a QD-Loday (resp. Lie) algebroid a left Loday (resp. Lie) quasi-algebroid (i.e. Equation such that Equation, for all X ∈ F; then, he proves the following.

Theorem 11. Every QD-Loday algebroid (resp. QD-Lie) with rank ≥ 1 is a Loday algebroid (resp. Lie).

Theorem 12. Every QD-Loday algebroid of rank 1, is a QD-Lie algebroid.

Generation of Loday algebroids

As we have seen in the previous section, in order to get genuine examples of Loday quasi-algebroids, we must avoid that the two conditions Equation are satisfied simultaneously. To get examples of this situation, it is useful to know how to generate Loday brackets from operators with certain features. First of all, note that given a left Loday bracket [[ , ]] : F ×F →F, if we define

Equation

then we have that Equation is R-bilinear and for all X, Y,Z ∈ F,

Equation

thus, Equation is a right Loday bracket.

Analogously, given a right Loday bracket we can define a left Loday one, obtaining a correspondence between left and right Loday algebras.

Proposition 13. If (A, ·) is an associative R-algebra4 and, moreover, is endowed with an R-linear mapping D : A→Averifying

Equation

then one can define

[ , ] : A×A −→A, (a, b) −→ [a, b] := D(a) · b − b · D(a),

which satisfies the properties of R-bilinearity and the left Leibniz rule (so, it is a left Loday algebra).

Proof. Let us check first the R-bilinearity

Equation

Equation

For the left Leibniz rule, we have

Equation

Example 14. Some examples of such mappings D :A→A are the following.

(a) The identity D = Id. In this particular case we obtain a Lie algebra.

(b) A zero-square derivation D. Indeed, if this is the case,

Equation

(c) A projector D, that is, D is an algebra morphism and D2 = D. Then

Equation

Now, we can give a simple example of a left Loday quasi-algebroid which does not admit a right Loday quasialgebroid structure.

Example 15. Consider F = Ω(R6), which is an R-algebra with the exterior product ∧ and, moreover, a C (R6)- module (i.e. R = R, A = C (M)). Define

Equation

It is immediate that (Ω(R6), [[ , ]]) is a left Loday algebra, as d is a square-zero operator (see (b) above). Now, we have, for any α, β ∈ Ω(R6), f ∈ C (R6),

Equation

Thus, on the other hand, if Equation

Equation

but

Equation

and there is no Equation such that

Equation

Thus, (Ω(R6), [[ , ]]) has a left Loday quasi-algebroid structure, with anchor Equation, but it does not admit a right Loday quasi-algebroid structure. Note that Equation is (trivially) tensorial.

The following, less trivial, example was suggested to us by Y. Sheng. It shows that the kind of structures we are considering can appear in the more general context of higher order Courant algebroids (although here we just take n = 1 for simplicity) through the associated Dorfman bracket; see [11].

Example 16. LetM be a differential manifold. Consider the vector bundle Equation whose sections are endowed with the Dorfman bracket:

Equation

Then we have a left Loday algebra, as [[ , ]] is clearly R-bilinear and

Equation

for all X + α, Y + β, Z + γ ∈ Equation. Moreover, Equation and Y + β ∈ Equation. Then,

Equation

so (Equation ,[[ , ]]) is a left Loday quasi-algebroid, with anchor map the projection onto the first factor:

Equation

Note that in this case the anchor is tensorial: if f, g ∈ C (M) and X + α ∈ Equation, then

Equation

so (Equation, [[ , ]]) is indeed a left Loday algebroid. However, for Equation we find

Equation

and the term Equation clearly spoils the possibility that Equation is a quasi-derivation.

Left omni-Loday algebroids and omni-Lie algebroids

Having established the non-triviality of left Loday quasi-algebroids, we now turn to the question of whether an analogue of Weinstein’s omni-Lie algebra exists for these structures. As before, let A be an associative algebra, commutative and with unit element 1A over a ring R that is commutative and with unit element 1R. Also, let F be a faithful A-module.

Definition 17. Consider the product space gl(FF and define the bracket

Equation

where [ , ] is the commutator of endomorphisms.

Remark 18. It is straightforward to check that { , } is R-bilinear. However, it does not satisfy Jacobi’s identity (here Equation denotes cyclic sum), as we have

Equation

As stated in the introduction, the bracket { , } satisfies instead of the left Leibniz identity the following:

Equation

Now, let B : F ×F →F be an R-bilinear form. Define the “graph” of B as

Equation

where

Equation

Proposition 19. The graph FB is closed under { , } if and only if (F,B) is a left Loday algebra. Moreover, if B is antisymmetric, FB is closed if and only if (F,B) is a Lie algebra.

Proof. FB is closed with respect to { , } if, and only if, for all X, Y ∈ F we have

Equation

that is, for all Z ∈ F,

Equation

or

Equation

or equivalently

Equation

which is the left Leibniz identity, that is, (F,B) is a left Loday algebra.

For left Loday quasi-algebroids, we have the following.

Theorem 20. Let B : F ×F →F be an R-bilinear form and ρ :F →DerR(A) a morphism of R-modules. Then, (F,B) is a left Loday quasi-algebroid with anchor map ρ, if and only if FB is closed with respect to { , } and B is such that

Equation

Equation

Proof. If (F,B) is a left Loday quasi-algebroid, it is also a left Loday algebra and then, by Proposition 19, FB is closed under { , }. On the other hand, the condition of being quasi-algebroid implies that for all X,Z ∈ F and for all f ∈ A, we have

Equation

that is,

Equation

For the second implication, consider FB closed with respect to { , }, so (F,B) is a left Loday algebra (see Proposition 19). Moreover, for all X,Z ∈ F and for all f ∈ A,

Equation

so

Equation

that is, Equation is a quasi-derivation for all X ∈ F. Thus, (F,B) is a left quasi-algebroid with anchor map ρ.

Let us try to get rid of the “quasi” prefix.

Theorem 21. Let F be a free module of rank k > 1, B : F ×F →F an R-bilinear form and ρ :F →DerR(A) a morphism of R-modules. Suppose that FB is closed with respect to { , } and that B is such that

Equation

Equation

Equation. Then (F,B, ρ) is a left Loday algebroid.

Proof. On one hand, applying (a) then (b),

Equation

and on the other hand, first applying (b),

Equation

Therefore, Equation, or equivalently

Equation (5.1)

Let {Yj}j∈I be a basis of F. Taking X = Yi and Z = Yj for some distinct i, j ∈ I in (5.1), we have

Equation

so

Equation (5.2)

for all f, g ∈ A.

Now, let Equation and f ∈ A. By the R-linearity of ρ and (5.2),

Equation

that is, ρ is tensorial. Thus, (F,B, ρ) is a left Loday algebroid.

Remark 22. However, we can not say anything about the converse, as Example 15 shows (there, we have a left Loday algebroid and the first condition (a) above is trivially satisfied while (b) is not).

We also can avoid the “quasi” prefix if we add the condition of antisymmetry to B, thus entering into the realm of Lie structures.

Theorem 23. Let F be a free module of rank k > 1, B : F ×F →F an R-bilinear form, and ρ :F →DerR(A) a morphism of R-modules. Then, (F,B, ρ) is a Lie algebroid if and only if FB is closed with respect to { , }, B is antisymmetric and, for all X,Z ∈ F and f ∈ A, the following holds:

Equation

(that is, ad X ∈ QDerR(A)).

Proof. If (F,B, ρ) is a Lie algebroid, (F,B) is a Lie algebra, that is, (F,B) is a left (and right) Loday algebra and B is antisymmetric, so Proposition 19 tells us that FB is closed with respect to { , }. Now, let X,Z ∈ F, f ∈ A; then we have

Equation

which is the same as

Equation

If now is FB closed with respect to { , }, Proposition 19 again tells us that (F,B) is a left Loday algebra, but as B is also antisymmetric, (F,B) is a Lie algebra.

On the other hand, the hypothesis of Theorem 20 is satisfied, so we know that adX is a quasi-derivation for all X ∈ F and (F,B) is a Lie quasi-algebroid with anchor map ρ.

To finish, let us check (see Theorem 21) that for all X, Y,Z ∈ F and for all f ∈ A the following holds:

Equation

But, by the antisymmetry of B,

Equation

So (F,B, ρ) is a Lie algebroid.

The preceding results motivate the following definition.

Definition 24. Let A be an associative, commutative algebra with unit element 1A over a commutative ring with unit element 1R. Let F be a free A-module of rank k > 1. We call (gl(FF, { , }) the left omni-Loday algebroid determined by F.

Remark 25. Note that if (F,B, ρ) is a left Loday algebroid then, in particular, it is a left Loday quasi-algebroid and thus FB ⊂ gl(FF is closed with respect to { , }, as by Theorem 20: every left Loday algebroid can be seen as a closed subspace of left omni-Loday algebroid.

Remark 26. In the case of Lie algebroids, we have the same situation as in the preceding remark: given an Rbilinear F-valued form B : F ×F →F such that it is antisymmetric and satisfies adLX ∈ QDerR(F), by Theorem 23 there is a correspondence between Lie algebroids (F,B, ρ) and closed subspaces FB, but this time given by an “if and only if” statement. Thus, we could call (gl(FF, { , }) an omni-Lie algebroid as well.

It is worth noting that a different definition for omni-Lie algebroids (based on the notion of Courant structures on the direct sum of the gauge Lie algebroid and the bundle of jets of a vector bundle E over a manifold M) has been presented very recently in [1]. It would be interesting to know if this definition is equivalent to ours.

Acknowledgments

The authors want to express their gratitude to Prof. M. K. Kinyon and Prof. Y. Sheng for their useful comments and for providing references to previous work on Loday algebras. The second author’s work was supported by a CONACyT grant, project CB-2007-1 code J2-78791.

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