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In this paper, we study the relation of the algebraic properties of the higher-order Courant bracket and Dorfman bracket on the direct sum bundle *TM*♁∧^{p}*T*M** *for an m-dimensional smooth manifold M, and a Lie 2-algebra which is a “categorified” version of a Lie algebra. We prove that the higher-order Courant algebroids give rise to a semistrict Lie 2-algebra, and we prove that the higher-order Dorfman algebroids give rise to a hemistrict Lie 2-algebra. Consequently, there is an isomorphism from the higher-order Courant algebroids to the higher-order Dorfman algebroids as Lie 2-algebras homomorphism.

The notion of Courant algebroid was introduced in [

But these articles just introduced the Courant algebroids and Dorfman algebroids. And they did not find the relation between the higher-order Courant algebroids and the higher-order Dorfman algebroids.

The standard Courant algebroid is the direct sum bundle

However, many experts know that on the direct sum bundle

which we call the higher-order Courant bracket. We have proved that the Jacobi identity holds up to an exact term.

So in our paper, we introduce the higher-order Courant algebroids and Dorfman algebroids, and find their relation. In Section 2, we review the higher-order Courant bracket and higher-order Courant Dorfman bracket. Also we review the basic definitions about the Dorfman algebroids and Lie 2-algebras. In Section 3, we intro- duce emphatically the equivalence between the higher-order Courant algebroids and higher-order Dorfman algebroids. In Thm. 3.1, we construct a Lie 2-algebra which is “semistrict”, meaning that the bracket is skew- symmetric, but the Jacobi identity holds only up to isomorphism, where the Lie bracket of observables is given instead by the higher-order Courant bracket. In Thm. 3.2, we construct another Lie 2-algebra with the same objects and morphisms, where the Lie bracket of observables is given instead by the higher-order Dorfman bracket. In Thm. 3.3, we show that these two Lie 2-algebras are isomorphic.

In this section, we introduce the higher-order Courant and Dorfman bracket on the direct sum bundle

First, there is a natural

The higher-order Courant bracket satisfies some similar properties as the Courant bracket.

Theorem 2.1. [

1)

2)

3)

4)

Second, we introduce the following higher-order Dorfman bracket,

The higher-order Dorfman bracket also satisfies similar properties as the usual Dorfman bracket.

Theorem 2.2. [

2) The Dorfman bracket

Consequently,

3) The pairing (2) and the higher-order Dorfman bracket is compatible in the following sense,

Third, we review some definitions about the Lie 2-algebra.

Definition 2.3. [

1) A chain map

2) A chain homotopy

3) An antisymmetric chain homotopy

In addition, the following diagrams are required to commute:

Definition 2.4. [

Definition 2.5. [

1) A Chain map

2) A chain homotopy

such that the following diagrams commute:

In this section, we construct a “semistrict” Lie 2-algebra and a “hemistric” Lie 2-algebra and give the relation between p-order Courant algebroids and p-order Dorfman algebroids. We all know that

where d is the usual exterior derivative of functions. To see that this chain complex is well-defined.

We make L into a semistrict Lie 2-algebra. For this, we use a chain map called the semi-bracket:

In degrees 1 and 2, we set it equal to zero:

Theorem 3.1.

1) The space of 0-chains is

2) The space of 1-chains is

3) The differential is the exterior derivative

4) The bracket is

5) The alternator is the bilinear map

6) The Jacobiator is the identity, hence given by the trilinear map

with

Proof. We note from Equation (1) that the semi-bracket is antisymmetric. Since both S and the degree 1 chain map are zero, the alternator defined above is a chain homotopy with the right source and target. So again, we just need to check that the Lie 2-algebra axioms hold. The following identities can be checked by simple calculation, and the commutativity of the last diagram follows:

Since the Jacobiator is antisymmetric and the alternator is the identity, the first and second diagrams com- mute as well. The third diagram commutes because all the edges are identity morphisms. □

Next, the hemistrict Lie 2-algebra comes with a bracket called the hemi-bracket:

In degree 1, it is given by:

To see that the hemi-bracket is in fact a chain map, it suffices to check it on hemi-brackets of degree 1:

Theorem 3.2.

1) The space of 0-chains is

2) The space of 1-chains is

3) The differential is the exterior derivative

4) The bracket is

5) The alternator is the bilinear map

6) The Jacobiator is the identity, hence given by the trilinear map

Proof That S is a chain homotopy with the right source and target follows from thm. (2.2) and the fact that:

Thm. (2.2) also says that the Jacobi identity holds. The following equations then imply that J is also a chain homotopy with the right source and target:

So, we just need to check that the Lie 2-algebra axioms hold. The first and the last two diagrams commute since each edge is the identity. The commutativity of the second diagram from thm. (2.2) (3) is shown as follows:

The third diagram says that

Theorem 3.3.

Proof. We show that the identity chain maps with appropriate chain homotopies define Lie 2-algebra ho- momorphisms and that their composites are the respective identity homomorphisms. There is a homomorphism

We check that the two diagrams in the definition of a Lie 2-algebra homomorphism commute. Noting that the chain map

To check the commutativity of the second diagram we only need to perform the following calculation:

The second diagram meet commutativity if and only if the following equation established:

Clockwise from the upper left to the lower right corner:

Counterclockwise from the upper left to the lower right corner:

Because of the commutativity of the second diagram we must have the the following equation:

because of

if we choose

□

We give warmest thanks to Zhangju Liu and Yunhe Sheng for useful comments and discussion. We also thanks the referees for very helpful comments.

Research was partially supported by NSF of China (11126338, 11461047, 11201218).

Yanhui Bi,Fengying Han,Meili Sun, (2016) A Class of Lie 2-Algebras in Higher-Order Courant Algebroids. Journal of Applied Mathematics and Physics,04,1254-1259. doi: 10.4236/jamp.2016.47131