Abstract 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 for an m-dimensional smooth manifold M, and a Lie 2-algebra which is a &quot;categorified&quot; 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

Fengying Han

Meili Sun

Yanhui Bi

CiteSeerX

Journal of Applied Mathematics and Physics, 2016, 4, 1254-1259 
Published Online July 2016 in SciRes. http://www.scirp.org/journal/jamp 
http://dx.doi.org/10.4236/jamp.2016.47131  
 
 
A Class of Lie 2-Algebras in Higher-Order 
Courant Algebroids 
Yanhui Bi, Fengying Han, Meili Sun 
College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, China 
    
 
Received 9 May 2016; accepted 12 July 2016; published 15 July 2016 
 
 
 
Abstract 
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 pTM 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. 
 
Keywords 
Higher-Order Courant Algebroids, Higher-Order Dorfman Algebroids, Lie 2-Algebra 
 
 
1. Introduction 
The notion of Courant algebroid was introduced in [1] to study the double of Lie bialgebroids. Equivalent 
definition was given by Roytenberg [2]. In resent years, with the development and exploration of the theory of 
categorified Lie algebras, or “Lie 2-algebras”, Courant algebroids have been far and wide studied from several 
aspects and have been found many applications in the theory of Manin pairs and moment maps [3] [4]; 
generalized complex structures [5]; L∞ -algebras and symplectic supermanifolds [2]; gerbes [6] as well as BV 
algebras and topological field theories. 
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 *TM T M⊕ . The standard Courant bracket is given 
by  
 
1, [ , ] ( ).
2X Y Y X
X Y X Y L L di diα β β α α β+ + = + − + −                      (1) 
However, many experts know that on the direct sum bundle *pTM T M⊕∧ , there is also a similar bracket 
operation, i.e.  
 
1, [ , ] ( ), , ( ), , ( ),
2
p
X Y Y XX Y X Y L L di di X Y M Mα β β α α β α β+ + = + − + − ∀ ∈ ∈ΩX  
How to cite this paper: Bi, Y.H., Han, F.Y. and Sun, M.L. (2016) A Class of Lie 2-Algebras in Higher-Order Courant Algebroids. 
Journal of Applied Mathematics and Physics, 4, 1254-1259. http://dx.doi.org/10.4236/jamp.2016.47131  
Y. H. Bi et al. 
 
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. 
2. Preliminaries 
In this section, we introduce the higher-order Courant and Dorfman bracket on the direct sum bundle 
*p pTM T M⊕∧  and associated properties. Also we review the definition of Lie 2-algebras. 
First, there is a natural 1 *p T M−∧ -valued nondegenerate symmetric pairing ( , )+⋅ ⋅  on the direct sum bundle 
p ,  
1( , ) ( ), , ( ), , ( ).
2
p
X YX Y i i X Y M Mα β β α α β++ + = + ∀ ∈ ∈ΩX                 (2) 
The higher-order Courant bracket satisfies some similar properties as the Courant bracket.  
Theorem 2.1. [7] For any 11 2 3, , ( ), ( ), ( ),
p pe e e f C M Mξ∞ −∈Γ ∈ ∈Ω  we have 
1) 
 1 2 3 1 2 3, , . . ( , , ),e e e c p dT e e e+ =    where 
3 1 *: p pT T M−∧ → ∧  is defined by  
 1 2 3 1 2 3
1( , , ) (( , , ) . .).
3
T e e e e e e c p+= − +  
2) 
 1 2 1 2 1 2 1 2, , ( )( ) ( , ) .e fe fe e e f e df e eρ += + − ∧  
3) 
 1 2 1 2, [ ( ), ( )].e e e eρ ρ ρ=  
4) 
   1( ) 2 3 1 2 1 2 3 2 1 3 1 3
( , ) ( , ( , ) , ) ( , , ( , ) ) .eL e e e e d e e e e e e d e eρ + + + += + + +   
Second, we introduce the following higher-order Dorfman bracket,  
 1 2 1 2 1 2 1 2{ , } , ( , ) , , ( ).
pe e e e d e e e e+= + ∀ ∈Γ                         (3) 
The higher-order Dorfman bracket also satisfies similar properties as the usual Dorfman bracket. 
 
Theorem 2.2. [7] 1) For any 1 2, ( ), ( ),
pe e f C M∞∈Γ ∈  we have  
1 2 1 2 1 2 1 2 1 2 2 1 1 2{ , } { , } ( )( ) ,{ , } { , } ( )( ) 2( , ) .e fe f e e e f e fe e f e e e f e df e eρ ρ += + = − + ∧  
2) The Dorfman bracket { , }⋅ ⋅  is a Leibniz bracket, i.e. for any 1 2 3, , ( ),
pe e e ∈Γ    
1 2 3 1 2 3 2 1 3{ ,{ , }} {{ , }, } ( ,{ , }) .e e e e e e e e e += +  
Consequently, ( ,{ , }, )p ρ⋅ ⋅  is a Dorfman algebroid. 
3) The pairing (2) and the higher-order Dorfman bracket is compatible in the following sense,  
1( ) 2 3 1 2 3 2 1 3
( , ) ({ , }, ) ( ,{ , }) .eL e e e e e e e eρ + + += +                         (4) 
Third, we review some definitions about the Lie 2-algebra.  
Definition 2.3. [8] A Lie 2-algebra is a 2-term chain complex of vector spaces 0 1= ( )
dL L L←  equipped 
with the following structure: 
1) A chain map [ , ] : L L L⋅ ⋅ ⊗ →  which called the bracket; 
2) A chain homotopy : [ , ] [ , ]S σ⋅ ⋅ ⇒ − ⋅ ⋅   which called the alternator; 
3) An antisymmetric chain homotopy : [ ,[ , ]] [[ , ], ] [ ,[ , ]] ( 1)J σ⋅ ⋅ ⋅ ⇒ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⊗  which called the Jacobiator. 
In addition, the following diagrams are required to commute: 
 
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Y. H. Bi et al. 
 
 
 
Definition 2.4. [8] A Lie 2-algebra for which the Jacobiator is the identity chain homotopy is called 
hemistrict. One for which the alternator is the identity chain homotopy is called semistrict.  
Definition 2.5. [8] Given Lie 2-algebras L and 'L  with bracket, alternator and Jacobiator [ , ],⋅ ⋅  ,S J  and 
' ' '[ , ] , ,S L⋅ ⋅  respectively, a homomorphism from L to 'L  consists of: 
1) A Chain map ': ,L Lφ →  and 
2) A chain homotopy ': [ , ] ( ) [ , ]φ φ φΦ ⋅ ⋅ ⊗ ⇒ ⋅ ⋅   
such that the following diagrams commute: 
 
3. The Equivalence between p-Order Courant Algebroid and p-Order Dorfman  
Algebroid 
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 
 
( , , , )p ρ⋅ ⋅  is a p- 
order Courant algebroid, and ( ,{ , }, )p ρ⋅ ⋅  is a p-order Dorfman algebroid. We shall construct two Lie 2- 
algebras associated to p : one hemistrict and one semistrict. Then we shall prove these are isomorphic. Both 
these Lie 2-algebras have the same underlying 2-term complex, namely:  
0 0 01( ) 0 0dp pL M−= ←Ω ← ← ←  
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:  
 
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Y. H. Bi et al. 
 
 
, : .L L L⋅ ⋅ ⊗ →  In degree 0, the semi-bracket is given as in Equation (1)  
 
1, [ , ] ( ), , ( ), , ( ),
2
p
X Y Y XX Y X Y L L di di X Y M Mα β β α α β α β+ + = + − + − ∀ ∈ ∈ΩX  
In degrees 1 and 2, we set it equal to zero: 
     1 1
1, , , , 0.
2 X
e i d eξ ξ ξ ξ η= = − =  
Theorem 3.1. 
 
( , , , )p ρ⋅ ⋅  is a p-order Courant algebroid, there is a semistrict Lie 2-algebra 
 
( , , , )pL ρ⋅ ⋅  
where 
1) The space of 0-chains is ( );pΓ   
2) The space of 1-chains is 1( );p M−Ω  
3) The differential is the exterior derivative 1: ( ) ( );p pd M−Ω → Γ   
4) The bracket is 
 
, ;⋅ ⋅  
5) The alternator is the bilinear map 1: ( ) ( ) ( )p p pS M−Γ ×Γ →Ω   defined by 
1 2,
0;e eS =  
6) The Jacobiator is the identity, hence given by the trilinear map 1: ( ) ( ) ( ) ( )p p p pJ M−Γ ×Γ ×Γ →Ω     
with 
1 2 3, , 1 2 3 1 2 3
1( , , ) ( , , . .).
3e e e
J T e e e e e e c p= − = 〈 〉 +   
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:  
1 2 3 4 2 3 4 2 1 3 4 1 2 3 4 1 2
4 1 3 2 4 3 1 2 4 1 2 3 4 1 3 4 1 2 3
4 , , , , 1 [ , ], , ,[ , ], , , 3
,[ , ], [ , ], , , ,[ , ] , , 2 [ , ], ,
([ , ] 1) ([ , ] 1) ( ) [ , ]
( ) ([ , ] 1) .
e e e e e e e e e e e e e e e e e
e e e e e e e e e e e e e e e e e e e
e J J e J J J e
J J J J e J
+ + +
= + + +
  
 
 
1 2 3 4, , , ( ),
pe e e e∀ ∈Γ   we have  
     
   
     
   
4 1 2 3 4 2 1 3 4 1 2 3
4 1 2 3 4 2 3 1
4 1 2 3 4 1 3 2 4 1 2 3
4 3 1 2 4 1 3 2
( , , ), ( , , , ) ( , , , )
, ( , , ) ( , , ),
( , , , ) ( , , ), ( , , , )
( , , , ) ( , , , ).
dT e e e e dT e e e e dT e e e e
e dT e e e dT e e e e
dT e e e e dT e e e e dT e e e e
dT e e e e dT e e e e
+ +
+ +
= + +
+ +
 
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: { , }: .L L L⋅ ⋅ × →  In degree 
0, the hemi-bracket is given as Dorfman bracket:  
{ , } [ , ] X YX Y X Y L i dα β β α+ + = + −  
In degree 1, it is given by: { , } ,{ , } 0.XX i d Yα ξ ξ η β+ = + =  In degree 2, we necessarily have { , } 0ξ η = , 
where , ( ),pX Yα β+ + ∈Γ   while 1, ( ).p Mξ η −∈Ω  
To see that the hemi-bracket is in fact a chain map, it suffices to check it on hemi-brackets of degree 1:  
{ , } { , }, { , } 0 { , }.X Xd X di d L d X d d X d Xα ξ ξ ξ α ξ ξ α ξ α+ = = = + + = = +  
Theorem 3.2. ( ,{ , }, )p ρ⋅ ⋅  is a p-order Dorfman algebroid, there is a hemistrict Lie 2-algebra ( ,{ , }, )pL ρ⋅ ⋅  
where 
1) The space of 0-chains is ( );pΓ   
2) The space of 1-chains is 1( );p M−Ω  
3) The differential is the exterior derivative 1: ( ) ( );p pd M−Ω → Γ   
4) The bracket is { , };⋅ ⋅  
5) The alternator is the bilinear map 1: ( ) ( ) ( )p p pS M−Γ ×Γ →Ω   defined by 
1 2, 1 2
2( , ) ;e eS e e += −  
6) The Jacobiator is the identity, hence given by the trilinear map 1: ( ) ( ) ( ) ( )p p p pJ M−Γ ×Γ ×Γ →Ω    
with 
1 2 3, ,
0.e e eJ =   
Proof That S is a chain homotopy with the right source and target follows from thm. (2.2) and the fact that:  
1 2 11 2 2 1 , 1 1 ,
{ , } { , } , { , } { , } .e e e de e e e dS e e S ξξ ξ= − − = − −  
 
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Y. H. Bi et al. 
 
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:  
1 2 1 2 2 1{ ,{ , }} {{ , }, } { ,{ , }},e e e e e eξ ξ ξ= +  
1 2 1 2 1 2{ ,{ , }} {{ , }, } { ,{ , }} 0,e e e e e eξ ξ ξ= = =  
1 2 1 2 1 2{ ,{ , }} {{ , }, } { ,{ , }} 0.e e e e e eξ ξ ξ= = =  
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:  
1 2 3 2 1 3 2 3{ , }, ,{ , } 1 2 3 2 1 3 2 3 1 ,
2({ , }, ) ( ,{ , }) 2 ( , ) { , }e e e e e e X e eS S e e e e e e L e e e S+ + ++ = + = =  
The third diagram says that 
1 2 3 2 3 1,{ , } { , },
0e e e e e eS S− =  and this follows from the fact that the alternator is 
symmetric: 
1 2 2 1, ,
.e e e eS S= .                                                                   □ 
Theorem 3.3. 
 
( , , , )p ρ⋅ ⋅  and ( ,{ , }, )p ρ⋅ ⋅  are isomorphic as Lie 2-algebras.  
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 
 
: ( , , , ) ( ,{ , }, )p pϕ ρ ρ⋅ ⋅ → ⋅ ⋅   with the identity chain map and the chain homotopy given by 
1 2, 1 2
( , ) .e e e e +Φ =  
This is a chain homotopy follows from the bracket relation 
  1 21 2 1 2 ,
{ , } , ,e ee e e e d= + Φ  noted in Equation (3) 
together with the equations  
   1 11 , 1 1 , 1
, { , }, , { , }e ee d e e d eξ ξξ ξ ξ ξ+ Φ = + Φ =  
We check that the two diagrams in the definition of a Lie 2-algebra homomorphism commute. Noting that the 
chain map φ  is the identity, the commutativity of the first diagram is easily checked by recalling that 
1 2, 1 2
2( , )e eS e e +=  and 1 2
'
,e eS  are the identities. Noting that any edge given by the bracket for ( ,{ , }, )p ρ⋅ ⋅  in 
degree 1 is the identity and that 
1 2 3, ,e e e
J  is the identity, we easily check the commutativity of the first diagram:  
 1 2 1 2 1 2 1 2
'
, , 1 2 2 1 1 2 , , 1 2( ) ( , ) , ( , ) ({ , }).e e e e e e e eS e e e e d e e S e eφ φ φ φφ φ φ φ φ+Φ = − + = −Φ   
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:  
   
 
1 2 1 3 1 2 31 2 3 2 1 3
1 2 3 2 31 2 3
'
, 3 2 , , , 1 2 3, , , ,
, , 1 , 1 2 3, ,
( ) ({ , } { , }) ({ ,{ , }})
{ , }({ ,{ , }})
e e e e e e ee e e e e e
e e e e ee e e
e e J e e e
J e e e e
Φ +Φ Φ + Φ
= Φ Φ
 
 
 
Clockwise from the upper left to the lower right corner:  
   
   
   
   
1 2 1 3 1 2 31 2 3 2 1 3
1 2 1 31 2 3 2 1 3
1 2 3 2 1 3
'
, 3 2 , , , 1 2 3, , , ,
, 3 2 , 1 2 3 2 1 3, , , ,
1 2 3 2 1 3 1, , , ,
( ) ({ , } { , }) ({ ,{ , }})
( ) ({ , } { , })({{ , }, } { ,{ , }})
( )({ , , } { , , }) { ( ,
e e e e e e ee e e e e e
e e e ee e e e e e
e e e e e e
e e J e e e
e e e e e e e e
e e e e e e d e e
Φ +Φ Φ + Φ
= Φ +Φ Φ + Φ +
= Φ +Φ + +
 

       
2 3 2 1 3
1 2 3 2 1 3 1 2 3 2 1 3 1 2 3 2 1 2
) , } { , ( , ) }
, , , , ( , , ) ( , , ) { ( , ) , } { , ( , ) }.
e e d e e
e e e e e e d e e e d e e e d e e e e d e e
+ +
+ + + +
+
= + + + + +   
   
 
Counterclockwise from the upper left to the lower right corner:  
 
 
 
   
     
1 2 3 2 31 2 3
1 2 3 1 2 3
1 2 3
, , 1 , 1 2 3, ,
, , 1 2 3 1 2 3, ,
, , 1 2 3 1 2 3 1 2 3
1 2 3 2 1 3 1 2 3 1 2 3 1
{ , }({ ,{ , }})
({ , , }) { , ( , ) }
( , , ) ( , , ) { , ( , ) }
, , , , ( , , ) ( , , ) {
e e e e ee e e
e e e e e e
e e e
J e e e e
J e e e e d e e
J e e e d e e e e d e e
e e e e e e dT e e e d e e e e
+
+ +
+
Φ Φ
= Φ +
= + +
= + + + +
 

 
 
   
    2 3, ( , ) }d e e +
 
 
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Y. H. Bi et al. 
 
Because of the commutativity of the second diagram we must have the the following equation:  
 
   
1 2 3 1 2 3 1 2 3
1 2 3 2 1 3 1 2 3 2 1 3
( , , ) ( , , ) { , ( , ) }
( , , ) ( , , ) { ( , ) , } { , ( , ) }
dT e e e d e e e e d e e
d e e e d e e e d e e e e d e e
+ +
+ + + +
+ +
= + + +
 
   
 
     
1 2 3 1 2 3 2 1 3 1 2 3
2 1 3 1 2 3 1 2 3
1 2 3 2 1 3 1 2 3
1 3 2 3
( , , ) ( , , ) ( , , ) { ( , ) , }
{ , ( , ) } ( , , ) { , ( , ) }
( , , ) ( , , ) ( , , )
( , ) ( , )Y X
dT e e e d e e e d e e e d e e e
e d e e d e e e e d e e
d e e e d e e e d e e e
dL e e dL e e
+ + +
+ + +
+ + +
+ +
= + +
+ − −
= + −
+ −
 
because of 
 1 2 3 1 2 3
1( , , ) (( , , ) . .),
3
T e e e e e e c p+= − +  we get the following calculations:  
1 2 3 1 2 3 2 1 32 ( , , ) ( ,{ , }) ( ,{( , )})dT e e e d e e e d e e e+ += −                        (5) 
if we choose 1 2 3, , ( ),
pe X e Y e Zα β γ= + = + = + ∈Γ    
1 2 3 2 1 3
1 2 3
1( ,{ , }) ( ,{( , )}) ( 2 . .)
2
1 12( . .)
4 2
2 ( , , )
Y Z Z Y Y Z
Y Z Z Y Y Z
d e e e d e e e di di di di di i d c p
di di di di di i d c p
dT e e e
α α α
α α α
+ +− = − − + +
= − − + +
=  
□ 
Acknowledgements 
We give warmest thanks to Zhangju Liu and Yunhe Sheng for useful comments and discussion. We also thanks 
the referees for very helpful comments. 
Funding 
Research was partially supported by NSF of China (11126338, 11461047, 11201218). 
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ley. arXiv:math.DG/9910078 
[3] Alekseev, A. and Kosmann-Schwarzbach, Y. (2000) Manin Pairs and Moment Maps. Journal of Differential Geometry, 
56, 133-165. 
[4] Li-Bland, D. and Meinrenken, E. (2009) Courant Algebroids and Poisson Geometry. International Mathematics Re-
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A Class of Lie 2-Algebras in Higher-Order Courant Algebroids

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A Class of Lie 2-Algebras in Higher-Order Courant Algebroids

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