Journal of Alied Mathematics and Physics, 06, 4, 54-59 Published Online July 06 in SciRes htt://wwwscirorg/journal/jam htt://dxdoiorg/046/jam0647 A Class of Lie -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 06; acceted July 06; ublished 5 July 06 Abstract In this aer, we study the relation of the algebraic roerties of the higher-order Courant bracket and Dorfman bracket on the direct sum bundle TM T M for an m-dimensional smooth manifold M, and a Lie -algebra which is a categorified version of a Lie algebra We rove that the higher-order Courant algebroids give rise to a semistrict Lie -algebra, and we rove that the higher-order Dorfman algebroids give rise to a hemistrict Lie -algebra Consequently, there is an isomorhism from the higher-order Courant algebroids to the higher-order Dorfman algebroids as Lie -algebras homomorhism Keywords Higher-Order Courant Algebroids, Higher-Order Dorfman Algebroids, Lie -Algebra Introduction The notion of Courant algebroid was introduced in [] to study the double of Lie bialgebroids Equivalent definition was given by Roytenberg [] In resent years, with the develoment and exloration of the theory of categorified Lie algebras, or Lie -algebras, Courant algebroids have been far and wide studied from several asects and have been found many alications in the theory of Manin airs and moment mas [] [4]; generalized comlex structures [5]; L -algebras and symlectic suermanifolds []; gerbes [6] as well as BV algebras and toological 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 X α, Y β = [ X, Y ] LXβ LYα ( diyα dixβ) () * However, many exerts know that on the direct sum bundle TM T M, there is also a similar bracket oeration, ie X α, Y β = [ X, Y ] LXβ LYα ( diyα dixβ), X, Y X ( M ), α, β Ω ( M ), How to cite this aer: Bi, YH, Han, FY and Sun, ML (06) A Class of Lie -Algebras in Higher-Order Courant Algebroids Journal of Alied Mathematics and Physics, 4, 54-59 htt://dxdoiorg/046/jam0647
which we call the higher-order Courant bracket We have roved that the Jacobi identity holds u to an exact term So in our aer, we introduce the higher-order Courant algebroids and Dorfman algebroids, and find their relation In Section, 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 -algebras In Section, we introduce emhatically the equivalence between the higher-order Courant algebroids and higher-order Dorfman algebroids In Thm, we construct a Lie -algebra which is semistrict, meaning that the bracket is skewsymmetric, but the Jacobi identity holds only u to isomorhism, where the Lie bracket of observables is given instead by the higher-order Courant bracket In Thm, we construct another Lie -algebra with the same objects and morhisms, where the Lie bracket of observables is given instead by the higher-order Dorfman bracket In Thm, we show that these two Lie -algebras are isomorhic Preliminaries In this section, we introduce the higher-order Courant and Dorfman bracket on the direct sum bundle * TM T M and associated roerties Also we review the definition of Lie -algebras * First, there is a natural TM-valued nondegenerate symmetric airing (, ) on the direct sum bundle, ( X α, Y β) = ( ixβ iyα), XY, X ( M), α, β Ω ( M) () The higher-order Courant bracket satisfies some similar roerties as the Courant bracket Theorem [7] For any e, e, e Γ( ), f C ( M), ξ Ω ( M), we have ) e, e, e c = * dt( e, e, e), where T: TM is defined by T( e, e, e) = (( e, e, e) c ) ) e, fe = fe, e ρ ( e)( f ) e df ( e, e) ) ρe, = [ ρ( e), ρ( e)] 4) Lρ ( e ) ( e, e) = ( e, e d( e, e), e) ( e, e, e d( e, e) ) Second, we introduce the following higher-order Dorfman bracket, { e, e } = e, e d( e, e ), e, e Γ( ) () The higher-order Dorfman bracket also satisfies similar roerties as the usual Dorfman bracket Theorem [7] ) For any e, e Γ( ), f C ( M), we have { e, fe} = f{ e, e} ρ( e)( f) e,{ fe, e} = f{ e, e} ρ ( e)( f) e df ( e, e) ) The Dorfman bracket {,} is a Leibniz bracket, ie for any e, e, e Γ( ), { e,{ e, e}} = {{ e, e}, e} ( e,{ e, e }) Consequently, (,{, }, ρ) is a Dorfman algebroid ) The airing () and the higher-order Dorfman bracket is comatible in the following sense, L ( e, e ) = ({ e, e }, e ) ( e,{ e, e }) (4) ρ ( e ) Third, we review some definitions about the Lie -algebra d Definition [8] A Lie -algebra is a -term chain comlex of vector saces L =( L0 L) equied with the following structure: ) A chain ma [, ]:L L L which called the bracket; ) A chain homotoy S :[, ] [, ] σ which called the alternator; ) An antisymmetric chain homotoy J :[,[, ]] [[, ], ] [,[, ]] ( σ ) which called the Jacobiator In addition, the following diagrams are required to commute: 55
Definition 4 [8] A Lie -algebra for which the Jacobiator is the identity chain homotoy is called hemistrict One for which the alternator is the identity chain homotoy is called semistrict Definition 5 [8] Given Lie -algebras L and L with bracket, alternator and Jacobiator [, ], S, J and [, ], S, L resectively, a homomorhism from L to L consists of: ) A Chain ma φ : L L, and ) A chain homotoy Φ : [, ] (φ φ ) φ [, ] such that the following diagrams commute: The Equivalence between -Order Courant Algebroid and -Order Dorfman Algebroid In this section, we construct a semistrict Lie -algebra and a hemistric Lie -algebra and give the relation between -order Courant algebroids and -order Dorfman algebroids We all know that (,,, ρ ) is a order Courant algebroid, and (,{, }, ρ ) is a -order Dorfman algebroid We shall construct two Lie algebras associated to : one hemistrict and one semistrict Then we shall rove these are isomorhic Both these Lie -algebras have the same underlying -term comlex, namely: d 0 0 0 = L Ω ( M ) 0 0 where d is the usual exterior derivative of functions To see that this chain comlex is well-defined We make L into a semistrict Lie -algebra For this, we use a chain ma called the semi-bracket: 56
, : L L L In degree 0, the semi-bracket is given as in Equation () X α, Y β = [ X, Y ] LXβ LYα ( diyα dixβ), X, Y X ( M ), α, β Ω ( M ), In degrees and, we set it equal to zero: e, ξ = i dξ= ξ, e, ξη, = 0 X Theorem (,,, ρ) is a -order Courant algebroid, there is a semistrict Lie -algebra L (,,, ρ) where ) The sace of 0-chains is Γ( ); ) The sace of -chains is Ω ( M ); ) The differential is the exterior derivative d : Ω ( M) Γ( ); 4) The bracket is, ; 5) The alternator is the bilinear ma S : Γ ( ) Γ( ) Ω ( M) defined by S e, e = 0; 6) The Jacobiator is the identity, hence given by the trilinear ma J : Γ ( ) Γ ( ) Γ( ) Ω ( M) with Je, e, e = T( e, e, e) = ( e, e, e c ) Proof We note from Equation () that the semi-bracket is antisymmetric Since both S and the degree chain ma are zero, the alternator defined above is a chain homotoy with the right source and target So again, we just need to check that the Lie -algebra axioms hold The following identities can be checked by simle calculation, and the commutativity of the last diagram follows: e, e, e, e Γ( ), we have 4 ([ e, J ] ) ([ J, e ] ) ( J J ) [ J, e ] 4 e, e, e e4, e, e [ e4, e], e, e e4,[ e, e], e e4, e, e = ( J J J ) ([ J, e ] ) J e4,[ e, e], e [ e4, e], e, e e4, e,[ e, e] e4, e, e [ e4, e], e, e dt ( e4, e, e), e dt ( e4, e, e, e) dt ( e4, e, e, e) e4, dt( e, e, e) dt( e4, e, e), e = dt e4 e e e dt e4 e e e dt e4 e e e dt ( e4, e, e, e) dt ( e4, e, e, e) (,,, ) (,, ), (,,, ) Since the Jacobiator is antisymmetric and the alternator is the identity, the first and second diagrams commute as well The third diagram commutes because all the edges are identity morhisms Next, the hemistrict Lie -algebra comes with a bracket called the hemi-bracket: {, }: L L L In degree 0, the hemi-bracket is given as Dorfman bracket: { X α, Y β} = [ XY, ] LXβ id Y α In degree, it is given by: { X αξ, } = ix dξ,{ η, Y β} = 0 In degree, we necessarily have { ξη, } = 0, where X α, Y β Γ( ), while ξη, Ω ( M ) To see that the hemi-bracket is in fact a chain ma, it suffices to check it on hemi-brackets of degree : d{ X α, ξ} = di dξ = L dξ = { X α, dξ}, d{ ξ, X α} = 0 = { dξ, X α} X X Theorem (,{, }, ρ) is a -order Dorfman algebroid, there is a hemistrict Lie -algebra L(,{, }, ρ) where ) The sace of 0-chains is Γ( ); ) The sace of -chains is Ω ( M ); ) The differential is the exterior derivative d : Ω ( M) Γ( ); 4) The bracket is {, }; 5) The alternator is the bilinear ma S : Γ ( ) Γ( ) Ω ( M) defined by Se, e = ( e, e ) ; 6) The Jacobiator is the identity, hence given by the trilinear ma J : Γ ( ) Γ ( ) Γ( ) Ω ( M) with J e, e, e = 0 Proof That S is a chain homotoy with the right source and target follows from thm () and the fact that: { e, e } = { e, e } ds, { e, ξ} = { ξ, e } S ξ e, e e, d 57
Thm () also says that the Jacobi identity holds The following equations then imly that J is also a chain homotoy with the right source and target: { e,{ e, ξ}} = {{ e, e }, ξ} { e,{ e, ξ}}, { e,{ ξ, e }} = {{ e, ξ}, e } = { ξ,{ e, e }} = 0, { ξ,{ e, e }} = {{ ξ, e}, e } = { e,{ ξ, e }} = 0 So, we just need to check that the Lie -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 () () is shown as follows: S S = ({ e, e }, e ) ( e,{ e, e }) = L ( e, e ) = { e, S } { e, e}, e e,{ e, e} X e, e The third diagram says that Se,{ e, e} S{ e, e}, e= 0 and this follows from the fact that the alternator is symmetric: Se, e = S e, e Theorem (,,, ρ) and (,{, }, ρ) are isomorhic as Lie -algebras Proof We show that the identity chain mas with aroriate chain homotoies define Lie -algebra homomorhisms and that their comosites are the resective identity homomorhisms There is a homomorhism ϕ:(,,, ρ) (,{,}, ρ) with the identity chain ma and the chain homotoy given by Φ e, e = ( e, e ) This is a chain homotoy follows from the bracket relation { e, e} = e, e dφ e, e, noted in Equation () together with the equations e ξ e, ξ dφ = { e, ξ}, ξ, e dφ = { ξ, e}, ξ, e We check that the two diagrams in the definition of a Lie -algebra homomorhism commute Noting that the chain ma φ is the identity, the commutativity of the first diagram is easily checked by recalling that Se, e = ( e, e ) and S e, e are the identities Noting that any edge given by the bracket for (,{, }, ρ) in degree is the identity and that J is the identity, we easily check the commutativity of the first diagram: e, e, e e, e Φ φe, φe = = Φe, e φe, φe φ( S ) ( φe, φe ) e, e d( e, e ) S ({ φe, φe }) To check the commutativity of the second diagram we only need to erform the following calculation: The second diagram meet commutativity if and only if the following equation established: ( Φ Φ ) ({ Φ,,,, e, e, e } { e, Φe, e}) J e, e, e({ e,{ e, e}}) e e e e e e = Je, e, eφ,, { e, Φe, e}({ e,{ e, e}}) e e e Clockwise from the uer left to the lower right corner: ( Φ Φ ) ({ Φ,,,, e, e, e } { e, Φe, e}) J e, e, e({ e,{ e, e}}) e e e e e e = ( Φ, Φ,,, ) ({ Φ e, e, e } { e, Φ e, e})({{ e, e}, e} { e,{ e, e}}) e e e e e e = ( Φ Φ )({,, } {,, }) { (, e e e e e, e, e e, e, e e e d e e ), e} { e, de (, 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 uer left to the lower right corner: Je, e, e Φ { e,,, Φe, e }({ e,{ e, e}}) e e e = J Φ ({ e, e, e }) { e, d( e, e ) },, (,, e ) (,, ) {, (, ) } e, e, e e e e = J e e e d e e e e d e e e, e, = e, e, e e, e, e dt( e, e, e ) d( e, e, e ) { e, de (, e )} 58
Because of the commutativity of the second diagram we must have the the following equation: because of 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 ) } e de e de e e e de e dt( e, e, e ) = d( e, e, e ) d( e, e, e ) { d( e, e ), e } {, (, )} (,, ) {, (, )} = d( e, e, e) de (, e, e ) de (, e, e ) dl ( e, e ) dl ( e, e ) Y X T( e, e, e) = (( e, e, e) c ), we get the following calculations: dte (, e, e) = de (,{ e, e}) de (,{( e, e)}) (5) if we choose e = X α, e = Y β, e = Z γ Γ( ), d( e,{ e, e}) d( e,{( e, e)}) = ( diydizα dizdiyα diyizdα c ) = ( diydizα dizdiyα diyizdα c ) 4 = dt ( e, e, e ) Acknowledgements We give warmest thanks to Zhangju Liu and Yunhe Sheng for useful comments and discussion We also thanks the referees for very helful comments Funding Research was artially suorted by NSF of China (68, 46047, 08) References [] Liu, ZJ, Weinstein, A and Xu, P (997) Manin Triles for Lie Bialgebroids Journal of Differential Geometry, 45, 547-574 [] Roytenberg, D Courant Algebroids, Derived Brackets and Even Symlectic Suermanifolds PhD Thesis, UC Berkeley arxiv:mathdg/990078 [] Alekseev, A and Kosmann-Schwarzbach, Y (000) Manin Pairs and Moment Mas Journal of Differential Geometry, 56, -65 [4] Li-Bland, D and Meinrenken, E (009) Courant Algebroids and Poisson Geometry International Mathematics Research Notices,, 06-45 [5] Gualtieri, M Generalized Comlex Geometry arxiv:mathdg/040 [6] Ševera, P and Weinstein, A (00) Poisson Geometry with a -Form Background Progress of Theoretical Physics Sulements, 44, 45-54 Noncommutative Geometry and String Theory (Yokohama, 00) [7] Bi, YH and Sheng, YH (990) On Higher Analogues of Courant Algebroids Science China Mathematics A, Prerint [8] Roytenberg, D On weak Lie -Algebras arxiv:0746 59