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| In a field of [[mathematics]] known as [[differential geometry]], a '''Courant algebroid''' is a structure which, in a certain sense, blends the concepts of [[Lie algebroid]] and of [[quadratic Lie algebra]]. This notion, which plays a fundamental role in the study of Hitchin's [[generalized complex geometry|generalized complex structures]], was originally introduced by Zhang-Ju Liu, [[Alan Weinstein]] and Ping Xu in their investigation of doubles of [[Lie bialgebroid]]s in 1997.<ref>Z-J. Liu, A. Weinstein, and P. Xu: [http://arxiv.org/abs/dg-ga/9508013 Manin triples for Lie Bialgebroids], Journ. of Diff.geom. 45 pp.647–574 (1997).</ref> Liu, Weinstein and Xu named it after [[Theodore James Courant|Courant]], who had implicitly devised earlier in 1990<ref>T.J. Courant: ''Dirac Manifolds'', Transactions of the AMS, vol. 319, pp.631–661 (1990).</ref> the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on <math>TM\oplus T^*M</math>, called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids.
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| ==Definition==
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| A Courant algebroid consists of the data a vector bundle <math>E\to M</math> with a bracket <math>[.,.]:\Gamma E \times \Gamma E \to \Gamma E</math>, a non degenerate fiber-wise inner product <math>\langle.,.\rangle: E\times E\to M\times\R</math>, and a bundle map <math>\rho:E\to TM </math> subject to the following axioms,
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| :<math>[\phi, [\chi, \psi]] = [[\phi, \chi], \psi] + [\chi, [\phi, \psi]]</math>
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| :<math>[\phi, f\psi] = \rho(\phi)f\psi +f[\phi, \psi]</math>
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| :<math>[\phi,\phi]= \tfrac12 D\langle \phi,\phi\rangle</math>
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| :<math>\rho(\phi)\langle \psi,\psi\rangle= 2\langle [\phi,\psi],\psi\rangle </math>
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| where ''φ,ψ,χ'' are sections of ''E'' and ''f'' is a smooth function on the base manifold ''M''. ''D'' is the combination <math>\kappa^{-1}\rho^T d</math> with ''d'' the de Rham differential, <math>\rho^T</math> the dual map of <math>\rho</math>, and ''κ'' the map from ''E'' to <math>E^*</math> induced by the inner product.
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| ==Properties==
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| The bracket is not skew-symmetric as one can see from the third axiom. Instead it fulfills a certain Jacobi-identity (first axiom) and a Leibniz rule (second axiom). From these two axioms one can derive that the anchor map ''ρ'' is a morphism of brackets:
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| ::<math> \rho[\phi,\psi] = [\rho(\phi),\rho(\psi)] .</math>
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| The fourth rule is an invariance of the inner product under the bracket. Polarization leads to
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| ::<math> \rho(\phi)\langle \chi,\psi\rangle= \langle [\phi,\chi],\psi\rangle +\langle \chi,[\phi,\psi]\rangle .</math>
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| ==Examples==
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| An example of the Courant algebroid is the [[Dorfman bracket]]<ref>I.Y. Dorfman: ''Dirac structures of integrable evolution equations'', Phyics Letters A, vol.125, pp.240–246 (1987).</ref> on the direct sum <math>TM\oplus T^*M</math> with a twist introduced by Ševera,<ref>P. Ševera: [http://sophia.dtp.fmph.uniba.sk/~severa/letters/no1.ps Letters to A. Weinstein], unpublished.</ref> (1998) defined as:
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| ::<math> [X+\xi, Y+\eta] = [X,Y]+(\mathcal{L}_X\,\eta -i(Y) d\xi +i(X)i(Y)H)</math>
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| where ''X,Y'' are vector fields, '' ξ,η'' are 1-forms and ''H'' is a closed 3-form twisting the bracket. This bracket is used to describe the integrability of [[generalized complex geometry|generalized complex structures]].
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| A more general example arises from a Lie algebroid ''A'' whose induced differential on <math>A^*</math> will be written as ''d'' again. Then use the same formula as for the Dorfman bracket with ''H'' an ''A''-3-form closed under ''d''.
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| Another example of a Courant algebroid is a quadratic Lie algebra, i.e. a Lie algebra with an invariant scalar product. Here the base manifold is just a point and thus the anchor map (and ''D'') are trivial.
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| The example described in the paper by Weinstein et al. comes from a Lie bialgebroid, i.e. ''A'' a Lie algebroid (with anchor <math>\rho_A</math> and bracket <math>[.,.]_A</math>), also its dual <math>A^*</math> a Lie algebroid (inducing the differential <math>d_{A^*}</math> on <math>\wedge^* A</math>) and <math>d_{A^*}[X,Y]_A=[d_{A^*}X,Y]_A+[X,d_{A^*}Y]_A</math> (where on the RHS you extend the ''A''-bracket to <math>\wedge^*A</math> using graded Leibniz rule). This notion is symmetric in ''A'' and <math>A^*</math> (see Roytenberg). Here <math>E=A\oplus A^*</math> with anchor <math>\rho(X+\alpha)=\rho_A(X)+\rho_{A^*}(\alpha)</math> and the bracket is the skew-symmetrization of the above in ''X'' and ''α'' (equivalently in ''Y'' and ''β''):
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| :<math>[X+\alpha,Y+\beta]= ([X,Y]_A +\mathcal{L}^{A^*}_{\alpha}Y-i_\beta d_{A^*}X) +([\alpha,\beta]_{A^*} +\mathcal{L}^A_X\beta-i_Yd_{A}\alpha)</math>
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| ==Skew-symmetric bracket==
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| Instead of the definition above one can introduce a [[Bilinear form#Symmetric, skew-symmetric and alternating forms|skew-symmetric]] bracket as
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| : <math>[[\phi,\psi]]= \tfrac12\big([\phi,\psi]-[\psi,\phi]\big.)</math>
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| This fulfills a homotopic Jacobi-identity.
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| :<math> [[\phi,[[\psi,\chi]]\,]] +\text{cycl.} = DT(\phi,\psi,\chi)</math>
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| where ''T'' is
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| : <math>T(\phi,\psi,\chi)=\frac13\langle [\phi,\psi],\chi\rangle +\text{cycl.}</math> | |
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| The Leibniz rule and the invariance of the scalar product become modified by the relation <math> [[\phi,\psi]] = [\phi,\psi] -\tfrac12 D\langle \phi,\psi\rangle</math> and the violation of skew-symmetry gets replaced by the axiom
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| ::<math> \rho\circ D = 0 </math>
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| The skew-symmetric bracket together with the derivation ''D'' and the Jacobiator ''T'' form a [[strongly homotopic Lie algebra]].
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| == Dirac structures ==
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| Given a Courant algebroid with the inner product <math>\langle.,.\rangle</math> of split signature (e.g. the standard one <math>TM\oplus T^*M</math>), then a [[Dirac structure]] is a maximally isotropic integrable vector subbundle ''L → M'', i.e.
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| :<math> \langle L,L\rangle \equiv 0</math>,
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| :<math> \mathrm{rk}\,L=\tfrac12\mathrm{rk}\,E</math>,
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| :<math> [\Gamma L,\Gamma L]\subset \Gamma L</math>.
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| === Examples ===
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| As discovered by Courant and parallel by Dorfman, the graph of a 2-form ''ω'' ∈ ''Ω''<sup>2</sup>(''M'') is maximally isotropic and moreover integrable iff d''ω'' = 0, i.e. the 2-form is closed under the de Rham differential, i.e. a presymplectic structure.
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| A second class of examples arises from bivectors <math>\Pi\in\Gamma(\wedge^2 TM)</math> whose graph is maximally isotropic and integrable iff [Π,Π] = 0, i.e. Π is a [[Poisson manifold|Poisson bivector]] on ''M''.
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| == Generalized complex structures ==
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| (see also the main article [[generalized complex geometry]])
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| Given a Courant algebroid with inner product of split signature. A generalized complex structure ''L → M'' is a Dirac structure in the [[complexification|complexified]] Courant algebroid with the additional property
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| :<math> L \cap \bar{L} = 0</math>
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| where <math>\bar{\ }</math> means complex conjugation with respect to the standard complex structure on the complexification.
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| As studied in detail by Gualtieri<ref>M. Gualtieri: ''Generalized complex geometry'', Ph.D. thesis, Oxford university, (2004)</ref> the generalized complex structures permit the study of geometry analogous to [[complex manifold|complex geometry]].
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| === Examples ===
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| Examples are beside presymplectic and Poisson structures also the graph of a [[Complex Manifold#Almost complex structures|complex structure]] ''J'': ''TM'' → ''TM''.
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| ==References==
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| <references/>
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| * Dmitry Roytenberg: [http://arxiv.org/abs/math.DG/9910078 Courant algebroids, derived brackets, and even symplectic supermanifolds], PhD thesis Univ. of California Berkeley (1999)
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| [[Category:Differential geometry]]
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