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LA-Courant Algebroids and Their Applications

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LA-Courant Algebroids and Their Applications LA-Courant algebroids and their applications. by David Scott Li-Bland A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy arXiv:1204.2796v1 [math.DG] 12 Apr 2012 Graduate Department of Mathematics University of Toronto Copyright c 2012 by David Scott Li-Bland Abstract LA-Courant algebroids and their applications. David Scott Li-Bland Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2012 In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d; g)-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-Lagrangian) subbundles W ⊆ E of a Courant algebroid such that the Courant bracket endows W naturally with the structure of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the theory of q-Poisson (d; g)-structures. ii Dedication To Esther, the love of my life, and to Wesley, the apple of our eyes. Thanks be to God for his blessings. Acknowledgements I would like to thank my father, John Bland, for inspiring a love of mathematics in me. I would like to thank my supervisor, Eckhard Meinrenken, for his guidance, patience, advice, encouragement and help over the years. I would like to thank Pavol Severaˇ for many delightful conversations, explanations, perspectives, and the math he taught me. I would like to thank Alfonso Gracia-Saz and Rajan Mehta for teaching me supergeometry and the theory of double and LA-vector bundles. I would like to thank Alejandro Cabrera for many interesting discussions and teaching me about tangent prolongations. I would also like to thank Anton Alekseev, Henrique Bursztyn, Arlo Caine, Marco Gualtieri, Travis Li, Jiang Hua Lu, Brent Pym, and Ping Xu for many interesting conversations. Finally, I would like to thank my beautiful wife, Esther, for all her love, her sup- port, her encouragement, and her strength of character. She is the most wise, inspiring, fascinating, and wonderful person I have ever met. iii Contents 1 Introduction 1 1.0.1 A brief history . .1 1.0.2 Multiplicative Courant algebroids, and their infinitesimal versions LA-Courant algebroids . .4 1.0.3 pseudo-Dirac structures . .5 1.0.4 Applications . .7 2 Preliminaries 15 2.1 Linear relations . 15 2.2 Lie algebroids and Courant algebroids . 16 2.2.1 Poisson manifolds . 16 2.2.2 Lie algebroids . 19 2.2.3 Courant algebroids . 22 2.3 Multiplicative Courant algebroids, and multiplicative Manin pairs . 29 2.4 Double structures . 31 2.5 Double vector bundles . 31 2.5.1 The core . 34 2.5.2 Duals of double vector bundles . 37 2.5.3 Total kernels and quotients of double vector bundles . 40 2.6 Triple vector bundles . 41 2.6.1 Duality . 42 2.7 LA-vector bundles and double linear Poisson vector bundles . 44 2.7.1 LA-vector bundles . 44 2.7.2 Double linear Poisson vector bundles . 45 3 VB-Courant algebroids 47 3.1 VB-Courant algebroids . 47 3.2 Lie 2-algebroids . 57 3.3 Further examples . 59 3.4 Tangent prolongation of a Courant algebroid . 61 4 pseudo-Dirac structures 66 4.1 pseudo-Dirac structures . 66 4.1.1 Examples . 73 4.1.2 Forward and backward images of pseudo-Dirac structures . 74 iv 5 LA-Courant algebroids 80 5.1 LA-Courant algebroids . 80 5.1.1 Examples . 82 5.2 Poisson Lie 2-algebroids and multiplicative Courant algebroids . 89 6 Outlook 94 6.1 Reduction of Courant algebroids . 94 6.1.1 Reduction via LA-Dirac structures in T E ............. 94 6.1.2 Reduction via VB-Dirac structures in T E .............. 99 6.2 Integration of q-Poisson (d; g)-structures . 102 6.2.1 A canonical morphism . 102 6.2.2 Transfer of data . 106 flip 6.2.3 The morphism of Manin pairs (TL; T L ) 99K (d × T d; d × T g). 109 6.2.4 Integration corollaries . 111 A Technical proof for Courant algebroids 113 B Technical proof for double vector bundles 116 C Technical proofs for LA-Courant algebroids 119 Bibliography 123 v List of Figures 1.1 For functions c; c0 2 Z(S) vanishing on the submanifold S ⊆ M, the Hamiltonian vector fields XcjS and Xc0 jS may not be tangent to S. D ⊆ TM is the distribution spanned by the restriction of all such vector fields to S.......................................2 1.2 The submanifold S ⊆ M describes constraints on the physical system, while vector fields tangent to the foliation describe symmetries. .8 1.3 The subbundle D ⊆ TMjS of the ambient tangent space controls how functions on S are extended to M ......................9 −1 2.1 The vector ξ 2 TxB is tangent to the fibre qB=M (x), and therefore defines a core element of TB at x 2 M. If σ 2 Γ(B) is such that ξ = σ(x), then for any X 2 TM, σC (X) = X +T B=B ξ, (here the addition takes place in the vector space TxB). ............................ 37 2.2 For any section σ 2 Γ(B), the tangent lift σT 2 Γ(T B; T M) of σ takes X 2 TM to dσ(X) 2 TB........................... 38 vi Chapter 1 Introduction 1.0.1 A brief history Courant algebroids and Dirac structures were first introduced by Courant [28,30] as a geo- metric framework for Dirac's theory of Hamiltonian systems with constraints [34].Courant's original setup was generalized in [69], as a means of constructing doubles for Lie bial- gebroids. Courant algebroids have now found many uses, from the theory of moment maps [2,14,16,17,129] to generalized complex geometry [46,50]. The Dirac bracket We shall go into some more detail. In Hamiltonian mechanics, the phase space - the space of all possible states of a physical system - is described by a smooth manifold M. Smooth functions f 2 C1(M) on the phase space describe various quantities one might wish to measure, such as energy, position, or momentum. Additionally, M carries a Poisson structure: a bivector field π 2 X2(M), such that the bracket ff; gg := π(df; dg); f; g 2 C1(M) endows the vector space C1(M) with the structure of a Lie algebra. Noether's first theorem - that conserved quantities correspond to symmetries - arises as follows: the Poisson structure associates to any function f 2 C1(M) (a conserved quantity) the vector field Xf := g ! ff; gg 2 X(M) (describing the corresponding symmetry of phase space). Introducing constraints on this system corresponds to describing a submanifold S ⊆ M. Ideally, one would like the Poisson bracket {·; ·} on C1(M) to descend to a bracket 1 {·; ·}S on C (S) so that the restriction of functions to S is a morphism of Lie algebras, i.e. ∗ ∗ ∗ 1 fi f; i ggS = i ff; gg; f; g 2 C (M); where i : S ! M is the inclusion. Unfortunately, this is impossible in general, since the vanishing ideal Z(S) ⊆ C1(M) of S ⊆ M might not be a Lie algebra ideal. Equivalently, for an arbitrary function c 2 Z(S), the vector field Xc := fc; ·} 2 X(M) 1 Chapter 1. Introduction 2 (Xc)x x (Xc0 )x Figure 1.1 { For functions c; c0 2 Z(S) vanishing on the submanifold S ⊆ M, the Hamiltonian vector fields XcjS and Xc0 jS may not be tangent to S. D ⊆ TM is the distribution spanned by the restriction of all such vector fields to S. might not vanish when restricted to S (nor even be tangent to S). Let D ⊆ TMjS be the distribution spanned by the vector fields XcjS for c 2 Z(S) (see Fig. 1.1). A function f 2 C1(S) is called admissible if v · f = 0 for any vector v 2 D \ TS. Similarly, a function f~ 2 C1(M) is called an admissible extension of f if i∗f~ = f and v · f~ = 0 for any vector v 2 D. Dirac showed that there exists a Lie bracket {·; ·}DB (called the 1 Dirac bracket) on the subspace Cadm(S) of admissible functions such that ∗ ~ ∗ ∗ ~ ff; ggDB = fi f; i g~gDB = i ff; g~g (1.0.1) ~ 1 1 for any admissible extensions f; g~ 2 C (M) of f; g 2 Cadm(S). Remark 1.0.1. One may interpret the distribution D \ TS as describing some gauge transformations of the physical system: When D \ TS is of constant rank, then it is involutive, and thus, by Frobenius's theorem, defines a foliation. The admissible functions can then be interpreted as the algebra of functions on the leaf space. Courant algebroids In 1986 Courant [28, 30] described a geometric framework for Dirac's theory of con- strained Hamiltonian mechanics, which we shall briefly summarize. The Pontryagin Chapter 1. Introduction 3 bundle TM := TM ⊕ T ∗M carries a split signature metric defined by the natural paring h(v; µ); (w; ν)i := µ(w) + ν(v); (v; µ); (w; ν) 2 TM: Courant's insight was to consider Lagrangian subbundles of the Pontryagin bundle (i.e. L 2 TM such that L? = L), replacing the Poisson bivector field π 2 X2(M) with its graph gr(π]) := f(X; α) 2 TM ⊕ T ∗M j X = π(α; ·)g: He described a bracket, which is usually referred to as the Courant bracket in the litera- ture,1 [[(X; α); (Y; β)]] = [X; Y ]; LX β − ιY dα (1.0.2) on the space of sections Γ(TM), and showed that this bracket restricts to a Lie bracket on Γ gr(π]).A Dirac structure is defined to be an arbitrary Lagrangian subbundle L ⊆ TM which is involutive with respect the Courant bracket.
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