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The Classification and Dynamics of the Momentum Polytopes of the Su(3) Action on Points in the Complex Projective Plane with an Application to Point Vortices

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The Classification and Dynamics of the Momentum Polytopes of the Su(3) Action on Points in the Complex Projective Plane with an Application to Point Vortices The classification and dynamics of the momentum polytopes of the SU(3) action on points in the complex projective plane with an application to point vortices Shaddad, Amna 2018 MIMS EPrint: 2019.9 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 THE CLASSIFICATION AND DYNAMICS OF THE MOMENTUM POLYTOPES OF THE SU(3) ACTION ON POINTS IN THE COMPLEX PROJECTIVE PLANE WITH AN APPLICATION TO POINT VORTICES A THESIS SUBMITTED TO THE UNIVERSITY OF MANCHESTER FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF ENGINEERING AND PHYSICAL SCIENCE 2018 Amna Shaddad School of Mathematics Contents Abstract 9 Declaration 10 Copyright Statement 11 1 Introduction 12 Acknowledgments 22 2 Symplectic Geometry 24 2.1 Weyl Chambers . 24 2.1.1 Rudimentary Principles . 24 2.1.1.1 The notion of Lie algebra, linear Lie algebras and Lie algebra derivations . 24 2.1.1.2 Ideals, homomorphisms, representations, automor- phisms, solvability and nilpotency . 26 2.1.1.3 Engel's Theorem . 28 2.1.2 Semisimple Lie Algebras . 28 2.1.2.1 Lie's theorem, Jordan-Chevalley decomposition, Cartan's criterion, criterion for simplicity, simple ideals of L ...................... 28 2.1.2.2 Inner derivations, abstract Jordan decomposition, modules, Casimir elements of representation, Weyl's theorem, preservation of Jordan decomposition . 30 2.1.2.3 Weights, maximal vectors, irreducible module clas- sification . 31 2 CONTENTS 3 2.1.2.4 Maximal toral subalgebras, roots, centraliser of H, orthogonality, integrality and rationality properties 32 2.1.3 Root Systems . 34 2.1.3.1 Reflections, root systems, root pairs . 34 2.1.3.2 Weyl chambers, simple roots, the Weyl group and irreducible root systems . 36 2.1.3.3 Cartan matrices and their irreducible parts, weights 40 2.1.4 Reduction and Isomorphism . 42 2.2 Momentum Map . 43 2.2.1 Symplectic Manifolds . 43 2.2.2 Almost Complex Structures . 44 2.2.3 Symplectic and Hamiltonian Actions of R ......... 45 2.2.4 Lie Groups . 46 2.2.5 Orbit Space . 47 2.2.6 Momentum Maps . 48 2.2.6.1 Examples . 49 2.3 Coadjoint Orbit . 51 2.3.1 Coadjoint Equivariant Momentum Map . 52 2.3.2 Symplectic Reduction . 53 2.3.2.1 The orbit momentum map . 54 2.4 Morse-Bott Functions . 55 2.5 Convexity Properties of Momentum Mapping . 57 2.5.1 Introduction . 57 2.5.2 The Atiyah-Guillemin-Sternberg Convexity Theorem . 61 2.5.2.1 Preliminaries . 61 2.5.2.2 Convex Properties . 62 2.5.3 The non-abelian convexity theorem . 63 2.5.4 Reyer Sjamaar's re-examination . 64 2.5.5 Simple examples of projections of some coadjoint orbits to t∗ 65 2.5.6 A by-no-means-exhaustive bibliography of modern devel- opments . 66 2.5.6.1 Infinite dimensional Lie theory . 66 2.5.6.2 Linear and nonlinear symplectic actions . 66 2.5.6.3 The Local-Global Convexity Principle . 68 2.6 The Bifurcation Lemma . 68 4 CONTENTS 2.7 Dynamics . 72 2.7.1 Relative Equilibrium . 72 2.8 The Witt-Artin Decomposition . 74 3 SU(3) action on products of CP2 76 3.1 Some General Properties of SU(3) and CP2 ............. 76 3.2 Isotropy Subgroups for SU(3) Acting on (CP2)2 .......... 77 3.3 Isotropy Subgroups for SU(3) Acting on (CP2)3 .......... 79 3.4 Isotropy Subgroups for SU(n + 1) Acting on (CPn)3 ........ 80 4 The Momentum Map 82 4.1 The Fubini-Study Form . 82 4.2 Properties of the Momentum Map . 84 4.3 The Weighted Momentum Map . 86 5 Momentum Polytopes 87 5.1 Weighted Momentum Map of SU(3) Acting on (CP2)2 ....... 88 5.1.1 The Γ1 to Γ2 Ratio . 91 5.2 The Momentum Map for SU(3) Acting on (CP2)3 ......... 92 5.2.1 The Γ1 to Γ2 to Γ3 Ratio . 93 5.3 Momentum Polytopes for SU(3) Acting on (CP2)2 ......... 104 5.4 Transition Polytopes for SU(3) Acting on (CP2)2 ......... 109 5.5 Gradient of Edges of Momentum Polytopes . 111 5.6 Momentum Polytopes for SU(3) Acting on (CP2)3 ......... 114 5.7 Transition Polytopes for SU(3) Acting on (CP2)3 ......... 135 6 Definite and Indefinite Vertices 146 7 Dynamics 160 7.1 Generalised Vortex Dynamics . 160 7.1.1 Spherical Vortices . 161 7.1.2 Our Generalisation . 162 7.2 Symplectic Geometry of Vortex Systems . 163 7.2.1 Vortices on (CP2)2 ...................... 163 7.2.2 Vortices on (CP2)3 ...................... 164 7.3 The Space of Allowed Velocity Vectors R0 .............. 165 CONTENTS 5 8 Further Proposed Research 169 8.1 The Littlewood-Richardson Polytope . 169 8.2 Hamiltonian . 169 8.3 Further Generalisation . 170 8.4 Semi-Toric Systems . 170 Bibliography 171 List of Figures 1.1 The momentum polytopes of SU(3) y CP2 × CP2....................... 15 1.2 The momentum polytopes of SU(3) y CP2 × CP2 × CP2....................... 16 1.3 Example transition polytopes for SU(3) y CP2 × CP2 × CP2................ 17 sl(2; F) root system ................... 35 Some rank 2 root system ................... 35 & 36 Fundamental Weyl chamber example ................... 38 Dominant weights example ................... 41 Generic and Degenerate orbits of SU(3) ................... 65 5.1 The positive Weyl chamber ....................... 88 5.2 The Γ1 to Γ2 ratio .................................91 5.3 The Γ1 + Γ2 + Γ3 = ∆ plane on the Γ1 - Γ2 - Γ3-axes ...............................93 5.4 The Γ1 + Γ2 + Γ3 = ∆ plane flattened ....................................94 5.5 The Γ1+Γ2+Γ3 = ∆ plane and Γ3 = 0 shown and marked ............................94 5.6 Γ1 =6 0 and Γ2 = 0 also shown ................................95 5.7 Γ1 = Γ2 drawn and labelled ...................96 5.8 Γ2 = Γ3 added ..........................................97 5.9 Γ1 = Γ3 added ...............................98 5.10 Γ1 = -Γ2 and Γ3 = Γ1 + Γ2 included ...........................99 5.11 Γ1 = -Γ3 and Γ2 = Γ1 + Γ3 added .............................100 5.12 Focus between Γ2 = Γ3 and Γ1 = Γ3 .................................101 5.13 Regions A to H marked ......................................102 5.14 Γ1 > Γ2 > 0 momentum polytope ............................105 5.15 Γ2 > 0 > Γ1 and jΓ2j > jΓ1j momentum polytope ......................106 5.16 0 > Γ1 > Γ2 momentum polytope ...................................107 5.17 Γ1 > 0 > Γ2 and jΓ2j > jΓ1j.........................108 5.18 Γ1 = Γ2..........................................109 5.19 Γ1 = -Γ2.........................................110 6 CONTENTS 7 5.20 An Example polytope for SU(3) action on CP2 × CP2 × CP2 ........115 5.21 Momentum polytope A.....................................116 5.22 Momentum polytope A: First Step...................................117 5.23 Momentum polytope A: Second Step..............................118 5.24 Momentum polytope A: Third Step.................................119 5.25 Momentum polytope A: Fourth Step..............................120 5.26 Momentum polytope A: Fifth Step....................................121 5.27 Momentum polytope B.....................................123 5.28 Momentum polytope C.....................................124 5.29 Momentum polytope D.....................................125 5.30 Momentum polytope D: First Step.....................................126 5.31 Momentum polytope D: Second Step.....................................127 5.32 Momentum polytope D: Third Step.....................................128 5.33 Momentum polytope E...................................130 5.34 Momentum polytope F...................................131 5.35 Momentum polytope G...................................132 5.36 Momentum polytope H...................................133 5.37 Momentum polytope Zero...................................134 5.38 Transition polytope from A to B......................137 5.39 Transition polytope from C to H......................139 5.40 Transition polytope from E to F......................140 5.41 Transition polytope from C to E......................141 5.42 Transition polytope from F to H......................142 5.43 Transition polytope from F to G......................143 6.1 Momentum polytope for Γ1 = -5:5, Γ2 = 27:5 and Γ3 = 8:25...........151 6.2 Root diagram for SU(3)...............................152 6.3 Vertices of momentum polytope marked...........153 6.4 Definite and indefinite vertices shown...........154 6.5 First potential edge marked..............155 6.6 All potential edges marked....................157 6.7 Internal edge shown............................158 6.8 Final convex skeleton........................159 List of Tables R0 for two-vortex configurations......................................168 8 The University of Manchester Amna Shaddad Doctor of Philosophy The Classification and Dynamics of the Momentum Polytopes of the SU(3) Action on Points in the Complex Projective Plane with an Ap- plication to Point Vortices We have fully classified the momentum polytopes of the SU(3) action on CP2 × CP2 and CP2 × CP2 × CP2, both actions with weighted symplectic forms, and their corresponding transition momentum polytopes. For CP2 × CP2 the momentum polytopes are distinct line segments. The action on CP2 × CP2 × CP2 has 8 different momentum polytopes. The vertices of the momentum polytopes of the SU(3) action on CP2 × CP2 × CP2 fall into two categories: definite and indefinite vertices. The reduced space corresponding to momentum map image values at definite vertices is isomorphic to the 2-sphere. We show that these results can be applied to assess the dynamics by introducing and computing the space of allowed velocity vectors for the different configurations of two-vortex systems. 9 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 10 Copyright Statement i) The author of this thesis (including any appendices and/or schedules to this thesis) owns any copyright in it (the \Copyright") and s/he has given The University of Manchester the right to use such Copyright for any adminis- trative, promotional, educational and/or teaching purposes. ii) Copies of this thesis, either in full or in extracts, may be made only in accordance with the regulations of the John Rylands University Library of Manchester.
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