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Quantum Algebras Associated to Irreducible Generalized Flag Manifolds Quantum Algebras Associated to Irreducible Generalized Flag Manifolds by Matthew Tucker-Simmons A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley arXiv:1308.4185v1 [math.QA] 19 Aug 2013 Committee in charge: Professor Marc A. Rieffel, Chair Professor Nicolai Reshetikhin Professor Ori Ganor Fall 2013 Quantum Algebras Associated to Irreducible Generalized Flag Manifolds Copyright 2013 by Matthew Tucker-Simmons 1 Abstract Quantum Algebras Associated to Irreducible Generalized Flag Manifolds by Matthew Tucker-Simmons Doctor of Philosophy in Mathematics University of California, Berkeley Professor Marc A. Rieffel, Chair In the first main part of this thesis we investigate certain properties of the quantum sym- metric and exterior algebras associated to Type 1 representations of quantized universal enveloping algebras of semisimple Lie algebras, which were defined and studied by Beren- stein and Zwicknagl in [BZ08, Zwi09a]. We define a notion of a commutative algebra object in a coboundary monoidal category, and we prove that, analogously to the classical case, the quantum symmetric algebra associated to a module is the universal commutative algebra generated by that module. That is, the functor assigning to a module its quantum symmet- ric algebra is left adjoint to an appropriate forgetful functor, and likewise for the quantum exterior algebra. We also prove a strengthened version of a conjecture of Berenstein and Zwicknagl, which states that the quantum symmetric and exterior cubes exhibit the same amount of “collapsing” relative to their classical counterparts. More precisely, the difference in dimension between the classical and quantum symmetric cubes of a given module is equal to the difference in dimension between the classical and quantum exterior cubes. We prove that those quantum exterior algebras that are “flat deformations” of their classical analogues are Frobenius algebras. We also develop a rigorous framework for discussing continuity and limits of the structures involved as the deformation parameter q varies along the positive real line. The second main part of the thesis is devoted to quantum analogues of Clifford algebras and their application to the noncommutative geometry of certain quantum homogeneous spaces; this is where the thesis gets its name. We introduce the quantum Clifford algebra through its “spinor representation” via creation and annihilation operators on one of the flat quantum exterior algebras discussed in the first main part of the thesis. The proof that the spinor representation is irreducible, and hence that the creation and annihilation operators generate the full endomorphism algebra of the quantum exterior algebra, relies in an essential way on the Frobenius property for the quantum exterior algebra. We then use this notion of quantum Clifford algebra to revisit Krähmer’s construction in [Krä04] of a Dolbeault-Dirac-type operator on the canonical spinc structure over a quantized irreducible 2 flag manifold. This operator is of the form ð+ð∗, and we prove that ð can be identified with the boundary operator for the Koszul complex of a certain quantum symmetric algebra, which shows that ð2 = 0. This is a first step toward establishing a Parthasarathy-type formula for the spectrum of the square of the Dirac operator, and hence toward proving that it satisfies the technical conditions to be part of a spectral triple in the sense of Connes. Parts of this work appear in the preprints [CTS12, KTS13]. i Dedicated to Lisa Thus is our treaty written; thus is agreement made. Thought is the arrow of time; memory never fades. What was asked is given. The price is paid. –Robert Jordan, The Shadow Rising ii Contents Contents ii List of Tables iii 1 Introduction 1 1.1 Motivation..................................... 1 1.2 Lie groups, Lie algebras, and quantum groups . ....... 4 1.3 Quantum symmetric and exterior algebras . ...... 5 1.4 Quantum Clifford algebras and flag manifolds . ...... 10 1.5 Standardnotation................................ 15 2 Lie groups, Lie algebras, and quantum groups 17 2.1 SemisimpleLiealgebras . 17 2.2 Somesimpleexercises. .. .. 20 2.3 Parabolicsubalgebras. 23 2.4 Liegroups..................................... 35 2.5 Generalizedflagmanifolds . 35 2.6 Quantized enveloping algebras . ..... 36 2.7 Quantum Schubert cells and their twists . ...... 43 3 Quantum symmetric and exterior algebras 46 3.1 Preliminaries ................................... 46 3.2 Definitionsofthealgebras . 49 3.3 Canonical bases and continuity . .... 54 3.4 Quantum symmetric algebras are commutative . ....... 69 3.5 Flatnessandrelatedproperties . ..... 73 3.6 Collapsingindegreethree . 96 4 Quantum Clifford algebras and flag manifolds 106 4.1 Creationandannihilationoperators . ....... 106 4.2 The Dolbeault-Dirac operator on quantized flag manifolds .......... 117 4.3 The quantum Clifford algebra for CP2 ..................... 123 iii Bibliography 128 Index of notation 135 Index of terms 137 List of Tables 2.1 Dynkindiagramsandhighestroots . ...... 32 2.2 Parabolic Dynkin diagrams of simple Lie algebras . ........... 34 iv Acknowledgments There are many people that deserve thanks here for their contributions, direct or otherwise, to this thesis. First and foremost is my wonderful wife Lisa, who encouraged me along the way and especially helped to push me along in the final months. I can only hope that I will be as supportive of her during the completion of her doctorate as she has been during the completion of mine. I thank also my family, who have always inspired me with their dedication to knowledge, scholarship, and research, and who helped with key grammatical questions. On the mathematical side of things, I thank my advisor, Marc Rieffel, who helped me to find research questions to pursue, but in doing so gave me the freedom to explore my own interests. His helpful advice on matters mathematical and professional, and generous support over the years through NSF grant DMS-1066368, were instrumental in bringing this project to completion. I thank my co-advisor, Nicolai Reshetikhin, for his helpful comments and questions, and for pointing out useful references in the vast literature on quantum groups. It is a great pleasure to thank Ulrich Krähmer. Our chance meeting at the 2009 Spring School on Noncommutative Geometry at the IPM in Tehran led not only to our collaboration, upon which part of this work is based, but also to our friendship. I benefited greatly from many discussions with my professors and colleagues at Berkeley. I would like to recognize especially Professor Vera Serganova, who answered many questions on Lie theory and taught an excellent course on the subject. Scott Morrison helped me in learning to use Mathematica. I thank Alex Chirvasitu for his great willingness to discuss technical details of almost anything. Our collaboration was very enjoyable and led to part of the results of this thesis. Finally, I thank the Hillegass-Parker Co-op in Berkeley. My six years spent there taught me a lot about life and about people, and gave me the opportunity to interact with and learn from dozens of brilliant scholars, in disciplines ranging from history and political science to physics and materials science. The friends that I made there helped me in so many ways during this journey. I cannot possibly list them all, but I would be remiss not to mention at least these few: Megan Williams, Jesse Hart Fischer-Engel, Adam Ganes, Andrew Friedman, Tim Ruckle, and Sarah Daniels. 1 Chapter 1 Introduction In this brief chapter we describe the contents of the rest of this thesis. We begin in Section 1.1 by giving a broad outline of the background and motivation for the problems studied here. Then in Sections 1.2 to 1.4 we describe the content of the three following chapters. Finally in Section 1.5 we set notation that will be used throughout the entire thesis. 1.1 Motivation The original motivation for the results of this thesis was to further understand Krähmer’s construction in [Krä04] of a Dirac-type operator on certain quantum homogeneous spaces, and to make explicit certain aspects of that construction. The general setting is that of noncommutative geometry, in which the main theme is to understand a noncommutative algebra as the algebra of functions on a “quantum space,” with Hopf algebras playing the role of symmetry groups. This project lies at the crossroads of the representation-theoretic and metric approaches to noncommutative geometry. To set the stage for the results, we give some brief background on these two viewpoints before describing the contents of the thesis. 1.1.1 Homogeneous spaces, representation theory, and quantum groups The representation-theoretic viewpoint can be traced back to Felix Klein’s Erlangen Pro- gram [Kle93], in which he proposed that geometries should be classified and understood by their groups of symmetries. A modern perspective on this idea is that we should study ho- mogeneous spaces, that is, spaces of the form G/H, where G is a Lie group (or an algebraic group) and H is a closed (or Zariski-closed) subgroup. The fact that many spaces of independent interest can be realized in this way makes the study of homogeneous spaces attractive. For instance, spheres, projective spaces, Grassman- nians, and flag manifolds are all homogeneous spaces. CHAPTER 1. INTRODUCTION 2 The fact that homogeneous spaces carry a transitive group action allows one to make global constructions
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