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Structure of Classical W-Algebras Uhi Rinn

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Structure of Classical W-Algebras Uhi Rinn Structure of classical W-algebras by Uhi Rinn Suh Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY R\V June 2013 ANSFEW © Uhi Rinn Suh 2013. All rights reserved. Et A u th o r .............................................................. Department of Mathematics April 24, 2013 C ertified by ............. ........................... Victor G. Kac Professor Thesis Supervisor Accepted by ........................................... Paul Seidel Chair, Graduate Committee in Pure Mathematics 2 Structure of classical W-algebras by Uhi Rinn Suh Submitted to the Department of Mathematics on April 24, 2013, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Abstract The first part of the thesis provides three equivalent definitions of a classical finite W-algebra Wf"n(g, f) and two equivalent definitions of a classical affine W-algebra W(g, f, k) associated to a Lie algebra g, a nilpotent element f and k E C. A classical affine W-algebra W(g, f, k) has a Hamiltonian operator H and the H-twisted Zhu algebra of W(g, f, k) is the classical finite W-algebra Wi"(9, f). A classical finite (resp. affine) W-algebra is isomorphic to a polynomial (resp. differential polynomial) ring. I compute Poisson brackets (resp. Poisson A-brackets) between generating ele- ments of a classical finite (resp. affine) W-algebra when f is a minimal nilpotent. In the second part, I introduce a classical finite (resp. affine) fractional W-algebra Wf/"(g, Am, k) (resp. W(g, Am,k)), where Am = -fz- m - pz-m-1 E g((z)) for a certain p E g and an integer m > 0. If m = 0, then the algebra Wf/"(g, Am, k) (resp. W(g, Am, k)) is isomorphic to Wfi"(g, f) (resp. W(g, f, k)). I show that an affine fractional W-algebra W(g, Am, k) has a Hamiltonian operator H and the H- twisted Zhu-algebra of W(g, Am, k) is Wfi"(g, Am, k). As in ordinary W-algebras cases, a classical finite (resp. affine) fractional W-algebra is isomorphic to a poly- nomial (resp. differential polynomial) ring. In particular, I show explicit forms of generators and compute brackets (resp.A-brackets) between them when f is a mini- mal nilpotent. Using generalized Drinfel'd and Sokolov reduction, I find an infinite sequence of integrable systems related to an affine fractional W-algebra when Am is a semisimple element in g((z)). In the last part, I introduce generalized Drinfeld-Sokolov reductions and Hamilto- nian ODEs associated to classical finite W-algebras and finite fractional W-algebras. Also, I find integrals of motion of the Hamiltonian ODEs using Drinfel'd-Sokolov reductions. It is an open problem whether these equations are Lenard integrable. Thesis Supervisor: Victor G. Kac Title: Professor 3 4 Acknowledgments I would like to thank my thesis advisor, Victor Kac. He carefully listened about my work and whenever I got stuck he suggested new ways to resolve the difficulties. Also, he thoroughly reviewed the thesis and gave lots of good comments. His orga- nized guidance made this work possible. While I stayed in France, he helped me to adapt well to new environments. It was honored to be his student for past 5 years. Besides my thesis advisor, I had nice friends in my academic family: Jethro van Ekeren, Sylvain Carpentier and Daniel Valeri. We were good colleagues, good travel mates and good personal friends. I believe our relationship will be continued. Alberto De sole, who is my academic brother, showed interest in my research and counseled about my future. I also want to thank for his consideration while I was in France. It was pleased to learn from ingenious professors, including my thesis advisor Victor and former advisor Seok-Jin Kang in my college. Especially, I need to thank to Pavel Etingof, Gigliola Staffilani, who were my thesis committee members, and Ju-lee Kim who was my first year advisor. During my Ph.D. program, I could enjoy the life in the department thanks to my former and present officemates: Giorgia Fortuna, Sasha Tsymbaliuk, Jiayong Li, Jen- nifer Park, Roberto Svaldi, Alexander Moll, Timothy Nguyen and Nikola Kamburov. I also want to mention Vedran Sohinger, Mario De Franco and Eric Marberg to thank for their friendship. I was pleased to meet nice Korean friends : Eun Seon Cho, ShinYoung Kang, Yoon- suk Hyun, Hwanchul Yoo, Taedong Yun, Bonjun Koo, Stephanie Nam, Leebong Lee, Heesang Ju, Jiyoun Chang, Nicole Yang, Jungyun Kim. They made Boston to be my second hometown. Furthermore, I would like to thank my lifelong friends : Hyewon Jung, Yesul Choi, Rae Na, Okkyung Bang, Jinha Yoon, Seungeun Lee. They have motivated and in- spired me. 5 I would like to especially thank my family from the bottom of my heart. My mom and dad, Young Auk Kim and Myung Su Suh, always trust my decision and supportive of me. My sister, Rinn Ha Suh, is my best life counselor. My grandmother, Younghee Lee, raised me when I was very young and she has been always on my part. Finally, for the past five years, my fianc6 Younghun Hong was with me. I could happily overcome all the difficulties since he did not let me alone. 6 Contents 1 Introduction 9 2 (Ordinary) Classical W-algebras 17 2.1 Three equivalent definitions of classical finite W-algebras . .. .. 17 2.1.1 The first definition of classical finite W-algebras . .. .. .. 18 2.1.2 The second definition of classical finite W-algebras .. .. .. 21 2.1.3 The third definition of classical finite W-algebras . .. .. 24 2.1.4 Equivalence of the three definitions of classical finite W-algebras 28 2.1.5 Second proof of the equivalence of the three definitions .. 31 2.2 Two equivalent definitions of classical affine W-algebras . .. .. .. 39 2.2.1 The first definition of classical affine W-algebras . .. .. .. 39 2.2.2 The second definition of classical affine W-algebras .. .. .. 42 2.2.3 Equivalence of the two equivalent definitions of classical affine W -algebras .. .. .. .. .. .. .. .. .. 45 2.3 Generating elements of classical finite and affine W-algebras .. .. 54 2.3.1 Generating elements of a classical finite W-algebra and Poisson brackets between the generating elements . .. .. .. .. 54 2.3.2 Generating elements of a classical affine W-algebra and Poisson A-brackets between the generating elements .. .. .. .. 71 2.4 Relation between classical affine and finite W-algebras . .. .. .. 81 3 Fractional /V-algebras 87 3.1 Two equivalent definitions of affine fractional W-algebras .. .. 87 7 3.1.1 First definition of affine fractional W-algebras .... .... 90 3.1.2 Second definition of affine fractional W-algebras ... ..... 97 3.2 Generating elements of an affine fractional W-algebra and Poisson brackets between them .......................... 105 3.2.1 Generating elements of affine fractional W-algebras when g = s1 2 107 3.2.2 Generating elements of affine fractional W-algebras when g = sl, 109 3.2.3 Generating elements of affine fractional W-algebras when f is a minimal nilpotent ... .......... ........... 112 3.3 Two equivalent definitions of finite fractional W-algebras ...... 122 3.3.1 First definition of finite fractional W-algebras .... .... 122 3.3.2 Second definition of finite fractional W-algebras .... ... 126 4 Hamiltonian Equations and Classical W-algebras 133 4.1 Integrable systems associated to affine fractional W-algebras ..... 133 4.2 Generalized Drinfel'd-Sokolov reductions for finite W-algebras ... 143 4.2.1 Generalized Drinfel'd-Sokolov reductions for classical finite W- algebras .................. ............ 143 4.2.2 Generalized Drinfel'd-Sokolov reductions for finite fractional W-algebras ............... ............. 150 A Appendix 159 A.1 Lie superalgebras and Lie conformal superalgebras ........ ... 159 A.2 Vertex algebras and Poisson vertex algebras ......... ..... 160 A.3 Nonlinear Lie superalgebras and Nonlinear Lie conformal superalgebras 163 A.3.1 Nonlinear Lie superalgebras .... ............ ... 163 A.3.2 Nonlinear Lie conformal superalgebras . ........... 165 A.4 Zhu algebras ........... .................... 168 8 Chapter 1 Introduction In this thesis, we discuss classical W-algebras and related integrable systems. Drin- fel'd and Sokolov constructed W-algebras and associated integrable systems by the so called Drinfel'd-Sokolov reduction in [11]. In Section 2.1 and 2.2, we introduce several definitions of finite and affine classical W-algebras. De Sole and Kac explained in [4] three equivalent definitions of finite quantum W-algebras. Three definitions of finite classical W-algebras in Section 2.1 are analogue of the definitions in [4]. De Sole and Kac [4] also defined affine quantum W-algebras as vertex algebras by quantum BRST complexes. Since a Poisson vertex algebra (PVA, see the Appendix) is a quasi-classical limit of a one parameter family of vertex algebras, one may obtain affine classical W-algebras by classical limits of the quantum BRST complexes. Another possible way to define classical W-algebras is by Hamiltonian reductions. Gan and Ginzburg proved in [6] that a finite W-algebra Wf"(g, f) for a simple Lie algebra g and a nilpotent element f is (U(g)/ (X ± (f, )))ad, (1.1) where n is a nilpotent subalgebra of g and ( , ) is a bilinear form on g. Since an affine classical W-algebra is a PVA (see the Appendix), it has a Poisson A-bracket 9 structure. Hence we consider the space (S(C[9] 0 g)/ (X ± (f, X)))adA,, (1.2) and show that this definition is equivalent to the definition by the classical BRST complex. An affine classical W-algebra (1.2) is closely related to the classical W-algebra ob- tained by the Drinfel'd-Sokolov reduction in [11] (in the case when f is the principle nilpotent element of g).
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