2026-09-19T01:46:18Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/837002022-01-13T07:54:04Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Victor G. Kac.
Suh, Uhi Rinn
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2014-01-09T19:46:19Z
2014-01-09T19:46:19Z
2013
http://hdl.handle.net/1721.1/83700
864162264
Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 171-172).
The first part of the thesis provides three equivalent definitions of a classical finite W-algebra Wfin(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 [epsilon] 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 Wfin (9, f). A classical finite (resp. affine) W-algebra is isomorphic to a polynomial (resp. differential polynomial) ring. I compute Poisson brackets (resp. Poisson [lambda]-brackets) between generating elements 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 Wfin(g, Am, k) (resp. Wfin(g, Am,k)), where Am = -fz- m - pz-m-1 [epsilon] g((z)) for a certain p [epsilon] g and an integer m >/= 0. If m = 0, then the algebra Wfin(g, Am, k) (resp. W(g, Am, k)) is isomorphic to Wfin(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 Wfin(g, Am, k). As in ordinary W-algebras cases, a classical finite (resp. affine) fractional W-algebra is isomorphic to a polynomial (resp. differential polynomial) ring. In particular, I show explicit forms of generators and compute brackets (resp.[lambda]-brackets) between them when f is a minimal 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 Hamiltonian 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.
by Uhi Rinn Suh.
Ph.D.
172 pages
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Structure of classical W-algebras
Thesis
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<Abstract>The first part of the thesis provides three equivalent definitions of a classical finite W-algebra Wfin(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 [epsilon] 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 Wfin (9, f). A classical finite (resp. affine) W-algebra is isomorphic to a polynomial (resp. differential polynomial) ring. I compute Poisson brackets (resp. Poisson [lambda]-brackets) between generating elements 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 Wfin(g, Am, k) (resp. Wfin(g, Am,k)), where Am = -fz- m - pz-m-1 [epsilon] g((z)) for a certain p [epsilon] g and an integer m >/= 0. If m = 0, then the algebra Wfin(g, Am, k) (resp. W(g, Am, k)) is isomorphic to Wfin(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 Wfin(g, Am, k). As in ordinary W-algebras cases, a classical finite (resp. affine) fractional W-algebra is isomorphic to a polynomial (resp. differential polynomial) ring. In particular, I show explicit forms of generators and compute brackets (resp.[lambda]-brackets) between them when f is a minimal 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 Hamiltonian 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.</Abstract>
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