Structure of classical W-algebras

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2013

Author dc:creator

• Suh, Uhi Rinn

Advisor dc:contributor.advisor

• Victor G. Kac.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/83700

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/83700