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Factorization in unitary loop groups and reduced words in affine Weyl groups.

The purpose of this dissertation is to elaborate, with specific examples and calculations, on a new refinement of triangular factorization for the loop group of a simple, compact Lie group K, first appearing in Pickrell & Pittman-Polletta 2010. This new factorization allows us to write a smooth map from the unit circle into K (having a triangular factorization) as a triply infinite product of loops, each of which depends on a single complex parameter. These parameters give a set of coordinates on the loop group of K.The order of the factors in this refinement is determined by an infinite sequence of simple generators in the affine Weyl group associated to K, having certain properties. The major results of this dissertation are examples of such sequences for all the classical Weyl groups.We also produce a variation of this refinement which allows us to write smooth maps from the unit circle into the special unitary group of n by n matrices as products of 2n+1 infinite products. By analogy with the semisimple analog of our factorization, we suggest that this variation of the refinement has simpler combinatorics than that appearing in Pickrell & Pittman-Polletta 2010.

Identiferoai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/194348
Date January 2010
CreatorsPittman-Polletta, Benjamin Rafael
ContributorsPickrell, Doug, Pickrell, Doug, Pickrell, Doug, Glickenstein, David, Palmer, John, Flaschka, Hermann, Venkataramani, Shankar
PublisherThe University of Arizona.
Source SetsUniversity of Arizona
LanguageEnglish
Detected LanguageEnglish
Typetext, Electronic Dissertation
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.

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