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Permanent link (DOI): https://doi.org/10.7939/R3X34MZ75

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Conjugacy problems for "Cartan" subalgebras in infinite dimensional Lie algebras Open Access

Descriptions

Other title
Subject/Keyword
Lie algebras
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Yahorau, Uladzimir
Supervisor and department
Chernousov, Vladimir (Mathematical and Statistical Sciences)
Pianzola, Arturo (Mathematical and Statistical Sciences)
Examining committee member and department
Gille, Stefan (Mathematical and Statistical Sciences)
Kuttler, Jochen (Mathematical and Statistical Sciences)
Litvak, Alexander (Mathematical and Statistical Sciences)
Department
Department of Mathematical and Statistical Sciences
Specialization
Mathematics
Date accepted
2014-05-16T10:14:44Z
Graduation date
2014-11
Degree
Doctor of Philosophy
Degree level
Doctoral
Abstract
Chevalley's theorem on the conjugacy of split Cartan subalgebras is one of the cornerstones of the theory of simple finite dimensional Lie algebras over a field of characteristic 0. Indeed, this theorem affords the most elegant proof that the root system is an invariant of the Lie algebra. The analogous result for symmetrizable Kac-Moody Lie algebras is the celebrated theorem of Peterson and Kac. However, the methods they used are not suitable for attacking the problem of conjugacy in "higher nullity", i.e. for extended affine Lie algebras (EALA). In the thesis we develop a new cohomological approach which we use to prove 1) conjugacy of Cartan subalgebras in affine Kac-Moody Lie algebras; 2) conjugacy of maximal abelian ad-diagonalizable subalgebras (MADs) of EALA of finite type, coming as a part of the structure, where we assume that the centreless core is not isomorphic to sl_2(R), R is a ring of Laurent polynomials in more then 1 variables. We give a counterexample to conjugacy of arbitrary MADs in EALA. Some relevant problems on the lifting of automorphisms are discussed as well.
Language
English
DOI
doi:10.7939/R3X34MZ75
Rights
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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