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


Other title
Lie algebras
Type of item
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 of Mathematical and Statistical Sciences
Date accepted
Graduation date
Doctor of Philosophy
Degree level
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.
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