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

  • Author / Creator
    Yahorau, Uladzimir
  • 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.

  • Subjects / Keywords
  • Graduation date
    2014-11
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/R3X34MZ75
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
    English
  • Institution
    University of Alberta
  • Degree level
    Doctoral
  • Department
    • Department of Mathematical and Statistical Sciences
  • Specialization
    • Mathematics
  • Supervisor / co-supervisor and their department(s)
    • Chernousov, Vladimir (Mathematical and Statistical Sciences)
    • Pianzola, Arturo (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Litvak, Alexander (Mathematical and Statistical Sciences)
    • Kuttler, Jochen (Mathematical and Statistical Sciences)
    • Gille, Stefan (Mathematical and Statistical Sciences)