Orthosymplectic, Periplectic, and Twisted Super Yangians

  • Author / Creator
    Kettle, Bryan W
  • This dissertation establishes various structural and representation theoretic results in super Yangian theory.

    In its first part, this dissertation details the algebraic structure and representation theory for the Yangians of orthosymplectic Lie superalgebras. Addressing these Yangians via the RTT realization, we prove a Poincaré-Birkhoff-Witt-type theorem and provide a thorough study of the algebraic structure of their extended Yangians. The main result of this part, and of this dissertation, is the provision of many necessary conditions for the irreducible representations of these orthosymplectic Yangians to be finite-dimensional; furthermore, there is much progress made to address attaining sufficient conditions as well. These representation theoretic results are accomplished via the development of a highest weight theory, and such necessary conditions are given in terms of highest weights and tuples of Drinfel'd polynomials.

    The second part of this dissertation is devoted to the Yangians of periplectic Lie superalgebras and the twisted Yangians associated to symmetric superpairs of type AIII.

    Via the RTT formalism, we prove many structural results for the Yangians of type P strange Lie superalgebras that have only so far been established for the Yangians of type Q strange Lie superalgebras, including a proof of a Poincaré-Birkhoff-Witt-type theorem.

    The twisted super Yangians of type AIII are defined along with many structural properties established. We lay the foundation for the classification of their finite-dimensional irreducible representations by cultivating a highest weight theory and proving that all finite-dimensional irreducible modules must be highest weight.

  • Subjects / Keywords
  • Graduation date
    Fall 2023
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • 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.