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Representations of Twisted Yangians of Types B, C and D

 Author / Creator
 Wendlandt, Curtis

In the first part of this dissertation, we prove a generalization of a theorem of Drinfeld’s which allows one to rebuild the Yangian of an arbitrary simple Lie algebra starting from any of its finitedimensional modules satisfying a nontriviality condition. This is achieved using the socalled Rmatrix formalism, and the resulting realizing of the Yangian is called its Rmatrix presentation. When the underlying module is assumed to be irreducible, our result coincides with Drinfeld’s and, in particular, makes available a proof of his theorem  which has never appeared in the literature.
In addition, we provide a detailed study of the algebraic structure of the extended Yangian and prove several generalizations of results which are known to hold in the special case where the underlying module is the vector representation of a classical Lie algebra.
In the second part of this dissertation, we address the problem of classifying the finitedimensional irreducible representations for twisted Yangians associated to orthogonal and symplectic symmetric pairs of Lie algebras. We lay the foundation needed to solve this problem by developing a highest weight theory and proving that the highest weight of a finitedimensional irreducible module necessarily satisfies a set of relations involving a distinguished complex scalar and a tuple of polynomials whose set of roots are invariant under certain reflections.
Our main results on this topic provide a complete classification of finitedimensional irreducible modules for twisted Yangians associated to a large family of orthogonal and symplectic symmetric pairs.

 Graduation date
 Fall 2019

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
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