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Representations of Twisted Yangians of Types B, C and D
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- Author / Creator
- Wendlandt, Curtis
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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 finite-dimensional modules satisfying a non-triviality condition. This is achieved using the so-called R-matrix formalism, and the resulting realizing of the Yangian is called its R-matrix 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 finite-dimensional 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 finite-dimensional 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 finite-dimensional irreducible modules for twisted Yangians associated to a large family of orthogonal and symplectic symmetric pairs.
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- Graduation date
- Fall 2019
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- Type of Item
- Thesis
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- Degree
- Doctor of Philosophy
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- License
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