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Finite-dimensional representations of Yangians Open Access

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Other title
Subject/Keyword

Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Tan, Yilan
Supervisor and department
Guay, Nicolas (Math)
Examining committee member and department
Guay, Nicolas (Math)
Kuttler, Jochen (Math)
Cliff, Gerald (Math)
Pianzola, Arturo (Math)
Department
Department of Mathematical and Statistical Sciences
Specialization
Mathematics
Date accepted
2014-09-24T10:03:05Z
2014-11
Degree
Doctor of Philosophy
Degree level
Doctoral
Abstract
In this thesis, we study local Weyl modules of Yangians and a cyclicity condition for a tensor product of fundamental representations of a Yangian. Let $\g$ be a simple Lie algebra over $\C$ with rank $l$ and $\pi$ be a generic $l$-tuple of polynomials in $u$. We show that there exists a universal representation $W(\pi)$ of the Yangian $\yg$, called the local Weyl module associated to $\pi$, such that every finite-dimensional highest-weight representation associated to $\pi$ is a quotient of $W(\pi)$. We prove that the dimension of $W(\pi)$ is bounded by the dimension of some local Weyl module of the current algebra $\g[t]$. Let $L=V_{a_1}(\omega_{b_1})\otimes V_{a_2}(\omega_{b_2})\otimes\ldots\otimes V_{a_k}(\omega_{b_k})$, where $V_{a_i}(\omega_{b_i})$ is the $b_i$-th fundamental representation of $\yg$. We prove that if $\operatorname{Re}(a_1)\geq\operatorname{Re}(a_2)\geq \ldots \geq \operatorname{Re}(a_k)$, then $L$ is a highest weight representation. By comparing the dimensions of $L$ and the upper bound of $W(\pi)$, we have $W(\pi)\cong L$. A cyclicity condition of the tensor product $L$ is also studied: $L$ is a highest weight representation if $a_j-a_i\notin S(b_i, b_j)$ for \$1\leq i
Language
English
DOI
doi:10.7939/R35D8NP3V
Rights
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. 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.
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2014-11-15T08:17:26.216+00:00
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