学术报告报告题目：Finitedimensional representations of Yangians
报告人：谭易兰助理教授（加州大学河滨分校）
报告时间：2016年12月20日（星期二）上午14:30-15:30
报告地点：4号楼4318室
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数学学院
2016年12月19日
报告摘要：
Yangians$Y(\mathfrak{g})$ and quantum affine algebras$U_q(\hat{\mathfrak{g}})$ form two of the most important classes ofquantum groups. Their highest weight representation theory has manyapplications in mathematics and physics.One fundamental problemregarding the highest weight representation of a Yangian$Y(\mathfrak{g})$ is to determine the structure of thefinite-dimensional irreducible representations. Everyfinite-dimensional irreducible representation $L$ of$Y(\mathfrak{g})$ is a highest weight representation, and its highestweight is encapsulated by an $l$-tuple of monic polynomials$\pi=\big(\pi_1(u),\ldots,\pi_l(u)\big)$, where$\pi_i(u)=\prod\limits_{j=1}^{m_i}(u-a_{i,j})$; $\pi_i(u)$ is calleda Drinfeld polynomial. If there exists an $i\in I=\{1,\ldots, l\}$such that $\pi_i(u)=\big(u-a\big)\big(u-(a+1)\big)\ldots\big(u-(a+m-1)\big)$ and $\pi_j(u)=1$ for $j\neq i$, then $V(\pi)$ iscalled the Kirillov-Reshetikhin module. In this case, $\pi$ isdenoted by $\pi_{a,m}^{(i)}$. An irreducible representation is calledfundamental if there is an $i$ such that $\pi_i(u)=u-a$ and$\pi_{j}(u)=1$ for any $j\neq i$. In this case, we denote therepresentation by $V_{a}(\omega_i)$. %The structure of$V_{a}(\omega_i)$ is known, see \cite{ChPr4}.V. Chari and A. Pressleyshowed that the finite-dimensional irreducible representationof $Y(\mathfrak{g})$ associated to $\pi$ is a subquotient of a tensorproduct of fundamental representations $\bigotimes\limits_{i,j}V_{a_{i,j}}(\omega_i)$, where the tensor factors takes any order.
Inthis talk, we first provide a sufficient condition on the cyclicityof the tensor product. Then we give the definition of local Weylmodule of Yangians and show every local Weyl module is isomorphic toan ordered tensor product of fundamental representations of $Y(\mathfrak{g})$. In the end, we give an independent proof ofa cyclicity condition for a tensor product of Kirillov-Reshetikhinmodules of Yangians.