学术报告 报告题目：Finite dimensional 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 of quantum groups. Their highest weight representation theory has many applications in mathematics and physics.One fundamental problem regarding the highest weight representation of a Yangian $Y(/mathfrak{g})$ is to determine the structure of the finite-dimensional irreducible representations. Every finite-dimensional irreducible representation $L$ of $Y(/mathfrak{g})$ is a highest weight representation, and its highest weight 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 called a 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 $V(/pi)$ is called the Kirillov-Reshetikhin module. In this case, $/pi$ is denoted by $/pi_{a,m}^{(i)}$. An irreducible representation is called fundamental 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 the representation by $V_{a}(/omega_i)$. %The structure of $V_{a}(/omega_i)$ is known, see /cite{ChPr4}.V. Chari and A. Pressley showed that the finite-dimensional irreducible representation of $Y(/mathfrak{g})$ associated to $/pi$ is a subquotient of a tensor product of fundamental representations $/bigotimes/limits_{i,j} V_{a_{i,j}}(/omega_i)$, where the tensor factors takes any order.
In this talk, we first provide a sufficient condition on the cyclicity of the tensor product. Then we give the definition of local Weyl module of Yangians and show every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of $Y(/mathfrak{g})$. In the end, we give an independent proof of a cyclicity condition for a tensor product of Kirillov-Reshetikhin modules of Yangians.