Mathematical Research Letters

Volume 1 (1994)

Number 4

The center of a quantum affine algebra at the critical level

Pages: 469 – 480



Jintai Ding (Yale University)

Pavel Etingof (Yale University)


We construct central elements in a completion of the quantum affine algebra $U_q(\hat\frak g)$ at the critical level $c=-g$ from the universal $R$-matrix ($g$ being the dual Coxeter number of the simple Lie algebra $\frak g$), using the method of Reshetikhin and Semenov-Tian-Shansky \cite{RS}. This construction defines an action of the Grothendieck algebra of the category of finite-dimensional representations of $U_q(\hat\frak g)$ on any $U_q(\hat\frak g)$-module from category $\Cal O$ with $c=-g$. We explain the connection between the central elements from \cite{RS} and transfer matrices in statistical mechanics. In the quasiclassical approximation this connection was explained in \cite{FFR}, and it was mentioned that one could generalize it to the quantum case to get Bethe vectors for transfer matrices. Using this connection, we prove that the central elements from \cite{RS} (for all finite dimensional representations) applied to the highest weight vector of a generic Verma module at the critical level generate the whole space of singular vectors in this module. We also compute the first term of the quasiclassical expansion of the central elements near $q=1$, and show that it always gives the Sugawara current with a certain coefficient.

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