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The center of a quantum affine algebra at the critical level. (English) Zbl 0833.17008

Summary: We construct central elements in a completion of the quantum affine algebra \(U_q (\widehat {\mathfrak g})\) at the critical level \(c= -g\) from the universal \(R\)-matrix (\(g\) being the dual Coxeter number of a simple Lie algebra \({\mathfrak g}\)), using the method of N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky [(*) Lett. Math. Phys. 19, 133-142 (1990; Zbl 0692.22011)]. This construction defines an action of the Grothendieck algebra of the category of finite-dimensional representations of \(U_q (\widehat {\mathfrak g})\) on any \(U_q (\widehat {\mathfrak g})\)-module from category \({\mathcal O}\) with \(c= -g\). We explain the connection between the central elements from [(*), loc. cit.]and transfer matrices in statistical mechanics. In the quasiclassical approximation this connection was explained in [B. L. Feigin, E. V. Frenkel and N. Yu. Reshetikhin, Commun. Math. Phys. 166, 27- 62 (1994; Zbl 0812.35103)], 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 [(*), loc. cit.](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 the 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 central coefficient.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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