## Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras.(English)Zbl 0925.17022

Summary: We prove Drinfeld’s conjecture that the center of a certain completion of the universal enveloping algebra of an affine Kac-Moody algebra at the critical level is isomorphic to the Gelfand-Dikii algebra, associated to the Langlands dual algebra. The center is identified with a limit of the $$W$$-algebra, defined by means of the quantum Drinfeld-Solokov reduction.

### MSC:

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