## Quantization of the shift of argument subalgebras in type A

### Vyacheslav Futorny and Alexander Molev

#### Abstract

Given a simple Lie algebra $$\mathfrak{g}$$ and an element $$\mu\in\mathfrak{g}^*$$, the corresponding shift of argument subalgebra of $$\text{S}(\mathfrak{g})$$ is Poisson commutative. In the case where $$\mu$$ is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of $$\text{U}(\mathfrak{g})$$. We show that if $$\mathfrak{g}$$ is of type $$A$$, then this property extends to arbitrary $$\mu$$, thus proving a conjecture of Feigin, Frenkel and Toledano Laredo. The proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level.

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 Thursday, May 1, 2014