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## Abstract

<p>Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by $\mathbb{C}^\times$-equivariant Poisson automorphisms.</p> <p>We go on to study these definitions in the context of nilpotent Slodowy slices. First, we give a complete description of the cases in which the finite $W$-algebra is a universal filtered quantization of the slice, building on the work of Lehn–Namikawa–Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices.</p> <p>The subregular slices in non-simply laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type, we prove that the finite $W$-algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally, we apply this result to give a presentation of the subregular finite $W$-algebra of type $\mathsf{B}$ as a quotient of a shifted Yangian.</p>

Original language | English |
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Pages (from-to) | 1-35 |

Journal | Journal of Noncommutative Geometry |

Volume | 18 |

Issue number | 1 |

Early online date | 28 Oct 2023 |

DOIs | |

Publication status | Published - 15 Feb 2024 |

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