XXXIX Workshop on Geometric Methods in Physics | 27.06-3.07.2021 |

IX School on Geometry and Physics | 21-25.06.2021 |

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## Yuly Billig## Representations of Lie algebras of vector fields on smooth affine varietiesWe study a category of representations of the Lie algebras of vector fields on a smooth affine algebraic variety X that admit a compatible action of the algebra of polynomial functions on X. We investigate two classes of simple modules in this category: gauge modules and Rudakov modules, and establish a covariant pairing between modules of these two types. We state a conjecture that every module in this category, which is finitely generated over the algebra of functions is a gauge module. We give a proof of this conjecture when X is the affine space. This is a joint work with Slava Futorny, Jonathan Nilsson, Andre Zaidan, Colin Ingalls and Amir Nasr. |

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