Tag - W-algebras

Anne Moreau: Associated varieties and finite extension of vertex algebras

20th March, 2023

Motivated by recent works joint with Tomoyuki Arakawa and Jethro Van Ekeren on collapsing levels, we conjectured that if W is a finite extension of a vertex subalgebra V, then the natural morphism between the corresponding associated varieties is dominant. In the case where W is a simple W-algebra and V is its simple affine vertex algebra, the conjecture is deeply related with the singularities of nilpotent Słodowy slices. In this talk, I will explain some results toward the conjecture and interesting examples.

Alexander Molev: Symmetrization map, Casimir elements and Sugawara operators

2nd October, 2020

The canonical symmetrization map is a 𝔤-module isomorphism between the symmetric algebra S(𝔤) of a finite-dimensional Lie algebra 𝔤 and its universal enveloping algebra U(𝔤). This implies that the images of 𝔤-invariants in S(𝔤) are Casimir elements. For each simple Lie algebra 𝔤 of classical type we consider basic 𝔤-invariants arising from the characteristic polynomial of the matrix of generators. We calculate the Harish-Chandra images of the corresponding Casimir elements. By using counterparts of the symmetric algebra invariants for the associated affine Kac-Moody algebras we obtain new formulas for generators of the centres of the affine vertex algebras at the critical level. Their Harish-Chandra images are elements of classical W-algebras which we produce in an explicit form.