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.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
