The canonical symmetrization map is a π€-module isomorphism between the symmetric algebra S(π€) of a finite-dimensional Lie algebra π€ and its universal enveloping algebra U(π€). This implies that the images of π€-invariants in S(π€) are Casimir elements. For each simple Lie algebra π€ of classical type we consider basic π€-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 SΓ£o Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
