Casimir elements from the Brauer-Schur-Weyl duality

TL;DR

This paper constructs Casimir elements for orthogonal and symplectic Lie algebras using Brauer algebra, computes their images under Harish-Chandra isomorphism, and shows they generate the centers of universal enveloping algebras.

Contribution

It introduces a new method to generate algebraically independent Casimir elements via Brauer algebra and explicitly describes their images under Harish-Chandra isomorphism.

Findings

Casimir elements form algebraically independent generators of the centers.

Explicit calculation of images under Harish-Chandra isomorphism.

Identification of the Pfaffian-type element as a generator in the orthogonal case.

Abstract

We consider Casimir elements for the orthogonal and symplectic Lie algebras constructed with the use of the Brauer algebra. We calculate the images of these elements under the Harish-Chandra isomorphism and thus show that they (together with the Pfaffian-type element in the even orthogonal case) are algebraically independent generators of the centers of the corresponding universal enveloping algebras.