On Segal--Sugawara vectors and Casimir elements for classical Lie algebras

TL;DR

This paper derives new formulas for the generators of the centers of affine vertex algebras at the critical level for classical Lie algebras, including Capelli-type determinants and Harish-Chandra images of Casimir elements.

Contribution

It provides novel formulas for centers of affine vertex algebras and explicit calculations for classical Lie algebras, enhancing understanding of their algebraic structures.

Findings

New formulas for generators of centers in classical types

Explicit Capelli-type determinant formulas for symplectic Lie algebras

Calculated Harish-Chandra images of Casimir elements

Abstract

We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.