Symmetrisation and the Feigin-Frenkel centre

TL;DR

This paper provides explicit formulas for generators of the Feigin-Frenkel centre for Lie algebras of types B, C, D, and G2, utilizing symmetrisation maps and symmetric invariants to simplify their description.

Contribution

It introduces a new method using symmetrisation maps to explicitly describe the Feigin-Frenkel centre across different Lie algebra types.

Findings

Explicit formulas for types B, C, D, and G2

Reduction of FF-centre questions to symmetric invariants

General results on symmetrisation map's role in FF-centre

Abstract

For simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the Feigin-Frenkel centre. These formulas make use of the symmetrisation map as well as some well-chosen symmetric invariants of $\mathfrak g$. There are some general results on the role of the symmetrisation map in the explicit description of the FF-centre. The advantage of our method is that it reduces questions about elements of the FF-centre to questions on the structure of the symmetric invariants in a type-free way. As an illustration, we deal with type G$_2$ by hand.