Symmetrisation and the Feigin–Frenkel centre

Abstract

For complex simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the commutative subalgebra \(\mathfrak z(\hat {\mathfrak g})\subset {{\mathcal {U}}}(t^{-1}\mathfrak g[t^{-1}])\) known as the Feigin–Frenkel centre. These formulas make use of the symmetrisation map as well as of some well-chosen symmetric invariants of \(\mathfrak g\). There are some general results on the rôle of the symmetrisation map in the explicit description of the Feigin–Frenkel centre. Our method reduces questions about elements of \(\mathfrak z(\hat {\mathfrak g})\) 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. One of our technical tools is the map \({\sf m}\!\!: {{\mathcal {S}}}^{k}(\mathfrak g)\to \Lambda ^{2}\mathfrak g \otimes {{\mathcal {S}}}^{k-3}(\mathfrak g)\) introduced here. As the results show, a better understanding of this map will lead to a better understanding of \(\mathfrak z(\hat {\mathfrak g})\).