Symmetrisation and the Feigin–Frenkel centre

Compositio mathematica

article

Oksana Yakimova¹ 

[1] Institut für Mathematik, Friedrich-Schiller-Universität Jena

关键词: Kac–Moody algebra;    symmetric invariants;    Segal–Sugawara vectors;    16S30;    17B67;    17B20;    17B35;    17B63;   

DOI  :  10.1112/S0010437X22007485

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

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 $\mathfrak z(\hat {\mathfrak g})$ .

【 授权许可】

CC BY