1 results
Symmetrisation and the Feigin–Frenkel centre
- Part of
-
- Journal:
- Compositio Mathematica / Volume 158 / Issue 3 / March 2022
- Published online by Cambridge University Press:
- 19 May 2022, pp. 585-622
- Print publication:
- March 2022
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
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 
$\mathfrak z(\hat {\mathfrak g})$.
