Symmetrisation and the Feigin-Frenkel centre

Abstract: Let $G$ be a complex reductive group, set $\mathfrak g={\mathrm{Lie\,}}G$. The algebra ${\mathcal S}(\mathfrak g)^{\mathfrak g}$ of symmetric $\mathfrak g$-invariants and the centre ${\mathcal Z}(\mathfrak g)$ of the enveloping algebra ${\mathcal U}(\mathfrak g)$ are polynomial rings in ${\mathrm{rk\,}}\mathfrak g$ generators. There are several isomorphisms between them, including the symmetrisation map $\varpi$, which exists also for the Lie algebras $\mathfrak q$ with $\dim\mathfrak q=\infty$.

However, in the infinite dimensional case, one may need to complete ${\mathcal U}(\mathfrak q)$ in order to replace ${\mathcal Z}(\mathfrak q)$ with an interesting related object. Roughly speaking, the Feigin-Frenkel centre arises as a result of such completion in case of an affine Kac-Moody algebra. From 1982 until 2006, this algebra existed as an intriguing black box with many applications. Then explicit formulas for its elements appeared first in type ${\sf A}$, later in all other classical types, and it was discovered that the FF-centre is the centraliser of the quadratic Casimir element.

We will discuss the type-free role of the symmetrisation map in the description of the FF-centre and present new explicit formulas for its generators in types ${\sf B}$, ${\sf C}$, ${\sf D}$, and ${\sf G}_2$. One of our main technical tools is a certain map from ${\mathcal S}^{k}(\mathfrak g)$ to $\Lambda^2\mathfrak g \otimes {\mathcal S}^{k-3}(\mathfrak g)$.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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