Centers associated with the Borel subalgebra of certain simple Lie algebras

TL;DR

This paper explicitly describes the centers and semi-centers of the enveloping and symmetric algebras associated with Borel subalgebras of certain simple Lie algebras, extending previous studies.

Contribution

It provides explicit realizations and structural descriptions of the centers and semi-centers for these Lie algebra substructures, establishing key isomorphisms.

Findings

Explicit descriptions of centers and semi-centers for types G2, F4, Cn.

Structural isomorphisms between algebraic centers and Poisson centers.

Descriptions of centers as commutative rings.

Abstract

We continue the study in Ben-Shimol [1],[2] and consider a Borel subalgebra $\mathfrak{b}$ and its nil radical $\mathfrak{n}$ of the simple Lie algebras of types $G_2$, $F_4$, $C_n$ over arbitrary field. Let $\mathcal{L}\in\{\mathfrak{n}, \mathfrak{b}\}$. We establish here explicit realizations of the center $Z(\mathcal{L})$ and semi-center $Sz(\mathcal{L})$ of the enveloping algebra, the Poisson center $S(\mathcal{L})^{\mathcal{L}}$ and Poisson semi-center $S(\mathcal{L})^{\mathcal{L}}_{\operatorname{si}}$ of the symmetric algebra. We describe their structure as commutative rings and establish isomorphisms $Z(\mathcal{L})\cong~S(\mathcal{L})^{\mathcal{L}}$, $Sz(\mathcal{L})\cong S(\mathcal{L})^{\mathcal{L}}_{\operatorname{si}}$.