Centers associated with the Borel subalgebra of certain simple Lie algebras

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 $Z(\mathcal{L})\cong~S(\mathcal{L})^{\mathcal{L}}$, $Sz(\mathcal{L})\cong S(\mathcal{L})^{\mathcal{L}}_{\operatorname{si}}$.