Centers associated with the Borel subalgebra of the general linear Lie algebra

TL;DR

This paper explicitly describes the centers and semi-centers of the enveloping and symmetric algebras associated with Borel subalgebras of the general linear and special linear Lie algebras over any field, establishing their structures and isomorphisms.

Contribution

It provides explicit realizations and structural descriptions of the centers and semi-centers of these algebras, including isomorphisms between them, for Borel subalgebras of GL and SL.

Findings

Explicit realizations of centers and semi-centers of enveloping algebras.

Structural descriptions of these centers as commutative rings.

Isomorphisms between the centers and Poisson centers.

Abstract

We consider a Borel subalgebra $\fg$ of the general linear algebra and its subalgebra $\BB$ which is a Borel subalgebra of the special linear algebra, over arbitrary field. Let $\cL\in\{\fg, \BB\}$. We establish here explicit realizations of the center $Z(\cL)$ and semi-center $Sz(\cL)$ of the enveloping algebra, the Poisson center $S(\cL)^{\cL}$ and Poisson semi-center $S(\cL)^{\cL}_{\si}$ of the symmetric algebra. We describe their structure as commutative rings and establish isomorphisms $Z(\cL)\cong S(\cL)^{\cL}$, $Sz(\cL)\cong S(\cL)^{\cL}_{\si}$