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

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