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 $Z(\cL)\cong S(\cL)^{ \cL}$, $Sz(\cL)\cong S(\cL)^{ \cL}_{ \si}$
Categories: math.RT math.RA