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Centers associated with the Borel subalgebra of the general linear Lie algebra (1401.0755v1)
Published 4 Jan 2014 in math.RT and math.RA
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 $Z(\cL)\cong S(\cL){\cL}$, $Sz(\cL)\cong S(\cL){\cL}_{\si}$