Infinite Kostant cascades and centrally generated primitive ideals of U(n) in types A∞, C∞

We study the center of U(n), where n is the locally nilpotent radical of a splitting Borel subalgebra of a simple complex Lie algebra g = sl(infinity) (C), so(infinity)(C), sp(infinity) (C). There are infinitely many isomorphism classes of Lie algebras n, and we provide explicit generators of the center of U(n) in all cases. We then fix n with largest possible center of U(n) and characterize the centrally generated primitive ideals of U(n) for g = sl(infinity) (C), sp(infinity) (C) in terms of the above generators. As a preliminary result, we provide a characterization of the centrally generated primitive ideals in the enveloping algebra of the nilradical of a Borel subalgebra of sl(n)(C), sp(2n)(C). (C) 2015 Elsevier Inc. All rights reserved.