Date:	Fri, October 31, 2014
Time:	11:15
Place:	Seminar Room (120), Research I

Abstract: Let \(\mathfrak{g}(n)\) be a classical Lie algebra of rank \(n\), \(\mathfrak{n}(n)\) be a maximal nilpotent subalgebra of \(\mathfrak{g}(n)\), and \(U\big(\mathfrak{n}(n)\big)\) be the enveloping algebra of \(\mathfrak{n}(n)\). B. Kostant described the center \(Z\big(\mathfrak{n}(n)\big)\) of \(U\big(\mathfrak{n}(n)\big)\) in terms of so-called cascades of strictly orthogonal roots. In particular, he proved that \(Z\big(\mathfrak{n}(n)\big)\) is a polynomial ring.
Denote by \(\displaystyle\mathfrak{g}=\lim_{\longrightarrow}\mathfrak{g}(n)\) the direct limit of \(\mathfrak{g}(n)\) under the natural embeddings, and by \(\mathfrak{n}\) a maximal locally nilpotent subalgebra of \(\mathfrak{g}\). The root system of \(\mathfrak{g}\) is the direct limit of the root systems of \(\mathfrak{g}(n)\). Using infinite analogues of Kostant's cascades, I will describe the center \(Z(\mathfrak{n})\) of the enveloping algebra \(U(\mathfrak{n})\). In particular, I will show that \(Z(\mathfrak{n})\) is a polynomial ring (possibly, in infinitely many variables).
In the finite-dimensional case, the Dixmier map establishes a bijection between the set of primitive ideals of \(U\big(\mathfrak{n}(n)\big)\) (i.e., the set of the annihilators of simple \(\mathfrak{n}\)-modules) and the set of coadjoint orbits on \(\mathfrak{n}(n)^*\). It turns out that centrally generated ideals correspond to regular orbits (i.e., orbits of maximal dimension). I will describe an approach to the definition of the Dixmier map in the infinite-dimensional setting. In particular, I will define a regular orbit on \(\mathfrak{n}^*\) and prove that each centrally generated ideal of \(U(\mathfrak{n})\) it the image of a certain regular orbit under the analogue of the Dixmier map.
The talk is based on the joint work with I. Penkov.