相关论文: Primitive ideals of $\operatorname{U}(\frak{sl}(\i…

The purpose of this Ph.D. thesis is to study and classify primitive ideals of the enveloping algebras $U(\mathfrak{o}(\infty))$ and $U(\mathfrak{sp}(\infty))$. Let $\mathfrak{g}(\infty)$ denote any of the Lie algebras $\mathfrak{o}(\infty)$…

We study (two-sided) ideals $I$ in the enveloping algebra $\U(\frak g_\infty)$ of an infinite-dimensional Lie algebra $\frak g_\infty$ obtained as the union (equivalently, direct limit) of an arbitrary chain of embeddings of simple…

We provide a review of results on two-sided ideals in the enveloping algebra U$(\frak g(\infty))$ of a locally simple Lie algebra $\frak g(\infty)$. We pay special attention to the case when $\frak g(\infty)$ is one of the finitary Lie…

A distinguished family of completely prime primitive ideals in the universal enveloping algebra of a reductive Lie algebra ${\mathfrak g}$ over ${\mathbb C}$ are those ideals constructed from one-dimensional representations of finite…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$, $\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, and $U(\mathfrak{n})$ be the universal enveloping algebra of $\mathfrak{n}$.…

For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions $K$, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, $\widehat{U(\mathfrak{g})_{K}},$ is the…

Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider…

We give a criterion for the annihilator in U$(\frak{sl}(\infty))$ of a simple highest weight $\frak{sl}(\infty)$-module to be nonzero. As a consequence we show that, in contrast with the case of $\frak{sl}(n)$, the annihilator in…

We study the centrally generated primitive ideals of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the (locally) nilpotent radical of a (splitting) Borel subalgebra of a simple complex Lie algebra…

We prove that all finite W-algebras associated with nilpotent elements e in a complex semisimple Lie algebra g have finite-dimensional representations. In order to obtain this result we establish a connection between primitive ideals of…

Inner ideals of simple locally finite dimensional Lie algebras over an algebraically closed field of characteristic 0 are described. In particular, it is shown that a simple locally finite dimensional Lie algebra has a non-zero proper inner…

We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. For primitive ideals we obtain an analogue of Duflo's…

For any countable directed graph E we describe the primitive ideal space of the corresponding generalized Cuntz-Krieger algebra C*(E).

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is…

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the `Irreducibility theorem' and `Duflo theorem', to much wider classes of algebras. Our general version of Irreducibility theorem says…

We explore general intrinsic and extrinsic conditions that allow the transitivity of the relation of being a Lie ideal, in the sense that if a Lie algebra $\mathfrak{h}$ is a subideal of a Lie algebra $\mathfrak{g}$ (i.e. there exist Lie…

A finite-dimensional Lie algebra $L$ over a field $F$ of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an $A$-algebra if every nilpotent subalgebra is abelian. This paper is a…

We study the algebra $U_{\zeta}$ obtained via Lusztig's `integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra $\frak {g=sl}_2$ modulo the two-sided ideal generated by $K^l-1$. We show that $U_{\zeta}$ is a smash…

In this paper we study injective modules over universal enveloping algebras of finite-dimensional Lie algebras over fields of arbitrary characteristic. Most of our results are dealing with fields of prime characteristic but we also…

We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have…