Primitive Ideals Research Articles

A note on Jacobson rings and polynomial rings

As is well known, if R R is a ring in which every prime ideal is an intersection of primitive ideals, the same is true of R [ X ] R[X] . The purpose of this paper is to give a general theorem which shows that the above result remains true when many other classes of prime ideals are considered in place of primitive ideals.

Primitive ideals and nilpotent orbits in type G2

Primitive ideals and nilpotent orbits in type G2

A criterion for an ideal to be induced

1.1. Let g be a complex semisimple Lie algebra. In the study of the primitive spectrum Prim U(g) of the enveloping algebra U(g) of g, an important question which remains open is to determine those primitive ideals which are completely prime. One such family is provided by those ideals which are annihilators of modules induced from a one-dimensional module of a parabolic subalgebra p. For the purposes of this article we call such ideals induced (with respect to p).

Primitive ideals with bounded approximate units in L1-algebras of exponential lie groups

AbstractLet G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.

Primitive ideals and representations

Certain theorems of Joseph and King relating primitive ideals to characters in category O , are extended to the case of arbitrary g-modules with finite dimensional n-cohomology. A derived functor construction, analogous to a construction of Zuckerman in the case of a nilpotent Lie algebra, is used to extend the results. Applications to Harish-Chandra modules are also given.

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT1 primitive ideal space if the groupC*-algebra,C*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC*(G), denoted by PrimG. This means of course that every primitive ideal inC*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT1 primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT1 primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC*(G) are always maximal which implies of course that PrimG isT1.

Primitive ideals and integral points on cones

Primitive ideals and integral points on cones

�ber die zentrale Picard-Gruppe und die Einheiten lokaler Quaternionenordnungen

Let M be any order of a quaternion algebra Q over a local field F. The group Mx of units and the central Picard group Picent (M) of M are investigated. The group index (Lx:Mx) and, if F is of characteristic ≠2, also the norms of all primitive ideals of M and the order of Picent (M) are determined explicitly, where L denotes any maximal order of Q containing M. Applications of these results to local densities of ternary quadratic forms and to class numbers and type numbers of global quaternion orders are indicated.

Differential operators on homogeneous spaces. III

LetG be a semi-simple complex algebraic group with Lie algebra ϱ and flag varietyX=G/B. For each primitive idealJ with trivial central character in the enveloping algebraU(ϱ) we define a characteristic variety in the cotangent bundle ofX, which projects under the Springer resolution mapT * X→ϱ onto the closure of a nilpotent orbit. We prove that this characteristic variety is theG-saturation of the characteristic variety of a highest weight module with annihilatorJ. We conjecture that it is irreducible forG=SL n . Our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of Weyl group representations, as achieved by A. Joseph. The presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space, and to study more generally characteristic varieties of Harish-Chandra modules.

A Characterization of Algebras of Invariant-Coinvariant Module Type

K. R. Fuller has characterized rings of left invariant module type. An algebra is said to be of invariant-coinvariant module type if each of its indecomposable modules is quasi-injective or quasi-projective. In this note we shall give a characterization of algebras of invariant-coinvariant module type, which distinguishes this class from that of algebras of local-colocal type. It seems of interest that the distributivity of second radicals of primitive ideals appears in our characterization.

Characters of nilpotent groups

AbstractThe characters (extremal positive definite central functions) of discrete nilpotent groups are studied. The relationship between the set of characters of G and the primitive ideals of the group C*-algebra C*(G) is investigated. It is shown that for a large class of nilpotent groups these objects are in 1–1 correspondence. One proof of this exploits the fact that faithful characters of certain nilpotent groups vanish off the finite conjugacy class subgroup. An example is given where the latter property fails.

On the Jacobson radical of associative 2-graded rings

In this note definitions of the Jacobson radical of 2-graded rings are given which are analogous to the usual definitions for associative rings in terms of maximal modular one-sided ideals and of primitive ideals. We show that these definitions lead to the same ideal and are left-right symmetric. The relationship between the Jacobson radical of the 2-graded ring and that of the associated associative ring is also investigated. Preliminaries

Primitive Ideals of Enveloping Algebras Over Commutative Rings

1. Introduction. In this paper, we will study primitive ideals and the Jacobson radical of enveloping algebras over commutative rings. We also consider group rings of polycyclic-by-finite groups in the final section, on which we postpone comment until the end of the introduction. Throughout the paper, any Lie algebra g over a commutative ring R is assumed to be a free R-module of finite rank, and we denote the corresponding enveloping algebra by UR (g), or U(g). Our principal purpose is to prove that if R is a domain of characteristic 0, then UR (g) has many primitive ideals-in any case, sufficiently many for the Jacobson radical of UR (g) to be (0). This combines with an easier result in characteristic p to yield: THEOREM 1.1. Let g be a non-zero Lie algebra over a commutative

A characterization of algebras of invariant-coinvariant module type

K. R. Fuller has characterized rings of left invariant module type. An algebra is said to be of invariant-coinvariant module type if each of its indecomposable modules is quasi-injective or quasi-projective. In this note we shall give a characterization of algebras of invariant-coinvariant module type, which distinguishes this class from that of algebras of local-colocal type. It seems of interest that the distributivity of second radicals of primitive ideals appears in our characterization.

Primitive ideals and orbital integrals in complex exceptional groups

In [2], two related problems were studied for complex classical groups: determination of the primitive spectrum of the enveloping algebra (as a set), and Fourier inversion of unipotent orbital integrals. The first of these was solved completely (extending work of Joseph and others in SL(n, C) and small groups). Although our techniques presumably solve the second as well, up to determination of normalizing constants, we carried out the calculations only for special unipotent classes (see [ 181). We had, of course, tried to apply the same methods to the exceptional groups, but they seem to be inadequate. Since that work was done, however, Brylinski and Kashiwara and Beilinson and Bernstein have established the Kazhdan-Lusztig conjecture, giving character formulas for irreducible highest weight modules [3, 71. Because of a conjecture of Joseph proved in [24], this determines in principle the primitive ideals with a fixed regular integral infinitesimal character: they are in one-to-one correspondence with what Kazhdan and Lusztig call left cells in the Weyl group. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated, requiring one to compute roughly 1 I%‘]* polynomials of degrees on the order of half the number of positive roots. This is very unsatisfactory. However, it

On primitive ideals in group rings

On primitive ideals in group rings

Primitive ideals and orbital integrals in complex classical groups

Primitive ideals and orbital integrals in complex classical groups

Prime ideals in normalizing extensions

In this paper we offer a new approach to the study of prime ideals in finite normalizing extensions of rings. We introduce a “primitivity machine” which translates certain problems on prime ideals to analogous questions on primitive ideals. We then use the well-known Clifford theory of irreducible modules to study these primitive ideals.

Sur les semi-caracteres des groupes de Lie resolubles connexes

Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by D (G) the space of C ∞ functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra U G c of the complexification of the Lie algebra of G for which the linear form ϑ ψ(π(u ∗ ϑ)) on D (G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra U G c .

On the characterization of primitive ideals in enveloping algebras

On the characterization of primitive ideals in enveloping algebras