Primitive Ideals Research Articles

Sheets and Topology of Primitive Spectra for Semisimple Lie Algebras

Let g be a semisimple Lie algebra. Consider the set X of primitive ideals of the enveloping algebra U(g), given the Jacobson topology. A basic open problem is to describe X as a countable union of algebraic varieties V with strata V as large as possible. Towards this aim the notion of a sheet in X is introduced here in analogy with the notion of sheet in the adjoint orbit space g/G. The sheets in X are classified and given a purely topological description. Moreover Goldie-rank is constant and maximal on a dense open subset of a sheet. For every positive integer n it is shown that there are only finitely many sheets in X with maximal Goldie-rank n. For n=1, which corresponds to completely prime ideals, an extension of the Dixmier-map (from some adjoint orbits to primitive ideals) is introduced with the aim of showing that the sheets with n=1 are homeomorphic to sheets in g/G and hence explicitly described as algebraic varieties. This goal is partly though not fully achieved. The proofs of the above results require a detailed knowledge of Goldie-rank polynomials and even a new difficult result concerning the support of their restrictions to the walls. An extension of a theorem of Soergel on the topology of X is also required, and for this a new approach is given, which is both simpler and more general.

Prime and Primitive Ideals of a Class of Iterated Skew Polynomial Rings

Hodges and Levasseur described the primitive spectra of quantized coordinate rings of SL3 [14], and then of SLn [15]. These results established a close connection between primitive ideals, torus action, and Poisson geometry. The proofs relied on explicit computations involving generators and relations. Subsequently, Joseph generalized the Hodges–Levasseur program to semisimple algebraic quantum groups [17]. Hodges et al. then expanded Joseph’s work to include certain multiparameter deformations [16]. These papers rely less on concrete calculations and more on deeper, more conceptual, techniques. It is a natural and important question, then, as to how the preceding theory might apply to other algebras, particularly other algebras arising in the study of quantum groups. Goodearl and Letzter established parallel results for certain iterated skew polynomial rings [13], with application to quantum Weyl algebras and to some quantum coordinate rings.

Pseudo-finite dimensional representations of $ sl(2,k) $

Let k be an algebraically closed field of characteristic zero and L = sl(2,k) the Lie algebra of 2 × 2 traceless matrices over k. It is shown that there exists a von Neumann regular extension \( U(L) \subseteq U'(L) \) of the universal enveloping algebra, which is an epimorphism in the category of rings. The article is devoted to the study of the simple representations of U'(L), which may be topologized via the Ziegler topology on the set of injective indecomposable representations of U'(L) or via the Jacobson topology on the set of primitive ideals. These two topologies coincide and the finite dimensional simple representations of L form a dense, discrete and open subset. The field of fractions K(L) of the universal enveloping algebra is another simple representation of U'(L). If the point K(L) is removed from the Ziegler spectrum of U'(L), one obtains a compact totally disconnected topological space, which has the cardinality of the continuum. It is also shown that the lattice of ideals of U'(L) is isomorphic to the lattice of open subsets. The epimorphic ring extension \( U(L) \subseteq U'(L) \) is used to find an axiomatization of the finite dimensional representations of L in the language of left U(L)-modules. A representation V of L is called pseudo-finite dimensional if it satisfies these axioms. It is shown that a representation V of L is pseudo-finite dimensional if and only if for every central idempotent \( e \in U'(L) \) for which \( eK(L) \neq 0 \), whenever the subrepresentation eV is nonzero, then it has a nonzero highest weight space.

PRIME IDEALS OF THE COORDINATE RING OF QUANTUM SYMPLECTIC SPACE

In [1], K. R. Goodearl and E. S. Letzter study prime and primitive ideals in certain iterated Ore extensions of an infinite field k of arbitrary characteristic, which include several quantized alge...

Primitive ideals of locally C*-algebras

This paper is concerned with the connection between the structure space of a locally C*-algebra and the set of its continuous topologically irreducible *-representations. Properties of primitive ideals in such algebras are further investigated, for instance, closed ideals are expressed as intersections of primitive ideals, by using that the Jacobson radical reduces to 0.

Higher relative primitive ideals

The main object of this note is to introduce a higher order analog of the so-called primitive ideal of Y relative to X introduced by Jiang– Pellikaan–Siersma, where X ⊃ Y are germs of analytic subspaces of (Cn, 0). Our treatment of the problem is ideal-theoretic throughout, using the notion of iterated higher differential operators. Some examples from singularity theory are worked out. We establish the connection between higher primitive ideals and (relative) symbolic powers of an ideal and give an effective algorithm to compute both.

Lie Derivations on Banach Algebras

Let D be a Lie derivation on a unital complex Banach algebra A. Then for every primitive ideal P of A, except for a finite set of them which have finite codimension greater than one, there exist a derivation d from A/P to itself and a linear functional τ on A such that QPD(a)=d(a+P)+τ(a) for all a∈A (where QP denotes the quotient map from A onto A/P). Moreover the preceding decomposition holds for all primitive ideals in the case where D is continuous.

The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras

We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of n × n n\times n matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras A A over infinite fields k k of arbitrary characteristic. Our main result is the verification, for A A , of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal P P of A A is primitive if and only if the center of the Goldie quotient ring of A / P A/P is algebraic over k k , if and only if P P is a locally closed point – with respect to the Jacobson topology – in the prime spectrum of A A . These equivalences are established with the aid of a suitable group H \mathcal {H} acting as automorphisms of A A . The prime spectrum of A A is then partitioned into finitely many “ H \mathcal {H} -strata” (two prime ideals lie in the same H \mathcal {H} -stratum if the intersections of their H \mathcal {H} -orbits coincide), and we show that a prime ideal P P of A A is primitive exactly when P P is maximal within its H \mathcal {H} -stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which H \mathcal {H} acts transitively on the set of rational ideals within each H \mathcal {H} -stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of A A . For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.

On Duflo's Theorem That Minimal Primitive Ideals Are Centrally Generated

Let g be a complex semisimple Lie algebra. Duflo's theorem states that each minimal primitive ideal of the enveloping algebra U(g) is generated by its intersection with the center. We prove the following generalization. For the “relative enveloping algebra” A=U(g)/I relative to a parabolic subalgebra q of g, where I denotes the annihilator of the induced module U(g)⊗U([q,q])C, let Z denote the center of A, let Z̃ denote its normalization, and let Ã=AZ̃ be the slight extension of A obtained by normalizing the center. We present a theorem that states that under certain conditions, which are always satisfied if g=sln, we have that each minimal primitive ideal of à is generated by its intersection with the center Z̃. Duflo's theorem is the special case where q is a Borel subalgebra (then I=0, Z̃=Z, and Ã=A=U(g)). Note that the normalization of the center is really necessary for the theorem: The corresponding statement for A and Z instead of à and Z̃ fails for g=sl3.

On the Orbit Method and the Homomorphism of Harish-Chandra

Let g be a complex semisimple Lie algebra with adjoint group G. Let U(g) be the enveloping algebra of g. Following an idea of J. Dixmier, (Sur la méthode des orbites, in “Proceedings de la conférence: Non commutative Harmonic analysis, Marseille–Luminy, 1978,” Lecture Notes in Mathematics, Vol. 728, Springer-Verlag, New York/Berlin) we construct a bijection from the set of regular coadjoint orbits onto the set of minimal primitive ideals in U(g), without using the notion of “polarization” (see J. Dixmier, Algèbres Enveloppantes, Gauthier-Villars, Paris, 1974). This is a partial answer to the problem posed by J. Dixmier in the former reference.

Cyclic subgroups of ideal class groups in real quadratic orders

The primary purpose of this paper is to provide general sucient conditions for any real quadratic order to have a cyclic subgroup of order n 2 N in its ideal class group. This generalizes results in the literature, including some seminal classical works. This is done with a simpler approach via the interplay between the maximal order and the non-maximal orders, using the underlying infrastructure via the continued fraction algorithm. Numerous examples and a concluding criterion for non-trivial class numbers are also provided. The latter links class number one criteria with new prime-producing quadratic polynomials. 1. Notation and preliminaries. We will be considering arbitrary real quadratic orders, so we ®rst introduce the notions of arbitrary discriminants and radicands. Let D0 6ˆ 1 be a square-free integer, and set 0 ˆ D0 if D0 1 mod 4†, 4D0 otherwise. Then 0 is called a fundamental discriminant with associated fundamental radicand D0. Let f 2 N, and set ˆ f 2 0. Then ˆ D if D0 1 mod 4† and f is odd, 4D otherwise, is a discriminant with conductor f , and associated radicand D ˆ f =2† D0 if D0 1 mod 4† and f is even, f 2 D0 otherwise, having underlying fundamental discriminant 0 with associated fundamental radicand D0. Let be a discriminant with associated radicand D. Then ! ˆ 1‡  D p †=2 if ˆ D 1 mod 4†,  D p if 0 mod 4†, is called the principal surd associated with . This will provide the canonical basis element for our orders. First we need notation for a Z-module: ‰ ; Š ˆ x‡ y : x; y 2 Z ; Glasgow Math. J. 41 (1999) 197±206. # Glasgow Mathematical Journal Trust 1999. Printed in the United Kingdom where , 2 K ˆ Q  p † ˆ Q  D0 p †, the real quadratic ®eld of discriminant 0 and radicand D0. For this reason, fundamental discriminants are often called ®eld discriminants. In particular, if we set O ˆ ‰1; ! Š, then this is an order in K. Also, the index jO 0 : O j ˆ f is the conductor associated with , where O 0 is the maximal order in K, sometimes called the ring of integers of K. In other words, the maximal order in K is the order with conductor f ˆ 1, having square-free radicand D0 and fundamental discriminant 0. We also need to be able to distinguish those Z-modules that are ideals in O ; (see [1, pp. 9±30]). Theorem 1.1 (Primitive ideals and norms). Let be a discriminant, and let I 6ˆ 0† be a Z-submodule of O . Then I has a representation of the form I ˆ ‰a; b‡ c! Š, where a; c 2 N and b 2 Z with 0 b 0 is a discriminant and I is an O -ideal with N I † 0 be a discriminant with associated radicand D ˆ t ‡ r for t 2 N and jrj ˆ 1; 4. If I 1 in C , with N I † <  p =2, then one of the following holds. 1. N I † ˆ t=2, where r ˆ 1 and t is even. 2. N I † ˆ 4, where r ˆ 4 and t is even. 3. N I † ˆ ty 2, where r ˆ y4 and t is odd. 4. N I † ˆ 1. A formula for the class number of an order is given by h ˆ h 0 0 f †=u; 1:1† where f is the conductor associated with ,

Structure spaces of regular [formula omitted

In this paper, we study the structure spaces of regular C ∗- algebras . A complex C ∗- algebra A is regular if the mapping ϕ between the set of regular C ∗- seminorms on A and the set of primitive ideals of A is onto. We discuss the case of regular C ∗- algebras where the structure spaces are not T 1.

Proof of inseparable primeC *-algebrasA withRR (A) = 0 being primitiveC *-algebras

First, that primeC *-algebras with countable primitive ideals are all primitiieC *-algebras is proved. Then the proof that primeC *-algebras with propertyRR(A) = 0 are all primitiveC *-algebras is given.

Extended Central Characters and Dixmier's Map

Let p be a parabolic subalgebra in a semisimple Lie algebra g. We study the moduleMp(λ) induced from a one-dimensional p-module of weight λ. LetU=U(g)/Ibe the quotient of the enveloping algebraU(g) by the annihilatorIof the generic module induced from p. LetZ̃denote the integral closure of the centerZofU, andŨ=Z̃U. ThenMp(λ) is not only aU-module, but even aŨ-module [1], so its central characterZ→C extends to a central characterZ̃→C. We prove that this extended central character is useful: It is unique if and only if Dixmier's map is injective. Moreover, the “modified Dixmier-map” of the author's work (1998,Abh. Math. Sem. Univ. Hamburg68, 25–44) from the sheetS/Gdetermined by p into the space XŨof minimal primitive ideals ofŨis a homeomorphismS/G→XŨgiven by O↦AnnMp(λ). As an application, we obtain the following result related to Duflo's theorem that minimal primitive ideals are induced and centrally generated: Each minimal primitive idealJofŨis induced and “almost” generated by its intersection m with the center, in the sense thatJ=mŨ. As further application of the main theorem we get results on the well-definedness of Dixmier's map.

Primitive and Poisson Spectra of Twists of Polynomial Rings

A family of flat deformations of a commutative polynomial ring S on n generators is considered, where each deformation B is a twist of S by a semisimple, linear automorphism σ of ℙ n−1, such that a Poisson bracket is induced on S. We show that if the symplectic leaves associated with this Poisson structure are algebraic, then they are the orbits of an algebraic group G determined by the Poisson bracket. In this case, we prove that if σ is 'generic enough', then there is a natural one-to-one correspondence between the primitive ideals of B and the symplectic leaves if and only if σ has a representative in GL(ℂ n ) which belongs to G. As an example, the results are applied to the coordinate ring $$\mathcal{O}_q (M_2 )$$ of quantum 2 × 2 matrices which is not a twist of a polynomial ring, although it is a flat deformation of one; if q is not a root of unity, then there is a bijection between the primitive ideals of $$\mathcal{O}_q (M_2 )$$ and the symplectic leaves.

Zur Topologie der Dixmier-Abbildung

LetG be the coadjoint group of a finite-dimensional complex Lie algebrag. Forg solvable, the Dixmier-map is known to be a homeomorphism $$\mathfrak{g}*/G \to \chi $$ of the orbit space $$\mathfrak{g}*/G$$ /G onto the space χ of primitive ideals in the enveloping algebra U(G) [6,15]. For $$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n $$ , the Dixmier-map is known to be a bijection (and in general not a homeomorphism) $$\mathfrak{g}*/G \to \chi ^l $$ with the space χl of all completely prime primitive ideals [7, 16]. Here we derive from a result ofW. SOERGEL [18], that this map issheet- wise a homeomorphism onto the image. Here a sheet is a maximal irreducible subset consisting of orbits of a fixed dimension; obviouslyg decomposes into finitely many sheets [3]. The results of this paper hold more generally forg semisimple, if one restricts to a sheet of polarizable orbits, where a Dixmier-map can be defined. Relative to a fixed polarization (a parabolic subalgebra)p ⊂g let I be the annihilator of the generic module induced from p. The „relative enveloping algebra“ $$U: = U(\mathfrak{g})/I$$ ) has been studied e.g. bySOERGEL [19, 18]. Its center Z is described here by a relative Harish-Chandra isomorphism of the normalization $$\tilde Z$$ with a suitable ring of group invariants (3.2). We study here the extension $$\tilde U$$ ofU by $$\tilde Z$$ . We suggest that this very mild central extension ofU generates good properties and is very suitable for the study of the Dixmier-map (cf.4.3,5.6). In particular, we conjecture in case $$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n $$ : Every minimal primitive ideal of $$\tilde U$$ is generated by a maximal ideal of the center. This would generalize for $$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n $$ a well known theorem ofM. Duflo (casep Borel, where $$\tilde U = U = U(\mathfrak{g})$$ ). AsJ. Dixmier communicated in a letter, the main result here is exactly what he had hoped for when he first introduced a notion of sheets many years ago.

Actions of Algebraic Groups on the Spectrum of Rational Ideals, II

We study rational actions of a linear algebraic groupGon an algebraV, and the induced actions on Rat(V), the spectrum of rational ideals ofV(a subset of Spec(V) which often includes all primitive ideals). This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions onV. In particular, we relate properties of a rational idealJand itsorb, the ideal (J:G)=⋂γ∈Gγ(J). The rational ideals ofVcontaining the orb ofJare precisely those in the Zariski-closureXof the orbit ofJin Rat(V). TheG-stratumofJconsists of all rational ideals inXwhose orbit is dense inX(i.e., whose orb is equal to the orb ofJ). We show that theG-stratum of a rational ideal consists of exactly oneG-orbit, and that rational ideals are maximal in their strata in a strong sense. These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by Goodearl and Letzter. We further show that the orbit ofJis open in its closure in Rat(V), provided thatJis locally closed. Among other results, we show that the semiprime ideal (J:G) is Goldie, and we relate the uniform and Gelfand–Kirillov dimensions ofV/JandV/(J:G).

Primitive ideals in the coordinate ring of quantum Euclidean space

A twisted group algebra kσP on a free Abelian group P with finite rank and a Poisson structure on kP are studied. As an application, the primitive spectrum of , the coordinate ring of quantum Euclidean space, is described and a Poisson algebra A is constructed so that there is a bijection between the primitive spectrum of and the symplectic spectrum of R.

On the Orbit Method for the Lie Algebra of Vector Fields on a Curve

Let g be the Lie algebra of vector fields on an affine smooth curve Σ. Our goal is to establish an orbit method for g. Since g is infinite-dimensional, we face some technical problems. Without having groups acting on g, we try nevertheless to define the notion of “orbits.” So, we focus our attention to a subspace g*fof g*. This subspace consists of the “finite-dimensional orbits.” To almost all λ in g*fit corresponds a simple induced representation of g whose annihilator is a primitive ideal. We conjecture that this ideal has a finite Gelfand–Kirillov codimension. What we are actually looking for is a bijection similar to Dixmier's bijection (in the finite-dimensional case) between the “orbits” of g*fand certain primitive ideals of the enveloping algebra of g.

The weak topology for $SG$-primitive ideals

The weak topology for $SG$-primitive ideals