K. R. Goodearl

Prime ideals in skew polynomial rings and quantized Weyl algebras

The concern of this paper is to investigate the structure of skew polynomial rings (Ore extensions) of the form T=R [θ; σ, δ] where σ and δ are both nontrivial, and in particular to analyze the prime ideals of T . The main focus is on the case that R is commutative noetherian. In this case, the prime ideals of T are classified, polynomial identities and Artin-Rees separation in prime factor rings are investigated, and cliques of prime ideals are studied. The second layer condition is proved, as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique. Futher, q -skew derivations on noncommutative coefficient rings are introduced, and some preliminary results on contractions of prime ideals of T are obtained in this setting. Finally, prime ideals in quantized Weyl algebras over fields are analyzed.

Stable range one for rings with many units

Abstract The main purpose of this paper is to prove the stable range 1 condition for a number of classes of rings and algebras. Using a modification of a computation of D.V. Tyukavkin, stable range 1 (and a bit more) is obtained from the following simple condition on a ring R: given any x, y ϵ R, there is a unit u ϵ R such that x − u and y − u-1 are both units. Verification of the latter condition then yields stable range 1 in a number of cases, e.g.: (1) any algebra over an uncountable field, in which all non-zero-divisors are units and there are no uncountable direct sums of nonzero one-sided ideals; (2) any algebra over an uncountable field, with only countably many primitive factor rings, all of which are artinian; (3) the endomorphism ring of any noetherian module over an algebra as in (2); (4) any algebraic algebra over an infinite field; (5) any integral algebra over a commutative ring which modulo its Jacobson radical is algebraic over an infinite field; (6) any von Nuemann regular algebra over an uncountable field, which has a rank function. Using other techniques, it is proved that finite Rickart C∗-algebras, strongly π-regular von Neumann regular rings, and strongly π-regular rings in which every element is a sum of a unit plus a central unit, all have stable range 1. Finally, for an arbitrary commutative ring some overrings with specified stable range properties are constructed, in particular a more or less canonical overring having stable range 1.

Catenarity in quantum algebras

Abstract Various quantum algebras are shown to be catenary, i.e., all saturated chains of prime ideals between any two fixed primes have the same length. Further, Tauvels formula relating the height of a prime ideal to the Gelfand-Kirillov dimension of the corresponding factor ring is established. These results are obtained for coordinate rings of quantum affine spaces, for quantized Weyl algebras, and for coordinate rings of complex quantum general linear groups, as well as for quantized enveloping algebras of maximal nilpotent subalgebras of semisimple complex Lie algebras.

PRIME SPECTRA OF QUANTUM SEMISIMPLE GROUPS

We study the prime ideal spaces of the quantized function algebras Rq [G], for G a semisimple Lie group and q an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras Rq [G] satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkars strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of Rq [G]. In the final section the results are specialized to the case G = SLn (C), where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n-space satisfy our axiom scheme when the group generated by the parameters is torsionfree. 0. INTRODUCTION AND BACKGROUND 0.1. Let P and Q be prime ideals of a noetherian ring R. We say that P is linked to Q, and write P -*Q, if there is an ideal A of R with PQ C A c P n Q such that (P n Q)/A is a nonzero torsionfree left (R/P)-module and a torsionfree right (R/Q)-module. This condition can be simplified in case R satisfies the second layer condition (see below). In that case, P -* Q if and only if (P n Q)/PQ is faithful as a left (R/P)-module and as a right (R/Q)-module [39; 10, 1.4]. The graph of links of R is the directed graph whose vertices are the points of spec R (the set of prime ideals of R), with a directed edge from P to Q if and only if P -* Q. The primes in the connected component of the graph of links containing P constitute the clique of P, denoted clique(P). There is a close connection between the graph of links of R and the representation theory of R (cf. [20], [4]); this connection is particularly strong when R satisfies the strong second layer condition, which requires that, whenever U is a cyclic uniform R-module with the annihilator Received by the editors November 4, 1994 and, in revised form, September 5, 1995. 1991 Mathematics Subject Classification. Primary 16D30, 16D60, 16P40, 17B37. The research of the second author was partially supported by a grant from the National Science Foundation (USA). Part of the work was carried out while he visited the University of Glasgow Mathematics Department during October 1993, supported by the Edinburgh and London Mathematical Societies. Work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the Department of Mathematics of the University of Washington, whom both thank for its hospitality. The travel costs of the first author were in part covered by a grant from the Carnegie Trust for the Universities of Scotland. (?)1996 American Mathematical Society

Quantum determinantal ideals

Introduction. Fix a base fieldk. The quantized coordinate ring of n×n matrices over k, denoted by q(Mn(k)), is a deformation of the classical coordinate ring of n×n matrices, (Mn(k)). As such, it is ak-algebra generated by n2 indeterminates Xij , for 1 ≤ i,j ≤ n, subject to relations which we state in (1.1). Here, q is a nonzero element of the field k. Whenq = 1, we recover (Mn(k)), which is the commutative polynomial algebra k[Xij ]. The algebra q(Mn(k)) has a distinguished elementDq , the quantum determinant , which is a central element. Two important algebras q(GLn(k)) and q(SLn(k)) are formed by invertingDq and settingDq = 1, respectively. The structures of the primitive and prime ideal spectra of the algebras q(GLn(k)) and q(SLn(k)) have been investigated recently (see, for example, [2], [7], and [10]). Results obtained in these investigations can be pulled back to partial results about the primitive and prime ideal spectra of q(Mn(k)). However, these techniques give no information about the closed subset of the spectrum determined by Dq . In this paper, we begin the study of this portion of the spectrum. In the classical commutative setting, much attention has been paid to determinantal ideals: that is, the ideals generated by the minors of a given size. In particular, these are special prime ideals of (Mn(k)) containing the determinant. Moreover, there are interesting geometrical and invariant theoretical reasons for the importance of these ideals (see, for example, [4]). In order to put our results into context, it may be useful to review some highlights of the commutative theory. LetMl,m(k) denote the algebraic variety of l×m matrices overk. For t ≤ n, the general linear group GL t (k) acts onMn,t (k)×Mt,n(k) via g ·(A,B) := (Ag−1,gB).

Winding-invariant prime ideals in quantum 3×3 matrices

Abstract A complete determination of the prime ideals invariant under winding automorphisms in the generic 3×3 quantum matrix algebra O q (M 3 (k)) is obtained. Explicit generating sets consisting of quantum minors are given for all of these primes, thus verifying a general conjecture in the 3×3 case. The result relies heavily on certain tensor product decompositions for winding-invariant prime ideals, developed in an accompanying paper. In addition, new methods are developed here, which show that certain sets of quantum minors, not previously manageable, generate prime ideals in O q (M n (k)) .

Quantum cluster algebras and quantum nilpotent algebras

Significance Cluster algebras are used to study in a unified fashion phenomena from many areas of mathematics. In this paper, we present a new approach to cluster algebras based on noncommutative ring theory. It deals with large, axiomatically defined classes of algebras and does not require initial combinatorial data. Because of this, it has a broad range of applications to open problems on constructing cluster algebra structures on coordinate rings and their quantum counterparts. A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras.

Semiclassical Limits of Quantized Coordinate Rings

This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are breakdowns in the connections when the symplectic leaves are not algebraic. This prompts replacement of the differential-geometric concept of symplectic leaves with the algebraic concept of symplectic cores, and a reformulated conjecture is proposed: The primitive spectrum of A should be homeomorphic to the space of symplectic cores in V, and to the Poisson-primitive spectrum of O(V). Various examples, including both quantized coordinate rings and enveloping algebras of solvable Lie algebras, are analyzed to support the choice of symplectic cores to replace symplectic leaves.

PRIME IDEALS INVARIANT UNDER WINDING AUTOMORPHISMS IN QUANTUM MATRICES

The main goal of the paper is to establish the existence of tensor product decompositions for those prime ideals P of the generic algebra A of quantum n by n matrices which are invariant under winding automorphisms of A. More specifically, every such P is the kernel of a map from A to (A^+/P^+) tensor (A^-/P^-) obtained by composing comultiplication, localization, and quotient maps, where A^+ and A^- are special localized quotients of A while P^+ and P^- are prime ideals invariant under winding automorphisms. Further, the algebras A^+ and A^-, which vary with P, can be chosen so that the correspondence sending (P^+,P^-) to P is a bijection. The main theorem is applied, in a sequel to this paper, to completely determine the winding-invariant prime ideals in the generic quantum 3 by 3 matrix algebra.

NOETHERIAN HOPF ALGEBRAS

A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.