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K. R. Goodearl

Publications and source records attributed to K. R. Goodearl.

At least 55 records · Page 3Linked to original sources
arXiv2005

Homological properties of quantized coordinate rings of semisimple groups

We prove that the generic quantized coordinate ring $\mathcal{O}_q(G)$ is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group $G$. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a new set of positive cases for a question of Brown and the first author.

math.QA↗
arXiv2005

The Closed-Point Zariski Topology for Irreducible Representations

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simple left modules over a ring R. Namely, if R is left noetherian (or satisfies the ascending chain condition for semiprimitive ideals), and if R is either a countable dimensional algebra (over a field) or a ring whose (Gabriel-Rentschler) Krull dimension is a countable ordinal, then each closed set of the refined Zariski topology is the union of a finite set with a Zariski closed set. The approach requires certain auxiliary results guaranteeing embeddings of factor rings into direct products of simple modules. Analysis of these embeddings mimics earlier work of the first author and Zimmermann-Huisgen on products of torsion modules.

math.RA↗
arXiv2005

Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space

The standard Poisson structure on the rectangular matrix variety M_{m,n}(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T of GL_{m+n}(C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of M_{m,n}(C), isomorphic to intersections of dual Schubert cells in the full flag variety of GL_{m+n}(C). Three different presentations of the T-orbits of symplectic leaves in M_{m,n}(C) are obtained - (a) as pullbacks of Bruhat cells in GL_{m+n}(C) under a particular map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GL_m(C) and GL_n(C). In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group S_{m+n}, such that inclusions of Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed row-echelon form. Finally, decompositions of generalized double Bruhat cells in M_{m,n}(C) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained.

math.QA↗
arXiv2004

Semilattices of groups and inductive limits of Cuntz algebras

We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form $(Z/nZ)\sqcup\{0\}$ (where 0 is a new zero element), for positive integers $n$. The key properties are the Riesz refinement property and the requirement that each element $x$ has finite order, that is, $(n+1)x=x$ for some positive integer $n$. Such monoids are necessarily semilattices of abelian groups, and part of our approach yields a characterization of the Riesz refinement property among semilattices of abelian groups. Further, we describe the monoids in question as certain submonoids of direct products $Λ\times G$ for semilattices $Λ$ and torsion abelian groups $G$. When applied to the monoids $V(A)$ appearing in the non-stable K-theory of C*-algebras, our results yield characterizations of the monoids $V(A)$ for C* inductive limits $A$ of sequences of finite direct products of matrix algebras over Cuntz algebras $O_n$. In particular, this completely solves the problem of determining the range of the invariant in the unital case of Rørdam's classification of inductive limits of the above type.

math.OA↗
arXiv2004

The complete dimension theory of partially ordered systems with equivalence and orthogonality

We develop dimension theory for a large class of structures called espaliers, consisting of a set $L$ equipped with a partial order $\leq$, an orthogonality relation $\perp$, and an equivalence relation $\sim$, subject to certain axioms. The dimension range of $L$ is the universal $\sim$-invariant homomorphism from $(L,\oplus,0)$ to a partial commutative monoid $S$, where $\oplus$ denotes orthogonal sum in $L$. Particular examples of espaliers include (i) complete Boolean algebras, (ii) direct summand lattices of nonsingular injective modules, (iii) complete, meet-continuous, complemented, modular lattices, and (iv) projection lattices in AW*-algebras. We prove that the dimension range of any espalier is a lower interval of a commutative monoid of continuous functions of the form $C(Ω_{I},Z_γ) \times C(Ω_{II},R_γ) \times C(Ω_{III},2_γ)$, where $γ$ is an ordinal and the $Ω_{*}$ are complete Boolean spaces, and where $Z_γ$, $R_γ$, $2_γ$, respectively, denote the unions of the interval $\{\aleph_ξ\mid 0\le ξ\le γ\}$ with the sets of nonnegative integers, nonnegative real numbers, and 0, respectively. Conversely, we prove that every lower interval of a monoid of the above form can be represented as the dimension range of an espalier arising from each of the contexts (i)--(iv) above. As corollaries in cases (ii) and (iv), we obtain complete descriptions (both function-theoretic and axiomatic) of the monoids $V(R)$, consisting of the isomorphism classes of finitely generated projective modules over a ring $R$.

math.GM↗
arXiv2003

Simplicity of noncommutative Dedekind domains

The following dichotomy is established: A finitely generated, complex Dedekind domain that is not commutative is simple. Weaker versions of this dichotomy are proved for Dedekind prime rings and hereditary noetherian prime rings.

math.RA↗
arXiv2003

Stable rank of corner rings

B. Blackadar recently proved that any full corner $pAp$ in a unital C*-algebra $A$ has K-theoretic stable rank greater than or equal to the stable rank of $A$. (Here $p$ is a projection in $A$, and fullness means that $ApA=A$.) This result is extended to arbitrary (unital) rings $A$ in the present paper: If $p$ is a full idempotent in $A$, then $sr(pAp) \geq sr(A)$. The proofs rely partly on algebraic analogs of Blackadar's methods, and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners $pAq$. The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if $B$ is isomorphic to the endomorphism ring of a finitely generated projective generator $P_A$ which can be generated by $n$ elements, then $sr(A) \leq n{\cdot}sr(B)-n+1$.

math.RA↗
arXiv2003

Fractional skew monoid rings

Given an action of a monoid $T$ on a ring $A$ by ring endomorphisms, and an Ore subset $S$ of $T$, a general construction of a fractional skew monoid ring $S^{\rm op} * A * T$ is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case $S$ is a subsemigroup of a group $G$ such that $G=S^{-1}S$, we obtain a $G$-graded ring $S^{\rm op} * A * S$ with the property that, for each $s\in S$, the $s$-component contains a left invertible element and the $s^{-1}$-component contains a right invertible element. In the most basic case, where $G$ is the additive group of integers and $S=T$ is the submonoid of nonnegative integers, the construction is fully determined by a single ring endomorphism $α$ of $A$. If $α$ is an isomorphism onto a proper corner $pAp$, we obtain an analogue of the usual skew Laurent polynomial ring, denoted by $A[t_+,t_-;α]$. Examples of this construction are given, and it is proven that several classes of known algebras, including the Leavitt algebras of type $(1,n)$, can be presented in the form $A[t_+,t_-;α]$. Finally, mild and reasonably natural conditions are obtained under which $S^{\rm op} * A * S$ is a purely infinite simple ring.

math.RA↗
arXiv2002

Quantized coordinate rings and related noetherian algebras

This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of the prime spectrum is governed by a partition into strata determined by the action of a suitable group of automorphisms of the algebra. We discuss this stratification in detail, as well as its use in determining the primitive spectrum -- under suitable conditions, the primitive ideals are precisely those prime ideals which are maximal within their strata. The discussion then turns to the global structure of the primitive spectra of quantized coordinate rings, and to the conjecture that these spectra are topological quotients of the corresponding classical affine varieties. We describe the solution to the conjecture for quantized coordinate rings of full affine spaces and (somewhat more generally) affine toric varieties. The final part of the paper is devoted to the quantized coordinate ring of $n\times n$ matrices. We mention parallels between this algebra and the classical coordinate ring, such as the primeness of quantum analogs of determinantal ideals. Finally, we describe recent work which determined, for the $3\times3$ case, all prime ideals invariant under the group of winding automorphisms governing the stratification mentioned above.

math.QA↗
arXiv2001

Winding-invariant prime ideals in quantum $3\times3$ matrices

A complete determination of the prime ideals invariant under winding automorphisms in the generic 3 by 3 quantum matrix algebra is obtained. Explicit generating sets consisting of quantum minors are given for all of these primes, thus verifying a general conjecture in the 3 by 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 the n by n quantum matrix algebra.

math.QA↗
arXiv2001

$K_0$ of purely infinite simple regular rings

We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if $R$ is a purely infinite simple ring, then $K_0(R)^+= K_0(R)$, the monoid of isomorphism classes of finitely generated projective $R$-modules is isomorphic to the monoid obtained from $K_0(R)$ by adjoining a new zero element, and $K_1(R)$ is the abelianization of the group of units of $R$. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we compute the Grothendieck groups of these examples. In particular, we prove that every countable abelian group is isomorphic to $K_0$ of some purely infinite simple regular ring. Finally, some known examples are analyzed within this framework.

math.RA↗
arXiv2001

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.

math.QA↗
arXiv1999

Quantized primitive ideal spaces as quotients of affine algebraic varieties

Given an affine algebraic variety V and a quantization A of its coordinate ring, it is conjectured that the primitive ideal space of A can be expressed as a topological quotient of V. Evidence in favor of this conjecture is discussed, and positive solutions for several types of varieties (obtained in joint work with E. S. Letzter) are described. In particular, explicit topological quotient maps are given in the case of quantum toric varieties.

math.QA↗
arXiv1999

The first fundamental theorem of coinvariant theory for the quantum general linear group

We prove First Fundamental Theorems of Coinvariant Theory for the standard coactions of the quantum general and special linear groups on tensor products of quantum matrix algebras. More precisely, let m,n,t be arbitrary positive integers, let A and B be the quantum coordinate rings of the $t \times t$ general and special linear groups over an arbitrary field K, and let $C_{m,t}$ and $C_{t,n}$ denote the quantum coordinate rings of $m \times t$ and $t \times n$ matrices over K. We first prove that the set of coinvariants for the coaction of A on $C_{m,t} \otimes C_{t,n}$ equals the image of the natural K-algebra map from the quantum coordinate ring of $m \times n$ matrices to $C_{m,t} \otimes C_{t,n}$ induced by comultiplication. The set of coinvariants for the coaction of B on $C_{m,t} \otimes C_{t,n}$ is shown to be the subalgebra generated by the above image together with a tensor product of two algebras generated by $t \times t$ quantum minors.

math.QA↗
arXiv1999

$K_1$ of separative exchange rings and C*-algebras with real rank zero

For any (unital) exchange ring $R$ whose finitely generated projective modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus B\cong B\oplus B$ implies $A\cong B$), it is shown that all invertible square matrices over $R$ can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism $GL_1(R) \to K_1(R)$ is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra $A$ with real rank zero, the topological $K_1(A)$ is naturally isomorphic to the unitary group $U(A)$ modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.

math.RA↗
arXiv1999

The graded version of Goldie's Theorem

The analogue of Goldie's Theorem for prime rings is proved for rings graded by abelian groups, eliminating unnecessary additional hypotheses used in earlier versions.

math.RA↗
arXiv1999

Quantum n-space as a quotient of classical n-space

Let $A$ denote the commutative polynomial ring in $n$ variables, over an algebraically closed field $k$, and let $R$ denote the standard multiparameter quantization of $A$ determined by a multiplicatively antisymmetric $n\times n$ matrix $(q_{ij})$. In this paper we prove, when -1 cannot be multiplicatively generated by the $q_{ij}$, that the primitive spectrum of $R$ is a topological quotient of $k^n$. Under the same hypothesis, we further prove that the prime spectrum of $R$ is a topological quotient of the prime spectrum of $A$.

math.RA↗