1,536 research outputs found
Differential Krull dimension in differential polynomial extensions
We investigate the differential Krull dimension of differential polynomials
over a differential ring. We prove a differential analogue of Jaffard's Special
Chain Theorem and show that differential polynomial extensions of certain
classes of differential rings have no anomaly of differential Krull dimension.Comment: 23 page
Prime spectra of quantized coordinate rings
This paper is partly a report on current knowledge concerning the structure
of (generic) quantized coordinate rings and their prime spectra, and partly
propaganda in support of the conjecture that since these algebras share many
common properties, there must be a common basis on which to treat them. The
first part of the paper is expository. We survey a number of classes of
quantized coordinate rings, as well as some related algebras that share common
properties, and we record some of the basic properties known to occur for many
of these algebras, culminating in stratifications of the prime spectra by the
actions of tori of automorphisms. As our main interest is in the generic case,
we assume various parameters are not roots of unity whenever convenient. In the
second part of the paper, which is based on joint work with E. S. Letzter in
[The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl
algebras (to appear in Trans. Amer. Math. Soc.)], we offer some support for the
conjecture above, in the form of an axiomatic basis for the observed
stratifications and their properties. At present, the existence of a suitable
supply of normal elements is taken as one of the axioms; the search for better
axioms that yield such normal elements is left as an open problem.Comment: 33 pages, to appear in Proceedings of Euroconference on Interactions
between Ring Theory and Representations of Algebras (Murcia, 1998). See also
http://www.math.ucsb.edu/~goodearl/preprints.html
Quantum Schubert cells via representation theory and ring theory
We resolve two questions of Cauchon and Meriaux on the spectra of the quantum
Schubert cell algebras U^-[w]. The treatment of the first one unifies two very
different approaches to Spec U^-[w], a ring theoretic one via deleting
derivations and a representation theoretic one via Demazure modules. The
outcome is that now one can combine the strengths of both methods. As an
application we solve the containment problem for the Cauchon-Meriaux
classification of torus invariant prime ideals of U^-[w]. Furthermore, we
construct explicit models in terms of quantum minors for the Cauchon quantum
affine space algebras constructed via the procedure of deleting derivations
from all quantum Schubert cell algebras U^-[w]. Finally, our methods also give
a new, independent proof of the Cauchon-Meriaux classification.Comment: 29 pages, AMS Latex, minor changes in v.
The Prime Spectrum of Quantum and the Poisson-prime Spectrum of its Semi-classical Limit
A bijection is defined between the prime spectrum of quantum
and the Poisson prime spectrum of , and we verify that and
both preserve inclusions of primes, i.e. that is in fact a
homeomorphism between these two spaces. This is accomplished by developing a
Poisson analogue of Brown and Goodearl's framework for describing the Zariski
topology of spectra of quantum algebras, and then verifying directly that in
the case of these give rise to identical pictures on both the quantum
and Poisson sides. As part of this analysis, we study the Poisson primitive
spectrum of and obtain explicit generating sets for all of
the Poisson primitive ideals.Comment: Minor updates, clarifications, etc throughout the text. To appear in
Transactions of the London Mathematical Societ
A sparse effective Nullstellensatz
We present bounds for the sparseness and for the degrees of the polynomials
in the Nullstellensatz. Our bounds depend mainly on the unmixed volume of the
input polynomial system. The degree bounds can substantially improve the known
ones when this polynomial system is sparse, and they are, in the worst case,
simply exponential in terms of the number of variables and the maximum degree
of the input polynomials.Comment: 24 pages, Latex2e, available at http://www.dm.uba.ar/tera/ and
http://hilbert.matesco.unican.es/tera
Prime Ideals of q-Commutative Power Series Rings
We study the "q-commutative" power series ring R:=k_q[[x_1,...,x_n]], defined
by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric
scalars q_{ij} in a field k. Our results provide a detailed account of prime
ideal structure for a class of noncommutative, complete, local, noetherian
domains having arbitrarily high (but finite) Krull, global, and classical Krull
dimension. In particular, we prove that the prime spectrum of R is normally
separated and is finitely stratified by commutative noetherian spectra.
Combining this normal separation with results of Chan, Wu, Yekutieli, and
Zhang, we are able to conclude that R is catenary. Following the approach of
Brown and Goodearl, we also show that links between prime ideals are provided
by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find
that R has only finitely many prime ideals and is a UFD (in the sense of
Chatters).Comment: Revised; to appear in Algebras and Representation Theory. In this
(final) version, catenarity is established, by combining normal separation
with results of Chan, Wu, Yekutieli, and Zhang
From quantum Ore extensions to quantum tori via noncommutative UFDs
All iterated skew polynomial extensions arising from quantized universal
enveloping algebras of Kac-Moody algebras are special examples of a very large,
axiomatically defined class of algebras, called CGL extensions. For the
purposes of constructing initial clusters for quantum cluster algebra
structures on an algebra R, and classification of the automorphisms of R, one
needs embeddings of R into quantum tori T which have the property that R
contains the corresponding quantum affine space algebra A. We explicitly
construct such an embedding A \subseteq R \subset T for each CGL extension R
using the methods of noncommutative noetherian unique factorization domains and
running a Gelfand-Tsetlin type procedure with normal, instead of central
elements. Along the way we classify the homogeneous prime elements of all CGL
extensions and we prove that each CGL extension R has an associated maximal
torus which covers the automorphisms of R corresponding to all normal elements.
For symmetric CGL extensions, we describe the relationship between our quantum
affine space algebra A and Cauchon's quantum affine space algebra generated by
elements obtained via deleting derivations.Comment: 35 pages, AMS Latex. Final version; accepted for Advances in
Mathematic
Zariski topologies on stratified spectra of quantum algebras
A framework is developed to describe the Zariski topologies on the prime and
primitive spectra of a quantum algebra in terms of the (known) topologies
on strata of these spaces and maps between the collections of closed sets of
different strata. A conjecture is formulated, under which the desired maps
would arise from homomorphisms between certain central subalgebras of localized
factor algebras of . When the conjecture holds, spec and prim are
then determined, as topological spaces, by a finite collection of (classical)
affine algebraic varieties and morphisms between them. The conjecture is
verified for , , and when is a non-root of unity and the base field is
algebraically closed.Comment: 23 pages; 4 xypic diagram
Semicrossed Products of Operator Algebras by Semigroups
We examine the semicrossed products of a semigroup action by
-endomorphisms on a C*-algebra, or more generally of an action on an
arbitrary operator algebra by completely contractive endomorphisms. The choice
of allowable representations affects the corresponding universal algebra. We
seek quite general conditions which will allow us to show that the C*-envelope
of the semicrossed product is (a full corner of) a crossed product of an
auxiliary C*-algebra by a group action.
Our analysis concerns a case-by-case dilation theory on covariant pairs. In
the process we determine the C*-envelope for various semicrossed products of
(possibly nonselfadjoint) operator algebras by spanning cones and
lattice-ordered abelian semigroups.
In particular, we show that the C*-envelope of the semicrossed product of
C*-dynamical systems by doubly commuting representations of
(by generally non-injective endomorphisms) is the full corner of a C*-crossed
product. In consequence we connect the ideal structure of C*-covers to
properties of the actions. In particular, when the system is classical, we show
that the C*-envelope is simple if and only if the action is injective and
minimal.
The dilation methods that we use may be applied to non-abelian semigroups. We
identify the C*-envelope for actions of the free semigroup by
automorphisms in a concrete way, and for injective systems in a more abstract
manner. We also deal with C*-dynamical systems over Ore semigroups when the
appropriate covariance relation is considered.Comment: 100 pages; comments and references update
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.Comment: 36 pages. See also http://www.math.ucsb.edu/~goodearl/preprints.html
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