76 research outputs found
Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves
The algebra of quantum matrices of a given size supports a rational torus
action by automorphisms. It follows from work of Letzter and the first named
author that to understand the prime and primitive spectra of this algebra, the
first step is to understand the prime ideals that are invariant under the torus
action. In this paper, we prove that a family of quantum minors is the set of
all quantum minors that belong to a given torus-invariant prime ideal of a
quantum matrix algebra if and only if the corresponding family of minors
defines a non-empty totally nonnegative cell in the space of totally
nonnegative real matrices of the appropriate size. As a corollary, we obtain
explicit generating sets of quantum minors for the torus-invariant prime ideals
of quantum matrices in the case where the quantisation parameter is
transcendental over .Comment: 16 page
On the representation of simple Riesz groups
In this paper we answer Open Problem 2 of Goodearl's book on
partially ordered abelian groups in the case of partially ordered sim-
ple groups. As a consequence, we obtain a version of the Theorem of
structure of dimension groups in the case of simple Riesz groups. Also,
we give a method for constructing torsion-free strictly perforated simple
Riesz groups of rank one, and we see that every dense additive subgroup
of Q can be obtained using this method
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 ðZ=nZÞ t f0g (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, ðn þ 1Þx ¼ 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 L G for semilattices L and torsion abelian groups G.
When applied to the monoids VðAÞ appearing in the non-stable K-theory of C*-algebras,
our results yield characterizations of the monoids VðAÞ for C* inductive limits A of sequences
of finite direct products of matrix algebras over Cuntz algebras On. 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
K0 of purely infinite simple regular rings
We extend the notion of a purely in nite 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 in nite simple ring, then K0(R)+ = K0(R), the monoid
of isomorphism classes of nitely generated projective R-modules is isomorphic to the
monoid obtained from K0(R) by adjoining a new zero element, and K1(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 K0 of some purely in nite simple regular ring.
Finally, some known examples are analyzed within this framework
Diagonalization of matrices over regular rings
Square matrices are shown to be diagonalizable over all known classes of (von Neumann) regular rings. This diagonalizability is equivalent to a cancellation property
for finitely generated projective modules which conceivably holds over all regular rings.
These results are proved in greater generality, namely for matrices and modules over
exchange rings, where attention is restricted to regular matrices
Diagonalization of matrices over regular rings
Square matrices are shown to be diagonalizable over all known classes of (von Neumann) regular rings. This diagonalizability is equivalent to a cancellation property
for finitely generated projective modules which conceivably holds over all regular rings.
These results are proved in greater generality, namely for matrices and modules over
exchange rings, where attention is restricted to regular matrices
Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space
The standard Poisson structure on the rectangular matrix variety Mm,n(C) is
investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂
GLm+n(C). These orbits, finite in number, are shown to be smooth irreducible locally closed
subvarieties of Mm,n(C), isomorphic to intersections of dual Schubert cells in the full flag
variety of GLm+n(C). Three different presentations of the T-orbits of symplectic leaves in
Mm,n(C) are obtained – (a) as pullbacks of Bruhat cells in GLm+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 GLm(C) and GLn(C). In presentation (a), the orbits
of leaves are parametrized by a subset of the Weyl group Sm+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 rowechelon
form. Finally, decompositions of generalized double Bruhat cells in Mm,n(C) (with
respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves
are obtained
- …