76 research outputs found

    Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

    Get PDF
    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 qq is transcendental over Q\mathbb{Q}.Comment: 16 page

    Simple modules over Hereditary Noetherian Prime rings

    Get PDF

    On the representation of simple Riesz groups

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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

    Free and Residually Artinian Regular Rings

    Get PDF

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

    Get PDF
    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
    corecore