Results 1  10
of
29
Model Theory and Modules
, 2006
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se ..."
Abstract

Cited by 64 (20 self)
 Add to MetaCart
The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted ModR, the full subcategory of finitely presented modules will be denoted modR, the
Quantized reduction as a tensor product
 QUANTIZATION OF SINGULAR SYMPLECTIC QUOTIENTS. BASEL: BIRKHÄUSER, 2001. EPRINT MATHPH/0008004
, 2008
"... Matched bimodules for rings may be composed through the (algebraic) bimodule tensor product, the canonical bimodule R → R ← R serving as a unit for ⊗R. We describe this picture also for C ∗algebras, von Neumann algebras, Lie groupoids, Poisson manifolds, and symplectic groupoids. This hinges on th ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
Matched bimodules for rings may be composed through the (algebraic) bimodule tensor product, the canonical bimodule R → R ← R serving as a unit for ⊗R. We describe this picture also for C ∗algebras, von Neumann algebras, Lie groupoids, Poisson manifolds, and symplectic groupoids. This hinges on the correct notion of bimodule, tensor product, and unit: for C ∗algebras B one has Hilbert (C ∗ ) bimodules with Rieffel’s tensor product and the canonical Hilbert bimodule over B, for von Neumann algebras one uses correspondences with Connes’s tensor product and the standard form, for (symplectic) Lie groupoids G one has regular (symplectic) bibundles with the Hilsum–Skandalis tensor product and the canonical bibundle over G, and for integrable Poisson manifolds P one deals with regular symplectic bimodules (dual pairs) with Xu’s tensor product and the sconnected and ssimply connected symplectic groupoid over P. Morita theory relates socalled equivalence bimodules to equivalence of representation theories. Subsequently, we study certain interconnections between the various constructions. The relation between Hilbert bimodules and correspondences is reviewed in detail. The notion of Marsden–Weinstein reduction makes sense for Poisson manifolds, C ∗algebras, and von Neumann algebras. Poisson manifolds and Lie groupoids join in the theory of Lie algebroids and symplectic groupoids. Finally, we note that the Poisson manifolds associated to Morita equivalent sconnected and ssimply connected Lie groupoids are Morita equivalent in the sense of Xu.
Module and Comodule Categories  A Survey
 Proc. of the Mathematics Conference (Birzeit University 1998), World Scientific
, 2000
"... The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a re ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule structure of algebras and comodule categories are sketched.
A Morita theorem for algebras of operators on Hilbert Space
"... Abstract. We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Ha ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Abstract. We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product ( = interior tensor product) with a strong Morita equivalence bimodule.
ASSOCIATIVE CONFORMAL ALGEBRAS WITH FINITE FAITHFUL REPRESENTATION
, 2004
"... Abstract. We describe irreducible conformal subalgebras of CendN and build the structure theory of associative conformal algebras with finite faithful representation. 1. ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
Abstract. We describe irreducible conformal subalgebras of CendN and build the structure theory of associative conformal algebras with finite faithful representation. 1.
Isomorphisms of function modules, and generalized approximation in modulus
, 1999
"... Abstract. For a function algebra A we investigate relations between the following three topics: isomorphisms of singly generated Amodules, Morita equivalence bimodules, and ‘real harmonic functions ’ with respect to A. We also consider certain groups which are naturally associated with a uniform al ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. For a function algebra A we investigate relations between the following three topics: isomorphisms of singly generated Amodules, Morita equivalence bimodules, and ‘real harmonic functions ’ with respect to A. We also consider certain groups which are naturally associated with a uniform algebra A. We illustrate the notions considered with several examples. 1. Introduction. By a uniform algebra or function algebra on a compact Hausdorff space Ω, we shall mean a subalgebra A of C(Ω) (the continuous complex valued functions on Ω) which contains constants and separates points. In most of this paper we are concerned with closed submodules of C(Ω) of the form Af, where f is a strictly positive and continuous function on Ω. In Part
Leavitt path algebras and direct limits
"... Abstract. An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplic ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the Ktheoretic monoid of equivalence classes of idempotent matrices.
Wide Morita contexts, relative injectivity and equivalence resutls
 J. Algebra
"... We extend Morita theory to abelian categories by using wide Morita contexts. Several equivalence results are given for wide Morita contexts between abelian categories, widely extending equivalence theorems for categories of modules and comodules due to Kato, Müller and Berbec. In the case of Grothen ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
We extend Morita theory to abelian categories by using wide Morita contexts. Several equivalence results are given for wide Morita contexts between abelian categories, widely extending equivalence theorems for categories of modules and comodules due to Kato, Müller and Berbec. In the case of Grothendieck categories we derive equivalence results by using quotient categories. We apply the general equivalence results to rings with identity, rings with local units, graded rings, DoiHopf modules and coalgebras.
STRUCTURE OF FPINJECTIVE AND WEAKLY QUASIFROBENIUS RINGS
, 1999
"... Abstract. In the present paper new criteria for classes of FPinjective and weakly quasiFrobenius rings are given. Properties of both classes of rings are closely linked with embedding of finitely presented modules in fpflat and free modules respectively. Using these properties, we describe classe ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. In the present paper new criteria for classes of FPinjective and weakly quasiFrobenius rings are given. Properties of both classes of rings are closely linked with embedding of finitely presented modules in fpflat and free modules respectively. Using these properties, we describe classes of coherent CF and FGFrings. Moreover, it is proved that the group ring R(G) is FPinjective (resp. weakly quasiFrobenius) if and only if the ring R is FPinjective (resp. weakly quasiFrobenius) and G is locally finite. An application of the duality context RRR to categories of finitely generated left and right Rmodules leads to to the case when R is noetherian selfinjective ring. Such rings are called quasiFrobenius (or QFrings). In turn, an Rduality for categories of finitely presented modules leads to classes
On the representation theory of Lie triple systems
 Trans. Amer. Math. Soc
"... Abstract. In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of Hodge (2001). In a final section, we begin a study of the cohomology of Lie triple systems. 1. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of Hodge (2001). In a final section, we begin a study of the cohomology of Lie triple systems. 1.