n-Summable Valuated p n -Socles and Primary Abelian Groups

Generalizing the classical concept of a valuated vector space, we introduce the notion of a valuated p ⁿ -socle. A valuated p ⁿ -socle is said to be n-summable if it is isometric to the valuated direct sum of countable valuated groups. Many properties of these objects are established, and in particular, they are shown to be completely classifiable using Ulm invariants, providing a strong connection with the theory of direct sums of countable abelian p-groups. The resulting theory is then applied to the category of primary abelian groups.     

Compact abelian group extensions of discrete dynamical systems

Summary This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.     

Brauer groups are not characterized by Ulm invariants

Two important invariants of a fieldF are its Brauer groupB(F) and its character groupX(F). IfF is countable, these are countable abelian torsion groups, and so are determined by their Ulm invariants. We show here that Ulm’s invariants do not determine Brauer groups or character groups of uncountable fields. An essential tool, which is entirely group theoretic in nature, is a fact about ultraproducts of torsion groups. Supported in part by NSF Grant No. DMS-8500883. Supported in part by NSA Grant No. MDA904-85-H-0014. Supported in part by NSF Grant No. DMS-8500929.     

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide

ES(G)=min{|H|:H≤ Gdense and essential}

TD(G)=min{|H|:H≤ Gtotally dense}

We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.     

Infinitary model theory of abelian groups

The paper is a survey of results in the model theory of abelian groups, dealing with two sorts of problems: finding invariants which classify groups up toL _λκ-equivalence; and determining whether certain classes of groups are definable inL _λκ. Research supported by NSF grant GP 43910     

Rings of invariants for modular representations of elementary abelian <Emphasis Type="Italic">p</Emphasis>-groups

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two-dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two-dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three-dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three-dimensional representation of an elementary abelian p-group is a complete intersection.     

Minimal degrees of invariants of (super)groups – a connection to cryptology

We investigate questions related to the minimal degree of invariants of finitely generated diagonalizable groups. These questions were raised in connection to security of a public key cryptosystem based on invariants of diagonalizable groups. We derive results for minimal degrees of invariants of finite groups, abelian groups and algebraic groups. For algebraic groups we relate the minimal degree of the group to the minimal degrees of its tori. Finally, we investigate invariants of certain supergroups that are superanalogs of tori. It is interesting to note that a basis of these invariants is not given by monomials.     

Geometric invariants of link cobordism

A geometric notion of a “derivative” is defined for 2-component links ofS ⁿ inS ⁿ⁺² and used to construct a sequenceβ ⁱ ,i=1,2,... of abelian concordance invariants which vanish for boundary links. Forn>1, these generalize the only heretofore known invariant, the Sato-Levine invariant. Forn=1, these invariants are additive under any band-sum and consequently provide new information about which 1-links are concordant to boundary links. Examples are given of concordance classes successfully distinguished by theβ ⁱ but not by their , Murasugi 2-height, Sato-Levine invariant or Alexander polynomial. Supported in part by a grant from the National Science Foundation.     

The duals of almost completely decomposable groups

In this paper, we classify the direct products of one-dimensional compact connected abelian groups by cardinal invariants dualizing Baer’s classification theorem of completely decomposable groups. Almost completely decomposable groups are finite rank torsion-free abelian groups which contain a completely decomposable group of finite index. An isomorphism theorem for their Pontrjagin dual groups is given by using the dual concept of a regulating subgroup.     

Quotients of partial abelian monoids

A hierarchy of partial abelian structures is considered. In an order of decreasing generality, these structures include partial abelian monoids (PAM), cancellative PAMs (CPAM), effect algebras (or D-posets), orthoalgebras, orthomodular posets (OMP) and orthomodular lattices (OML). If P is a PAM, the concepts of a congruence on P and a quotient P are defined. Similar definitions are given for quotients of higher level categories in the hierarchy. The notion of a Riesz ideal I on a CPAM P is defined and it is shown that I generates a congruence on P. The corresponding quotients P/I for categories in the hierarchy are studied. It is shown that a subset I of an OML is a Riesz ideal if and only if I is a p-ideal. Moreover, for effect algebras, we show that congruences generated by Riesz ideals are precisely those that are given by a perspectivity. The paper includes a large number of counterexamples and examples that illustrate various concepts. Received April 14, 1997; accepted in final form January 19, 1998.     

Groups with exponents I. Fundamentals of the theory and tensor completions

We revise R. Lyndon's notion of group with exponents [1]. The advantage of the revised notion is that, in the case of abelian groups, it coincides with the notion of a module over a ring. Meanwhile, the abelian groups with exponents in the sense of Lyndon form a substantially wider class. In what follows we introduce basic notions of the theory of groups with exponents; in particular, we present the key construction in the category of groups with exponents, that of tensor completion.The main results of the article are exposed in [2]; the notions of freeA-group and free product ofA-groups can be found in [3].This work was supported by the Russian Foundation for Fundamental Research (Grant 93-011-1159).Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1106–1118, September–October, 1994.     

General Heart Construction on a Triangulated Category (I): Unifying <Emphasis Type="Italic">t</Emphasis>-Structures and Cluster Tilting Subcategories

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories, when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively.     

Instanton Counting and Wall-Crossing for Orbifold Quivers

Noncommutative Donaldson–Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative ${{\mathcal N}=2}$ gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated with the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson–Thomas invariants in the Coulomb branch in terms of gauge theory variables and orbifold data. We introduce the notion of virtual instanton quiver associated with the natural symplectic charge lattice which governs the quantum wall-crossing behaviour of BPS states in this context. The McKay correspondence naturally connects our formalism with other approaches to wall-crossing based on quantum monodromy operators and cluster algebras.     

Essentially normal Hilbert modules and <Emphasis Type="Italic">K</Emphasis>-homology

This paper mainly concerns the essential normality of graded submodules. Essentially all of the basic Hilbert modules that have received attention over the years are p-essentially normal—including the d-shift Hilbert module, the Hardy and Bergman modules of the unit ball. Arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer. Some positive results have been obtained by Arveson and Douglas. However, the problem has been resistant. In dimensions d = 2, 3, this paper shows that the Arveson’s conjecture is true. In any dimension, the paper also gives an affirmative answer in the case of the graded principal submodule. Finally, the paper is associated with K-homology invariants arising from graded quotient modules, by which geometry of the quotient modules and geometry of algebraic varieties are connected. In dimensions d = 2, 3, it is shown that K-homology invariants determined by graded quotients are nontrivial. The paper also establishes results on p-smoothness of K-homology elements, and gives an explicit expression for K-homology invariant in dimension d = 2.     

The strongly invariant extending property for abelian groups

Abstract

We define three specific properties of abelian groups calling them strongly invariant extending property, intermediate fully invariant extending property and strongly intermediate fully invariant extending property. Some results concerning these concepts are proved as for the third one a complete necessary and sufficient condition is established. Our achievements somewhat extend results due to Birkenmeier et al. published in Comm. Algebra (2001).     

Bose-Mesner Algebras Related to Type II Matrices and Spin Models

A type II matrix is a square matrixW with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebra N(W) , which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebrasN(W) . Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.     

STACKED BASES FOR HOMOGENEOUS COMPLETELY DECOMPOSABLE GROUPS

Generalizing a theorem by P. Hill and C. Megibben, fixing a rational group R, we characterize by numerical invariants R-presentations of a group G, namely, short exact sequences of the form 0 → A → X → G → 0, where A and X are homogeneous completely decomposable groups of the same type R. This characterization sets afloat the class of the “uniquely R-presented groups”. This class is investigated in connection with the extension to arbitrary groups of the Warfield equivalence between categories of torsionfree abelian groups induced by the functors Hom(R, –) and R ? ?. As an application, the stacked bases theorem proved by J. Cohen and H. Gluck in 1970 is extended to arbitrary pairs of homogeneous completely decomposable abelian groups of the same type.