Module-theoretic Approach Research Articles

Dirac operators on noncommutative hypersurfaces

This paper studies geometric structures on noncommutative hypersurfaces within a module-theoretic approach to noncommutative Riemannian (spin) geometry. A construction to induce differential, Riemannian and spinorial structures from a noncommutative embedding space to a noncommutative hypersurface is developed and applied to obtain noncommutative hypersurface Dirac operators. The general construction is illustrated by studying the sequence Tθ2↪Sθ3↪Rθ4 of noncommutative hypersurface embeddings.

Editorial

Editorial

James Alexander Green. 26 February 1926—7 April 2014

James Alexander Green, known as Sandy, was a mathematician of great influence and distinction. He was an algebraist, famous for his work on modular representations of finite groups, and the development of the theory of polynomial representations of general linear groups. He was elected Fellow of the Royal Society of Edinburgh (1968) and Fellow of the Royal Society of London (1987). He was awarded prizes of the London Mathematical Society, a Senior Berwick Prize (in 1984) and the De Morgan Medal (in 2001). In his doctoral thesis, on semigroups, Sandy introduced fundamental relations, now known as ‘Green's relations’. He determined the characters of arbitrary finite general linear groups published 1955. Sandy then turned to representations of finite groups over fields of prime characteristic; his work laid the foundations for the module theoretic approach to the subject. His next highlight is his monograph on polynomial representations of GL n , published in 1980, which has become the basis for algebraic highest weight theory. Furthermore, in 1995 he proved a fundamental result on Hall algebras, establishing a connection between quantum groups and representations of finite-dimensional quiver algebras.

A module-theoretic approach to matroids

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.

A module minimization approach to Gabidulin decoding via interpolation

We focus on iterative interpolation-based decoding of Gabidulin codes and present an algorithm that computes a minimal basis for an interpolation module. We extend existing results for Reed-Solomon codes in showing that this minimal basis gives rise to a parametrization of elements in the module that lead to all Gabidulin decoding solutions that are at a fixed distance from the received word. Our module-theoretic approach strengthens the link between Gabidulin decoding and Reed-Solomon decoding, thus providing a basis for further work into Gabidulin list decoding.

Erratum to “A module-theoretic approach to abelian automorphism groups”

Erratum to “A module-theoretic approach to abelian automorphism groups”

Special issue on symbolic methods in multidimensional systems theory

In the seventies, the study of transfer matrices of linear time-invariant systems led to the development of the polynomial approach (Kailath, Rosenbrock, et al.). In this approach, univariate polynomial matrices play a central role (e.g., Hermite and Smith normal forms, Bezout equations etc.). When generalizing linear systems given by ordinary differential/difference equations to differential time-delay systems, systems with parameters, 2-D or 3-D filters and circuits, one had to face the case of systems described by means of matrices with entries in multivariate polynomial rings. These new systems were called 2-D or 3-D and, more generally, multidimensional systems (Bose, Lin, Pommaret, Oberst, Youla, Wood, et al.). For such systems, no normal forms such as Hermite and Smith forms exist. In order to handle these problems, the concept of Grobner bases (developed by Buchberger) was introduced in multidimensional systems theory. In many ways, the computation of these bases can be seen as an extension of both Gaussian elimination and the Euclidean algorithm. Grobner bases have been implemented in nearly all modern computer algebra systems. Another difficulty with multidimensional systems is the lack of a canonical first order representation. Several models (Roesser, Fornasini-Marchesini, reduced Spencer form etc.) have been studied extensively. The geometric approach to systems and control (Basile,Marro, Wonham, et al.) has recently been extended to these classes of multidimensional systems yielding promising results. The reduction to first order representations (realization problem) is of great importance to many applications of multidimensional systems theory such as uncertainty modeling for robust control. Special computer algebra packages have been developed for the effective manipulation of linear fractional transformations. In his pioneering work, Kalman developed a module-theoretic approach to the realization problem. Independently, the Japanese school of Sato, Kashiwara, and Kawai proposed an

A module-theoretic approach to abelian automorphism groups

There are several examples in the literature of finite non-abelian $p$-groups whose automorphism group is abelian. For some time only examples that were special $p$-groups were known, until Jain and Yadav [JY12] and Jain, Rai and Yadav [JRY13] constructed several non-special examples. In this paper we show how a simple module-theoretic approach allows the construction of non-special examples, starting from special ones constructed by several authors, while at the same time avoiding further direct calculations.

Tensor structure on kC-mod and cohomology

AbstractLet $\mathcal{C}$ be a finite category and let k be a field. We consider the category algebra $k\mathcal{C}$ and show that $k\mathcal{C}$-mod is closed symmetric monoidal. Through comparing $k\mathcal{C}$ with a co-commutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. In particular, we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove that the Hochschild cohomology rings of a certain type of finite category algebras are finitely generated.

Duality in system analysis for bond graph models

Duality as a general notion has been discussed previously in multiple domains. The field of system modeling, analysis and control has used it for quite some time. The duality between the controllability and the observability is well known, especially in the case of the linear time-invariant systems. But when it comes to the linear systems in general, time-invariant or otherwise, the definitions become ambiguous. Even though there have been papers which use the state space representation or the module theoretical approach, a unified description has not been found yet. This paper is meant to fill in this gap, by using the bond graph representation. The bond graph perspective offers a global overview because the bond graph is a graphical tool which can be seen both as a state space representation and as a module.

Tilting categories with applications to stratifying systems

In the study of standardly stratified algebras and stratifying systems, we find an object which is either a tilting module or one whose properties strongly remind us of a tilting module. This tilting module appeared already in Dlab and Ringel's work on quasi-hereditary algebras (see [V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras, in: Repr. Theory and Related Topics, in: London Math. Soc. Lecture Note Ser. 168 (1992) 200–224] and [C.M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991) 209–223]). Also, this tilting module appears on the work on standardly stratified algebras of I. Ágoston, D. Happel, E. Lukács, and L. Unger, and in the paper of M.I. Platzeck and I. Reiten (see [I. Ágoston, D. Happel, E. Lukács, L. Unger, Standardly stratified algebras and tilting, J. Algebra 226 (2000) 144–160] and [M.I. Platzeck, I. Reiten, Modules of finite projective dimension for standardly stratified algebras, Comm. Algebra 29 (3) (2001) 973–986]). Inspired by them, we introduce the notion of tilting category in order to give a unified approach of these situations for stratifying systems. To do so, we use the ideas of M. Auslander, O. Buchweitz and I. Reiten related to approximation theory (see [M. Auslander, R.O. Buchweitz, The homological theory of maximal Cohen–Macaulay approximations. Mem. Soc. Math. Fr. (N.S.) 38 (1989) 5–37; M. Auslander, I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991) 111–152]).

On some theoretical and computational aspects of Anatol Vieru?s periodic sequences

This article develops some aspects of Anatol Vieru’s compositional technique based on finite difference calculus of periodic sequences taking values in a cyclic group. After recalling some group-theoretical properties, we focus on the decomposition algorithm enabling to represent any periodic sequence taking values in a cyclic group as a sum of a reducible and a reproducible sequence. The implementation of this decomposition theorem in a computer aided music composition language, as OpenMusic [1] , enables to easily verify if a given periodic sequence is reducible or reproducible. In this special case, one of the two terms will be identically zero. This means that every periodic sequence has in itself a certain degree of reducibility and reproducibility. We also suggest how to use this result in order to explain some regularities of the distribution of numerical values in the case of the finite addition process and how to generalize the decomposition theorem by means of the Fitting Lemma. This opens the problem of the musical relevance of a generalized module-theoretical approach in Vieru’s theory of periodic sequences.

Linkage of modules

It is shown that the notion of linkage of algebraic varieties, introduced by Peskine and Szpiro, can be generalized to finitely generated modules over non-commutative noetherian semiperfect rings. Besides greater generality, the module-theoretic approach brings about new invariants of linkage and yields improved results and simpler proofs even in the traditional settings of commutative algebraic geometry and local algebra. Connections with Auslander–Reiten sequences, singularity theory, derived categories, local cohomology, Buchsbaum modules, and maximal Cohen–Macaulay approximations are discussed.

Module theoretic approach to controllability of convolutional systems

Both the behavioral approach proposed by Willems and the module theoretic approach proposed by Fliess provide a representation free theory of dynamical systems. According to these approaches, controllability is an intrinsic system property independent of the system representation. In the first part of this paper a comparison between the behavioral approach and the module theoretic approach to convolutional systems is proposed. Then, controllability of this class of systems is analyzed according to these two approaches. This concept is related to the existence of image representations. Finally, the same analysis is proposed for systems described by delay-differential equations.

Module theoretic approach to controllability analysis of convolutional systems

The behavioral approach proposed by Willems and the module theoretic approach introduced by Fliess have many points of contact and analogies. Controllability of convolutional systems is analyzed, various properties connected to controllability are introduced and the interpretation of these properties are given according to both the approaches.

Modules and Behaviours in nD Systems Theory

This paper is intended both as an introduction to the behavioural theory of nD systems, in particular the duality of Oberst and its applications, and also as a bridge between the behavioural theory and the module-theoretic approach of Fliess, Pommaret and others. Our presentation centres on Pommaret‘s notion of a system observable, and uses this concept to provide new interpretations of known behavioural results. We discuss among other subjects autonomous systems, controllable systems, observability, transfer matrices, computation of trajectories, and system complexity.

A module-theoretic approach to Clifford theory for blocks

This work concerns a generalization of Clifford theory to blocks of group-graded algebras. A module-theoretic approach is taken to prove a one-to-one correspondence between the blocks of a fully group-graded algebra covering a given block of its identity component, and conjugacy classes of blocks of a twisted group algebra. In particular, this applies to blocks of a finite group covering blocks of a normal subgroup.

Duality in time-varying linear systems: A module theoretic approach

Using elementary module theory, an intrinsic definition of the dual (or adjoint) of a generalized time-varying linear system is given. With this, the duality of controllability and observability is recovered from their intrinsic module theoretical definitions. The duality of state feedback and output injection is discussed both in the static case and for its quasistatic generalization. Related Brunovsky type canonical forms are derived in the quasistatic case. The corresponding controllability indices and their duals the observability indices are defined intrinsically.

Dipolynomial minimal bases and linear systems in AR representation

This paper deals with a module-theoretic approach to the dipolynomial matrix parametrization of discrete-time behavior systems as introduced by Willems. Canonical minimal-lag representations for these behaviors are constructed which are tighter than the corresponding polynomial representations.

Explicit Solutions of the Matrix Equation $\sum {A^i XD_i } = C$

A module theoretic approach is developed to study the linear matrix equation $\sum {A^i } XD_i = C$. If the solution X is unique it can be expressed in the form $X = \sum {A^k CG_k } $. The matrices $G_k $ can be determined from an auxiliary equation which involves the companion matrix of the characteristic polynomial of A. A connection is made with realizations of the inverse of the associated polynomial matrix $D(x) = \sum {D_i } x^i $. The more general equation $\sum {A_i } XD_i \doteq C$ is discussed under the assumption that the matrices $A_i $ are pairwise commutative.