Graded Limits of Simple Tensor Product of Kirillov–Reshetikhin Modules for

We study graded limits of simple -modules which are isomorphic to tensor products of Kirillov–Reshetikhin modules associated to a fixed fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.     

Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

On the polytope defined by Feigin, Fourier and Littelmann, associated to any highest weight corresponding to a rectangular partition, we define a crystal structure of type $A_n$. We show that this crystal is isomorphic to the one obtained from Kashiwara?s crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov–Reshetikhin crystals for the affine type $A_n^{(1)}$ via polytopes.     

A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings

The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module.     

On the (in)finiteness of the image of Reshetikhin–Turaev representations

We state a simple criterion to prove the infiniteness of the image of Reshetikhin–Turaev irreducible representations of the mapping class groups of surfaces. We use it to study some of the Reshetikhin–Turaev representations associated to the tori with one and two punctures and derive an alternative proof of the results of Funar (Pac J Math 188:251–274, 1999).     

On Fomin–Kirillov algebras for complex reflection groups

In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification.     

On the Koszul Modules of Exterior Algebras

We prove that the Koszul modules over an exterior algebra can be filtered by the cyclic Koszul modules. We also introduce the cyclic dimension vector as invariants for studying the Koszul modules over an exterior algebra.     

On the Convergence of Generalizations of the Sinc Approximations on the Privalov–Chanturia Class

Journal of Applied and Industrial Mathematics - We establish some condition in terms of the left or right modulus of continuity and the negative or positive modulus of variation of a function $$f...     

On Graded Distributive Modules

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On τ-Extending Modules

Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ-large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ-closed, τ-complement submodules and the other generalization of extending modules (τ-complemented, τ-CS, s−τ-CS modules). We are also interested in determining when a τ-divisible module is τ-extending. For a τ-extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that M = t(M) ? KM = \tau (M)\,\oplus\,K and K is τ (M)-injective. We also treat when a direct sum of τ-extending modules is τ-extending.     

On ⌖-supplemented Modules

Let R be a ring and M a right R-module. M is called -supplemented if every submodule of M has a supplement that is a direct summand of M, and M is called completely -supplemented if every direct summand of M is -supplemented. In this paper various properties of these modules are developed. It is shown that (1) Any finite direct sum of -supplemented modules is -supplemented. (2) If M is -supplemented and (D3) then M is completely -supplemented.     

On the Construction of d—Continuous Modules

In this paper we introduce: and study the notion of dual Gontinuous (d-Contmuous) modnles, A decomposrtion thcorem for a d-continuous modules proved, this generahzes all known decompositron theorems for quasi-projective modules.     

Cohen–Macaulay Loci of Modules

The Cohen–Macaulay locus of any finite module over a noetherian local ring A is studied, and it is shown that it is a Zariski-open subset of Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen–Macaulay are studied.     

On Flat and Coflat Modules

It is well known that a ring R is left hereditary iff every left ideal of R is projective, iff every submodule of a projective module is projective (ld R≤1), iff every quotient module of an injective module is injective (1cd R≤1), where 1d R and 1cd R means the left global dimension resp. codimension of the ring R. These rings may be generalized to those of weak left global dimension at most 1 (wld R≤1). The latter condition holds iff every left ideal of R is flat, iff every submodule     

Cohen–Macaulayness of Rees Algebras of Modules

Abstract

The notion of quasi-polynomials is very important in the theory of functional identities. For example, results on quasi-polynomials were tools in the solution of long-standing Herstein's Lie map conjectures. In this paper, we show that functional identities involving quasi-polynomial of degree one have only standard solutions on d-free sets.     

Generalizations of the Goulden–Jackson cluster method

We give several modifications of the Goulden–Jackson cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various ‘weights’ on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at the website for this paper.