arXiv subjects

Asghar Daneshvar

Publications and source records attributed to Asghar Daneshvar.

5 recordsLinked to original sources

arXiv2026

Quasi-integrable modules over affine Lie superalgebras (Critical level)

Representation theory of Lie (super)algebras has attracted significant research interest for many years, especially due to its applications in theoretical physics; in this regard, the representation theory of affine Lie (super)algebras is of central importance. To characterize simple modules over affine Lie (super)algebras, it is necessary to study the cases of nonzero and critical levels separately. Although a vast amount of research has been done on the representation theory of affine Lie (super)algebras $\mathcal{L}$, investigations concerning general modules at the critical level remain limited. In all existing studies, the characterization of the modules under consideration is reduced to the characterization of modules over some subalgebras of $\mathcal{L}$. Depending on the structure of the original modules, these subalgebras -- and the corresponding modules -- have different natures some of which are already known, while others need to be studied separately. In this paper, we give a complete characterization of the modules over specific subalgebras $\mathcal{G}$ of a twisted affine Lie superalgebra $\mathcal{L}$ that arise in the study of general zero level simple finite weight $\mathcal{L}$-modules. In particular, in the special case that $\\mathcal{G} = \mathcal{L}$, we obtain a complete characterization of quasi-integrable $\mathcal{L}$-modules of level zero.

math.RT↗

arXiv2024

On results of certain modules over untwisted affine Liesuperalgebras

Since 2020, finite weight modules have been studied over twisted affine Lie superalgebras. To complete the characterization of modules over affine Lie superalgebras, we need some information regarding modules over untwisted affine Lie superalgebras. There are several known results on representations of twisted affine Lie superalgebras that hold for untwisted cases, and their proofs are just a minor modification of the known ones. In this note, we gather these results for our further use.

math.RT↗

arXiv2024

Characterization and examples of commutative isoartinian rings

Noetherian rings have played a fundamental role in commutative algebra, algebraic number theory, and algebraic geometry. Along with their dual, Artinian rings, they have many generalizations, including the notions of isonoetherian and isoartinian rings. In this paper, we prove that the Krull dimension of every isoartinian ring is at most one. We then use this result to provide a characterization of isoartinian rings. Specifically, we prove that a ring $R$ is isoartinian if and only if $R$ is uniquely isomorphic to the direct product of a finite number of rings of the following types: (i) Artinian local rings; (ii) non-Noetherian isoartinian local rings with a nilpotent maximal ideal; (iii) non-field principal ideal domains; (iv) Noetherian isoartinian rings $A$ with $\Min A$ being a singleton and $\Min A \subsetneq \Ass A$; (v) non-Noetherian isoartinian rings $A$ with $\Min A$ being a singleton and $\Min A \subsetneq \Ass A$; (vi) non-Noetherian isoartinian rings $A$ with a unique element in $\Min A$ that is not maximal, and $\Min A=\Ass A$. Several examples of these types of rings are also provided.

math.AC↗

arXiv2016

Two Generalizations of the Wedderburn-Artin Theorem with Applications

We say that an $R$-module $M$ is {\it virtually simple} if $M\neq (0)$ and $N\cong M$ for every non-zero submodule $N$ of $M$, and {\it virtually semisimple} if each submodule of $M$ is isomorphic to a direct summand of $M$. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules. Our theory provides two natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring $R$: (i) Every finitely generated left (right) $R$-modules is virtually semisimple; (ii) Every finitely generated left (right) $R$-modules is a direct sum of virtually simple modules; (iii) $R\cong\prod_{i=1}^{k} M_{n_i}(D_i)$ where $k, n_1,\ldots,n_k\in \Bbb{N}$ and each $D_i$ is a principal ideal V-domain; and {\rm (iv)} Every non-zero finitely generated left $R$-module can be written uniquely (up to isomorphism and order of the factors) in the form $ Rm_1 \oplus\ldots\oplus Rm_k$ where each $Rm_i$ is either a simple $R$-module or a left virtually simple direct summand of $R$.

math.RA↗

arXiv2016

Virtually Semisimple Modules and a Generalization of the Wedderburn-Artin Theorem

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them {\it virtually semisimple modules}. A module $_RM$ is called {\it completely virtually semisimple} if each submodules of $M$ is a virtually semisimple module. A ring $R$ is then called {\it left} ({\it completely}) {\it virtually semisimple} if $_RR$ is a left (compleatly) virtually semisimple $R$-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring $R$ is left completely virtually semisimple if and only if $R \cong \prod _{i=1}^ k M_{n_i}(D_i)$ where $k, n_1, ...,n_k\in \Bbb{N}$ and each $D_i$ is a principal left ideal domain. Moreover, the integers $k,~ n_1, ...,n_k$ and the principal left ideal domains $D_1, ...,D_k$ are uniquely determined (up to isomorphism) by $R$.

math.RA↗