## The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.

Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

## We also investigate the tensor products of the C⁢[K±1]\mathbb{C}[K^{\pm 1}]-free modules with finite-dimensional simple modules over Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}), and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}).

A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

## Specifically, in a particular class of these rings, which have only two non-isomorphic simple modules, by means of the strong preinjective partition of R-ind, we enumerate all idempotent preradicals, all radicals and all torsion theories, and we also give a description of the structure of all preradicals.

Preradicals over left pure semisimple hereditary rings

## Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules.

Cyclotomic and Littlewood Polynomials Associated to Algebras

## We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy.

Classification of some vertex operator algebras of rank 3

## The approach is based on a combination of three simple modules: 1) flood frequency analysis (frequency and peak discharge), 2) estimation of inundation depth, and 3) damage and loss estimation.

Flood risk mapping for direct damage to residential buildings in Quebec, Canada

## For those deformations that satisfy a certain non-degeneracy condition, we describe the structure of certain simple modules of the deformations of the subcharacter algebra of a finite group.

Semisimplicity of the Deformations of the Subcharacter Algebra of an Abelian Group

## We investigate integral forms of certain simple modules over group algebras in characteristic 0 whose $p$-modular reductions have precisely three composition factors.

ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

## Based on the extracted features by the feature extraction module (FEM) and the central information of ships, AR²Det adopts two simple modules, ship detector (SDet) and center detector (CDet), to generate and improve the detection results, respectively.

AR²Det: An Accurate and Real-Time Rotational One-Stage Ship Detector in Remote Sensing Images

## In this paper, we propose two simple modules, group communication and context codec, that can be easily applied to a wide range of architectures to jointly decrease the model size and complexity without sacrificing the performance.

Group Communication With Context Codec for Lightweight Source Separation

10.1142/9789811230295_0006

## The Jordan-Hölder theorem for modules says that the ways in which a module can be built up from simple modules are essentially unique.

The Jordan-Hölder property, Grothendieck monoids and Bruhat inversions

10.1016/J.JALGEBRA.2021.06.007

## In the end, we show that objects in the category R are restricted V q ˜ -modules, and we classify simple modules in the category R.

Algebra of q-difference operators, affine vertex algebras, and their modules

10.22034/KJM.2021.203331.1578

## The primeness of simple modules over Leavitt path algebras is also discussed.

Primeness of Simple Modules over Path Algebras and Leavitt Path Algebras

10.1016/j.aim.2021.108052

## As main theorem, we prove that these bounds agree whenever the first non-trivial grade conditions are satified for simple modules, so that either invariant computes the finitistic dimension in this case.

The depth, the delooping level and the finitistic dimension

10.1142/s0219498822502292

## The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.

Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

10.1007/S00200-021-00487-7

## We derive consequences for exact structures generated by the simple modules and the modules with zero Jacobson radical.

Relatively divisible and relatively flat objects in exact categories: applications

10.1016/J.JALGEBRA.2021.01.025

## We use simple modules over the finite-dimensional solvable Lie algebras to construct many simple restricted modules over the Heisenberg-Virasoro algebra L.

Simple restricted modules for the Heisenberg-Virasoro algebra

10.1142/s0218196721500168

## Let A′ be the Auslander algebra of a finite dimensional basic connected Nakayama algebra A with radical cube zero and n simple modules.

Tilting modules over Auslander algebras of Nakayama algebras with radical cube zero

10.1515/forum-2020-0345

## We also investigate the tensor products of the C⁢[K±1]\mathbb{C}[K^{\pm 1}]-free modules with finite-dimensional simple modules over Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}), and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}).

A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

10.1109/tgrs.2021.3092433

## Based on the extracted features by the feature extraction module (FEM) and the central information of ships, AR²Det adopts two simple modules, ship detector (SDet) and center detector (CDet), to generate and improve the detection results, respectively.

AR²Det: An Accurate and Real-Time Rotational One-Stage Ship Detector in Remote Sensing Images

10.1080/00927872.2021.1888963

## Specifically, in a particular class of these rings, which have only two non-isomorphic simple modules, by means of the strong preinjective partition of R-ind, we enumerate all idempotent preradicals, all radicals and all torsion theories, and we also give a description of the structure of all preradicals.

Preradicals over left pure semisimple hereditary rings

## These representations are quotients of induced modules over the affine Kac–Moody algebra $\widehat{\mathfrak{s}\mathfrak{l}}_{n+1}$ and include in particular all admissible simple highest weight modules and all simple modules induced from $\mathfrak{s}\mathfrak{l}_2$.

Simple Modules for Affine Vertex Algebras in the Minimal Nilpotent Orbit

10.1142/s021949882250236x

## showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules).

Some variations of projectivity

10.1109/TASLP.2021.3078640

## In this paper, we propose two simple modules, group communication and context codec, that can be easily applied to a wide range of architectures to jointly decrease the model size and complexity without sacrificing the performance.

Group Communication With Context Codec for Lightweight Source Separation

10.29350/JOPS.2021.26.1.1210

## In this paper, we introduce and investigate the notion of projection invariant semisimple modules.

On Projection Invariant Semisimple Modules

10.2140/pjm.2021.312.421

## In the paper, we describe the Drinfel’d double structure of the n-rank Taft algebra and all of its simple modules, and then endow its R-matrices with an application to knot invariants.

Drinfeld doubles of the n-rank Taft algebras and a generalization of the Jones polynomial

10.1142/S0219498822502048

## Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl.

Structure and isomorphisms of quantum generalized Heisenberg algebras

10.1007/s10468-021-10080-8

## In the double quantum affine case, we show that simple weight-finite modules are classified by their ($t$-dominant) highest $t$-weight spaces, a family of simple modules over the subalgebra $\ddot{\mathrm{U}}_q^0(\mathfrak a_1)$ of $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ which is conjecturally isomorphic to a split extension of the elliptic Hall algebra.

Weight-Finite Modules Over the Quantum Affine and Double Quantum Affine Algebras of Type $\mathfrak a_{1}$

10.1007/s10468-021-10100-7

## For those deformations that satisfy a certain non-degeneracy condition, we describe the structure of certain simple modules of the deformations of the subcharacter algebra of a finite group.

Semisimplicity of the Deformations of the Subcharacter Algebra of an Abelian Group

10.1007/s10468-021-10098-y

## As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

Torsion Pairs and Ringel Duality for Schur Algebras

## We also give a classification of simple modules that decompose into a direct sum of simple finite-dimensional sl2-modules with finite multiplicities.

On representations of the centrally extended Heisenberg double of SL2

10.1007/978-3-030-21792-1_5

## We start by describing the Brauer tree, a combinatorial object that encodes first the decomposition matrix of the block, then Ext1 between simple modules in the block, and indeed the Morita equivalence type of the block (but not the source algebra).

Blocks with Cyclic Defect Groups

## We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $\mathbb{A}$ and $\mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young.

A geometric $q$-character formula for snake modules.

10.1142/s0219498821400016

## Then, we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case $n=2$.

Finite-dimensional Nichols algebras over dual Radford algebras

10.1142/s0129167x19500708

## We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into simple modules over the three-point affine algebras of genus zero.

Representations for three-point Lie algebras of genus zero

10.1109/ACCESS.2019.2934718

## Three basic simple modules allows to achieve complex models for a real mine.

On Modelling and Simulating Open Pit Mine Through Stochastic Timed Petri Nets

10.1007/978-3-030-32906-8_3

## In particular, we show the Heisenberg vertex operator algebra gives an example of when the level one Zhu algebra, and in fact all its higher level Zhu algebras, do not provide new indecomposable non simple modules for the vertex operator algebra beyond those detected by the level zero Zhu algebra.

The Level One Zhu Algebra for the Heisenberg Vertex Operator Algebra

10.1007/978-3-030-21792-1_9

## The first section deals with defining characteristic representations, introducing highest weight modules, Weyl modules, and building up to the Lusztig conjecture, with a diversion into Ext1 between simple modules for the algebraic group and the finite group.

Representations of Groups of Lie Type

10.15446/recolma.v53nsupl.84006

## EnglishThe paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras.

Graded modules over simple Lie algebras

10.1016/j.jalgebra.2020.02.036

## We give new improved bounds for the dominant dimension of Nakayama algebras and use those bounds to give a classification of Nakayama algebras with $n$ simple modules that are higher Auslander algebras with global dimension at least $n$.

On the classification of higher Auslander algebras for Nakayama algebras

10.4310/MRL.2019.V26.N2.A5

## We study the problem of indecomposability of translations of simple modules in the principal block of BGG category O for sl(n), as conjectured in [KiM1].

Indecomposable manipulations with simple modules in category $\mathcal{O}$

10.1016/j.optcom.2018.09.008

## The notion of (semi)bricks, regarded as a generalization of (semi)simple modules, appeared in a paper of Ringel in 1976.

Gluing support {\tau}-tilting modules over {\tau}-tilting finite algebras

10.1016/j.aim.2020.107172

## Thus, once a formula for the characters of the indecomposable tilting $G$-modules has been found, a formula for the simple modules has been also.

On character formulas for simple and tilting modules.

10.5772/INTECHOPEN.82309

## Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules.

Cyclotomic and Littlewood Polynomials Associated to Algebras

10.1515/jgth-2019-0130

## This generalises a result by Navarro for simple modules over finite p-solvable groups, which is the main motivation for this note.

A note on vertices of indecomposable tensor products

## Furthermore, using these concepts, we characterize some classical modules such as simple modules, S -Noetherian modules, and torsion-free modules.

On S-prime submodules

10.20527/EPSILON.V12I2.314

## Keywords : Leavitt path algebra, Graph 𝐴𝐴~, Chen simple modules, Prime modules.

SIFAT KEPRIMAAN MODUL SEDERHANA CHEN UNTUK GRAF

10.2140/ant.2020.14.1613

## We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy.

Classification of some vertex operator algebras of rank 3

## The simple objects of these categories are tensor modules as in the previously studied category, however, the choice of k provides more flexibility of nonsimple modules.

Integrable sl ( ∞ ) -modules and category O for gl ( m | n )

10.2140/involve.2019.12.1369

## A complete description of simple modules over $R$ is obtained by using the results of Irving and Gerritzen.

Nonsplit module extensions over the one-sided inverse of k[x]

10.1109/ICRA.2019.8793663

## Previous work on dynamic robot morphology has focused on simulation, combining simple modules, or switching between locomotion modes.

Self-Modifying Morphology Experiments with DyRET: Dynamic Robot for Embodied Testing

10.1016/J.IJDRR.2018.09.007

## The approach is based on a combination of three simple modules: 1) flood frequency analysis (frequency and peak discharge), 2) estimation of inundation depth, and 3) damage and loss estimation.

Flood risk mapping for direct damage to residential buildings in Quebec, Canada

10.1080/00927872.2019.1570233

## We describe the formal characters of some Weyl modules for simply connected and semisimple algebraic groups of type Dl over an algebraically closed field of characteristic where h is the Coxeter number, in the terms of the formal characters of simple modules.

Submodule structure of some Weyl modules and the cohomology for

10.1080/00927872.2018.1477951

## In previous work by the author, a class of finite-dimensional semisimple Hopf algebras was considered with respect to the question under what condition all but one isomorphism class of simple modules are one-dimensional.

On matched pairs defining Hopf algebras in a certain class

10.1007/s00209-020-02496-7

## We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux.

Quantum affine algebras and Grassmannians.

## We investigate integral forms of certain simple modules over group algebras in characteristic 0 whose $p$-modular reductions have precisely three composition factors.

ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS