# Introduction to Simple Modules

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## Simple Modules sentence examples within Dimensional Simple Modules

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

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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(sl2)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(sl2)U_{q}(\mathfrak{sl}_{2}).

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## Simple Modules sentence examples within Isomorphic Simple Modules

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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.

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

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## Simple Modules sentence examples within Three Simple Modules

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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.

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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.

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## Simple Modules sentence examples within Certain Simple Modules

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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.

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

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## Simple Modules sentence examples within Two Simple Modules

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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.

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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.

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

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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.

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

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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.

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

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

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

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

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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(sl2)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(sl2)U_{q}(\mathfrak{sl}_{2}).

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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.

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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.

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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 $\mathfrak{s}\mathfrak{l}_2$.

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

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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.

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

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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.

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

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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.

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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.

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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.

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

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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).

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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.

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

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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.

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

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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.

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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.

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

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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 $n$.

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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].

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

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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.

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

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

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

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

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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.

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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.

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

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

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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.

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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.

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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.

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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.

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

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The commutative rings whose simple modules are indigent or injective are fully determined.

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Furthermore, these simple modules when restricted as modules over N = 1 superconformal algebras coincide with those modules constructed in Yang et al.

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We establish the prescription for refined characters in higher rank minimal models from the dual $(A_{n-1},A_{m-1})$ theories in the large $m$ limit, and then provide evidence for Song's proposal to hold (at least) in some simple modules (including the vacuum module) at finite $m$.

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We obtain a large family of simple modules that have a basis consisting of Gelfand–Tsetlin tableaux, the action of the Lie algebra is given by the Gelfand–Tsetlin formulas and with all Gelfand–Tsetlin multiplicities equal 1.

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As a result, simple modules for the Schrödinger algebra which are locally finite over the positive part are completely classified.

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The finite-dimensional simple modules over H and K, are classified; they all have dimension 1, respectively $$\le 2$$≤2.

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Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang.

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Lobillo, and Gabriel Navarro 131 Injective hulls of simple modules over nilpotent Lie color algebras Can Hatipoğlu 149 U -rings generated by its idempotents Yasser Ibrahim and Mohamed Yousif 157.

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We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category.

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We study the properties that these R-matrices have with respect to simple modules with the hope that this is a first step towards determining the existence of a (quantum) cluster algebra structure on a natural quotient of , the -algebra defined by Enomoto and Kashiwara, which the VV algebras categorify.

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This article is a sequel to the recent three papers on “virtually semisimple modules and rings,” by Behboodi et al.

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At the end, we study abelian endoregular modules as subdirect products of simple modules.

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The main result shows that there are five d-dimensional simple modules over Poisson algebra A for any d ≥ 1.

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System behaviors generated by these simple modules include (1) linear growth and decline, (2) exponential growth and decline, (3) logistic growth, (4) overgrowth and collapse, (5) oscillations, and (6) time lags.

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