Indecomposable Modules Research Articles

Semilinear clannish algebras

AbstractWe define a class of associative algebras generalizing ‘clannish algebras’, as introduced by the second author, but also incorporating semilinear structure, like a skew polynomial ring. Clannish algebras generalize the well‐known ‘string algebras’ introduced by Butler and Ringel. Our main result is the classification of finite‐dimensional indecomposable modules for these new algebras.

The module structure of a group action on a ring

Consider a finite group G acting on a graded Noetherian k-algebra S, for some field k of characteristic p; for example S might be a polynomial ring. Regard S as a kG-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of S.

PRIME SERIAL RINGS WITH KRULL DIMENSION

A Characterization of Prime Serial rings with Krull dimention as rings with a unique maximum ideal. A description of the laltice of deals and laltice submodules of a module over ring with Krull dimension is a direct sum of indecomposable modules.

Modules over the affine-Virasoro algebra of Nappi-Witten type

In this paper, we determine a class of modules over the affine-Virasoro algebra of Nappi-Witten type, which are free of rank one when restricted to the subalgebra U(CL0⊕Cd0). We study when such modules are simple and determine the isomorphism classes. As a by-product, we obtain a class of indecomposable modules for the affine Nappi-Witten algebra. We also consider the tensor product of finitely many rank one free simple modules with an arbitrary simple restricted module. The simplicity and isomorphism classes are investigated.

It takes two spectral sequences

We study the representation theory of the Lie superalgebra gl(1|1), constructing two spectral sequences which eventually annihilate precisely the superdimension 0 indecomposable modules in the finite-dimensional category. The pages of these spectral sequences, along with their limits, define symmetric monoidal functors on Repgl(1|1). These two spectral sequences are related by contragredient duality, and from their limits we construct explicit semisimplification functors, which we explicitly prove are isomorphic up to a twist. We use these tools to prove branching results for the restriction of simple modules over Kac-Moody and queer Lie superalgebras to gl(1|1)-subalgebras.

Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages.

Tame and wild theorem for the category of modules filtered by standard modules

We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove a tame-wild dichotomy for the category F(Δ) of Δ-filtered modules for an arbitrary finite homological system (P,≤,{Δi}i∈P). This includes the case of standardly stratified algebras. Moreover, in the tame case, we show that given a fixed dimension d, for every d-dimensional indecomposable module M∈F(Δ), with the only possible exception of those lying in a finite number of isomorphism classes, the module M coincides with its Auslander-Reiten translate in F(Δ). Our proofs rely on the equivalence of F(Δ) with the module category of some special type of ditalgebra.

Skew group categories, algebras associated to Cartan matrices and folding of root lattices

For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic $p$ and the acting group is a cyclic $p$ -group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.

Reflexive modules over the endomorphism algebras of reflexive trace ideals

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring.

Annihilator Ideals of Indecomposable Modules of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. We describe all annihilator ideals of indecomposable [Formula: see text]-modules by generators. In particular, we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.

Finite Group Modular Field Extensions, Green Theory and Absolutely Indecomposable and Simple Modules

Finite Group Modular Field Extensions, Green Theory and Absolutely Indecomposable and Simple Modules

Representation rings of extensions of Hopf algebra of Kac-Paljutkin type

<p>In this paper, we focus on studying two classes of finite dimensional $ \Delta $-associative algebras, which are extensions of a family of $ 2n^2 $-dimensional Kac-Paljutkin type semi-simple Hopf algebras $ H_{2n^2} $. All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations.</p>

INDECOMPOSABLE MODULES IN THE GRASSMANNIAN CLUSTER CATEGORY CM(B5,10)

In this paper, we study indecomposable rank 2 modules in the Grassmannian cluster category CM(B5,10). This is the smallest wild case containing modules whose profile layers are 5-interlacing. We construct all rank 2 indecomposable modules with a specific natural filtration, classify them up to isomorphism, and parameterize all infinite families of non-isomorphic rank 2 modules

Denominator vectors and dimension vectors from triangulated surfaces

In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster variable in an acyclic cluster algebra coincides with the dimension vector of the corresponding module. In this paper, we give analogues of their results for cluster algebras from triangulated surfaces by comparing two kinds of intersection numbers of tagged arcs.

When does the Auslander–Reiten translation operate linearly on the Grothendieck group? – Part I

For a hereditary, finite-dimensional algebra A the Coxeter transformation extends the action of the Auslander–Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the category of finitely generated A-modules. It is natural to ask whether other algebras admit a similar linear extension. We show that this is indeed the case for all Nakayama algebras. Conversely, we show that finite-dimensional algebras with non-acyclic and connected quiver admitting such a linear extension are already cyclic Nakayama algebras.

Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work.

Tensor category [formula omitted] via minimal affine W-algebras at the non-admissible level [formula omitted

We prove that the Kazhdan-Lusztig category of slˆm at level k, KLk(slm), is a semi-simple, rigid braided tensor category for all even m≥4, and k=−m+12. Moreover, all modules in KLk(slm) are simple-currents and they appear in the decomposition of conformal embeddings glm↪slm+1 at level k=−m+12. For this we inductively identify minimal affine W-algebra Wk−1(slm+2,θ) as simple current extension of Lk(slm)⊗H⊗M, where H is the rank one Heisenberg vertex algebra, and M the singlet vertex algebra for c=−2. The proof uses previously obtained results for the tensor categories of singlet algebra. We also classify all irreducible ordinary modules for Wk−1(slm+2,θ). The semi-simple part of the category of Wk−1(slm+2,θ)–modules comes from KLk−1(slm+2), using quantum Hamiltonian reduction, but this W-algebra also contains indecomposable ordinary modules.

On some Ringel-Hall polynomials associated to tame indecomposable modules

Let k be a finite field and Q an acyclic quiver of tame type (i.e. of type A˜m,D˜m,E˜6,E˜7,E˜8). Consider the path algebra kQ and the category of finite dimensional right modules mod-kQ. Using Schofield induction combined with some reduction tools, we determine all the Ringel-Hall polynomials associated to indecomposable modules of defect ranging from −2 to 2. In this way we obtain the full list of Ringel-Hall polynomials corresponding to indecomposables in the tame cases A˜m, D˜m and a partial list in the tame cases E˜6, E˜7, E˜8.

Pure C3 modules

Within the paper, we study the class of modules in which the direct sum of two disjoint summands that is a pure submodule is also a summand. This class of modules is considered [Formula: see text] modules, a generalization of [Formula: see text] modules. In addition to pure-injective and [Formula: see text] modules that belong to the class of [Formula: see text] modules, which also include the semisimple, continuous, indecomposable, and regular modules. We gave new characterizations of many rings in respect of [Formula: see text] modules, namely semisimple rings, pure-semisimple rings, von Neumann regular rings, Noetherian rings, pure-hereditary rings, pure-[Formula: see text]-rings, etc. Moreover, we also discuss the [Formula: see text] envelope and [Formula: see text] cover of a module and introduce the pure continuous modules as the generalization of the continuous modules and also introduce the pure quasi-continuous modules as the generalization of the quasi-continuous modules.

Finite quadratic modules and lattices

ABSTRACT We give a new proof of the fact that any finite quadratic module can be decomposed into indecomposable ones. For any indecomposable finite quadratic module, we construct a lattice, and a positive definite lattice, both of which are of the least rank, whose discriminant module is the given one. The resulting lattices are given by their Gram matrices explicitly.