Modules For Algebras Research Articles

Hamiltonian extended affine Lie algebra and its representation theory

We introduce a class of extended affine Lie algebras called Hamiltonian Extended Affine Lie Algebras (HEALAs). They are so called because the corresponding derivation algebra is the classical Hamiltonian algebra. We classify the irreducible integrable modules for HEALA based on the classification of irreducible Jet modules for the Hamiltonian algebra (both zero level and non-zero level).

Banach convolution modules of group algebras on covariant functions of characters of normal subgroups

This paper investigates structure of Banach convolution modules induced by group algebras on covariant functions of characters of closed and normal subgroups. Suppose that G is a locally compact group with the group algebra L1(G). Let N be a closed and normal subgroup of G. Assume that T is the circle group and ξ:N→T is a character. Further, suppose that 1≤p<∞ and Lξp(G,N) is the Lp-space of all covariant functions of ξ on G. It is shown that Lξp(G,N) is a Banach L1(G)-module. We then study convolution module actions of group algebras on covariant functions of characters for the case of canonical normal subgroups in semi-direct product groups.

On the primality of totally ordered q-factorization graphs

AbstractWe introduce the combinatorial notion of a q-factorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with three decorations: a coloring and a weight map on vertices, and an exponent map on arrows (the exponent map can be seen as a weight map on arrows). Such graphs do not contain oriented cycles and, hence, the set of arrows induces a partial order on the set of vertices. In this first paper on the topic, beside setting the theoretical base of the concept, we establish several criteria for deciding whether or not a tensor product of two simple modules is a highest- $\ell $ -weight module and use such criteria to prove, for type A, that a simple module whose q-factorization graph has a totally ordered vertex set is prime.

Scalar extension Hopf algebroids

Given a Hopf algebra [Formula: see text], Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld [Formula: see text]-module algebra [Formula: see text] gives rise to an associative [Formula: see text]-bialgebroid structure on the smash product algebra [Formula: see text]. They also exhibited an antipode map making [Formula: see text] the total algebra of a Lu’s Hopf algebroid over [Formula: see text]. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair [Formula: see text] and [Formula: see text] of braided commutative Yetter–Drinfeld [Formula: see text]-module algebras, and output is a symmetric Hopf algebroid [Formula: see text] over [Formula: see text]. This construction does not require that the antipode of [Formula: see text] is invertible.

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

Generalized imaginary Verma and Wakimoto modules

We develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach unifies and generalizes all previously known results with imposed restrictions on inducing modules. We also define generalized imaginary Wakimoto modules which provide an explicit realization for generic generalized imaginary Verma modules.

Some Results on Submodules Using (μ,ν,ω)-Single-Valued Neutrosophic Environment

The use of a single-valued neutrosophic set (svns) makes it much easier to manage situations in which one must deal with incorrect, unexpected, susceptible, faulty, vulnerable, and complicated information. This is a result of the fact that the specific forms of material being discussed here are more likely to include errors. This new theory has directly contributed to the expansion of both the concept of fuzzy sets and intuitionistic fuzzy sets, both of which have experienced additional development as a direct consequence of the creation of this new theory. In svns, indeterminacy is correctly assessed in a way that is both subtle and unambiguous. Furthermore, membership in the truth, indeterminacy, and falsity are all completely independent of one another. In the context of algebraic analysis, certain binary operations may be regarded as interacting with algebraic modules. These modules have pervasive and complicated designs. Modules may be put to use in a wide variety of different applications. Modules have applications in a diverse range of industries and market subsets due to their adaptability and versatility. Under the umbrella of the triplet (μ,ν,ω) structure, we investigate the concept of svns and establish a relationship between it and the single-valued neutrosophic module and the single-valued neutrosophic submodule, respectively. The purpose of this study is to gain an understanding of the algebraic structures of single-valued neutrosophic submodules under the triplet structure of a classical module and to improve the validity of this method by analyzing a variety of important facets. In this article, numerous symmetrical features of modules are also investigated, which demonstrates the usefulness and practicality of these qualities. The results of this research will allow for the successful completion of both of these objectives. The tactics that we have devised for use in this article are more applicable to a wide variety of situations than those that have been used in the past. Fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets are some of the tactics that fall under this category.

(Co)homology of crossed products by weak Hopf algebras

We obtain a mixed complex simpler than the canonical one that computes the cyclic type homologies of a crossed product with invertible cocycle A×ρfH, of a weak module algebra A by a weak Hopf algebra H. This complex is endowed with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in [12]. When f takes its values in a separable subalgebra of A that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.

An approach to (mu ,nu ,omega )-single-valued neutrosophic submodules

Dealing with erroneous, unexpected, susceptible, flawed, vulnerable, and intricate information is simplified with the use of a single-valued neutrosophic set (svns). This is because of the fact that these types of information are more sensitive to error. This is due to the fact that these particular kinds of information are more prone to error. The ideas of fuzzy sets and intuitionistic fuzzy sets have both undergone further development as a direct result of the development of this new theory. In svns, indeterminacy is quantified in a way that is both obvious and unambiguous, and truth membership, indeterminacy membership, and falsity membership are all completely independent of one another. In algebraic analysis, certain binary operations can be thought of as interacting with algebraic modules. These modules are intricate and ubiquitous structures. There are many different applications for modules to be used in. Modules find use in an extremely wide variety of different kinds of businesses and market segments. We investigate the idea of (mu ,nu ,omega )-svns and relate it to (mu ,nu ,omega )-single-valued neutrosophic module and (mu ,nu ,omega )-single-valued neutrosophic submodule, respectively. The goals of this research are to comprehend the algebraic structures of a (mu ,nu ,omega )-single-valued neutrosophic submodule of a classical module and enhance the legitimacy of this technique by discussing numerous essential aspects. Both of these goals will be accomplished through the course of this study. The strategies that we have developed in this manuscript are more generalizable than those that have been utilized in the past. These strategies include fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets.

A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

For Iwahori-spherical representations of non-Archimedean general linear groups, Chan–Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori–Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan’s formula for the Harish-Chandra–Howe character as a Grothendieck group expansion of the same module.

Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks

We give a mathematical definition of irregular conformal blocks in the genus-zero WZNW model for any simple Lie algebra, using coinvariants of modules for affine Lie algebras whose parameters match up with those of moduli spaces of irregular meromorphic connections: the open de Rham spaces. The Segal–Sugawara representation of the Virasoro algebra is used to show that the spaces of irregular conformal blocks assemble into a flat vector bundle over the space of isomonodromy times à la Klarès, and we provide a universal version of the resulting flat connection generalising the irregular KZ connection of Reshetikhin and the dynamical KZ connection of Felder–Markov–Tarasov–Varchenko.

Restricted modules for gap-p Virasoro algebra and twisted modules for certain vertex algebras

This paper studies restricted modules of gap-p Virasoro algebra gp and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted gp-modules of level ℓ_ and the category of twisted modules of vertex algebra VNp(ℓ_,0), where Np is a new Lie algebra, ℓ_:=(ℓ0,0,⋯,0)∈C[p2]+1, ℓ0∈C is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted gp-modules of level ℓ_. More explicitly, we give a uniform construction of simple restricted gp-modules as induced modules. We present several equivalent characterizations of simple restricted gp-modules, as locally nilpotent (equivalently, locally finite) modules with respect to certain positive part of gp. Moreover, simple restricted gp-modules of level ℓ_ are classified. They are either highest weight modules or simple induced modules. At the end, we exhibit several concrete examples of simple restricted gp-modules of level ℓ_ (including Whittaker modules).

Construction of color Lie algebras from homomorphisms of modules of Lie algebras

A close relation between categories of color Lie algebras and sets of pairs of homomorphisms of modules of Lie algebras is discussed, which is applicable to construct color Lie algebras from Lie algebras and their modules. It is also shown that such construction yields equivalent categories of color Lie algebras under an equivalent relation between symmetric bicharacters. As an application we construct and classify all simple color Lie algebras from the three dimensional simple complex Lie algebra and its finite-dimensional simple modules.

On the Concepts of Two- Fold Fuzzy Vector Spaces and Algebraic Modules

The concept of two-fold algebras is generated by merging some special sets such as fuzzy sets, neutrosophic sets or plithogenic sets with various algebraic structures. The main objective of this research paper is to provide a strict mathematical definition and a general study of two-fold fuzzy vector spaces defined over the real numbers and also two-fold fuzzy vector spaces defined over the complex numbers, as well as two-fold fuzzy algebraic modules defined over commutative rings with unity. The elementary properties and algebraic operations of those structures will be explained through many theorems and mathematical proofs.

Killing Operator for the Kerr Metric

When D: E → F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) → F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 → F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 ο D = 0 implies ad (D) ο ad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations

&lt;abstract&gt; &lt;p&gt;The quantale module introduced by Abramsky and Vickers, engaged a large number of researchers. This research article focuses the combined behavior of rough set, soft set and an algebraic structure quantale module with the left action. In fact, the paper reflects the generalization of rough soft sets. This combined effect is totally dependent on soft binary relation including aftersets and foresets. Different soft substructures in quantale modules are defined. The characterizations of soft substructures in quantale modules based on soft binary relation are presented. Further, in quantale modules, we define soft compatible and soft complete relations in terms of aftersets and foresets. Furthermore, we use soft compatible and soft complete relations to approximate soft substructures of quantale modules and these approximations are interpreted by aftersets and foresets. This concept generalizes the concept of rough soft quantale modules. Additionally, we describe the algebraic relationships between the upper (lower) approximations of soft substructures of quantale modules and the upper (lower) approximations of their homomorphic images using the concept of soft quantale module homomorphism.&lt;/p&gt; &lt;/abstract&gt;

On Quasi Quadratic Modules of Lie Algebras

This study introduces the category of quasi-quadratic modules of Lie algebras and discusses the functorial relations between quasi-quadratic modules and quadratic modules of Lie algebras.

Finite coproducts in the category of quadratic modules of Lie algebras

In this study, we will construct finite coproduct objects in the category of quadratic modules of Lie algebras with a new approach using the idea of quasi-quadratic modules.

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z>1, is equal to the category OM(p) of C1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since OM(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory OM(p)T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories OM(p) and OM(p)T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

Affine Lie algebra representations induced from Whittaker modules

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary affine Kac-Moody algebras. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the affine Lie algebra modules. Both functors preserve irreducibility when the central charge is nonzero.