Modules For Algebras Research Articles

An extension of Jónsson‐Tarski representation and model existence in predicate non‐normal modal logics

AbstractWe give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of the extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas.

Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O(n) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators.

Gluing vertex algebras

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed account of the canonical algebra construction in the Deligne product C⊠Crev. Especially, we show that if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C)X′⊠X is a commutative algebra, where X′ is a representing object for the functor HomC(•⊗CX,1C) (assuming X′ exists) and the sum runs over all inequivalent simple objects of U. Conversely, let A=⨁i∈IUi⊠Vi be a simple commutative algebra in a Deligne product U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. We show that if the unit objects 1U and 1V form a commuting pair in A in a suitable sense, then there is a braid-reversed equivalence between (sub)categories of U and V that sends Ui to Vi⁎.These results apply when U and V are braided (vertex) tensor categories of modules for simple vertex operator algebras U and V, respectively: Given τ:Irr(U)→Obj(V) such that τ(U)=V, we glue U and V along U⊠V via τ to create A=⨁X∈Irr(U)X′⊗τ(X). Then under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A has a simple conformal vertex algebra structure that (conformally) extends U⊗V. As examples, we glue suitable Kazhdan-Lusztig categories at generic levels to construct new vertex algebras extending the tensor product of two affine vertex subalgebras, and we prove braid-reversed equivalences between certain module subcategories for affine vertex algebras and W-algebras at admissible levels.

On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.

POINTED HOPF ACTIONS ON CENTRAL SIMPLE DIVISION ALGEBRAS

We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups, generalized Taft algebras with non-nilpotent skew primitive generators, and an example of non-Cartan type.

Semi-infinite Plücker relations and arcs over toric degeneration

We study the algebra of Weyl modules in types $A$ and $C$ using the methods of arcs over toric degenerations and functional realization of dual space. We compute the generators and relations of this algebra and construct its basis.

Canonical Modules and Class Groups of Rees-Like Algebras

Rees-like algebras have played a major role in settling the Eisenbud–Goto conjecture. This paper concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.

Soft Relations Applied to the Substructures of Quantale Module and Their Approximation

This research article offers a study on a new relation of rough sets and soft sets with an algebraic structure quantale module by using soft reflexive and soft compatible relations. The lower approximation and upper approximation of subsets of quantale module are utilized by aftersets and foresets. As a sequel of this relation, different characterizations of rough soft substructures of quantale modules are obtained. To ensure the results, soft reflective and soft compatible relations are focused and these are interpreted by aftersets and foresets. Furthermore, the algebraic relations between upper (lower) approximation of substructures of quantale module and the upper (lower) approximations of their homomorphic images with the help of quantale module homomorphism are examined. In comparison with the different type of approximations in different type of algebraic structures, it is concluded that this new study is much better.

A topology on categorical algebras

On categorical algebras, we define an idempotent filter as an analogous version of Gabriel?s. We also construct topologies on 2-crossed modules of algebras and categorical algebras.

Restriction and induction functors for the cyclic group algebra modules

Restriction and induction functors for the cyclic group algebra modules

C∗‐index of observable algebra in the field algebra determined by a normal group

Let G be a finite group and H a normal subgroup. By D(H; G), we denote the crossed product of C(H) and , which is only a subalgebra of the quantum double D(G) of G. One can construct a C∗‐subalgebra of the field algebra of G‐spin models, such that is a D(H; G)‐module algebra. The concrete construction of D(H; G)‐invariant subalgebra of is given. Moreover, the C∗‐index of the conditional expectation from onto is calculated in terms of the quasi‐basis for zH.

On (ɱ,ɳ)-Strongly Fully Stably Banach Algebra Modules Related to an Ideal of Am ×ɳ

The aim of this paper is introducing the concept of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal. Some properties of (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal have been studied and another characterizations have been given. The relationship of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal that states, a B- -module Ӽ is (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal , if and only if for any two ɱ-element sub-sets and of Ӽɳ, if , for each j = 1, …, ɱ, i = 1,…, ɳ and implies Ạɳ( ) Ạɳ( have been proved..

V-universal Hopf algebras (co)acting on Ω-algebras

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of A, called [Formula: see text]-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study V-universal measuring coalgebras and V-universal comeasuring algebras between [Formula: see text]-algebras A and B, relative to a fixed subspace V of [Formula: see text]. By considering the case [Formula: see text], we derive the notion of a V-universal (co)acting bialgebra (and Hopf algebra) for a given algebra A. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the V-universal acting bi/Hopf algebra and the finite dual of the V-universal coacting bi/Hopf algebra under certain conditions on V in terms of the finite topology on [Formula: see text].

H-adic quantum vertex algebras in types B, C, D and their ϕ-coordinated modules

We introduce the h-adic quantum vertex algebras associated with the trigonometric R-matrices in types B, C and D, thus generalizing the well-known Etingof–Kazhdan construction in type A. We show that restricted modules for quantum affine algebras in types B, C and D are naturally equipped with the structure of ϕ-coordinated module for the aforementioned h-adic quantum vertex algebras.

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

Equivalences of (co)module algebra structures over Hopf algebras

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behaviour of codimensions of H-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example we prove this analogue for all unital H-module structures on the algebra $F[x]/(x^2)$ of dual numbers.

A note on modular Terwilliger algebras of association schemes

Let p denote a prime number. In this note, we focus on the modular Terwilliger algebras of association schemes defined in Hanaki (Graphs Combin, 2021, https://doi.org/10.1007/s00373-021-02363-0). We define the primary module of a modular Terwilliger algebra of an association scheme and determine all its composition factors up to isomorphism. We then characterize the \(p'\)-valenced association schemes in terms of numerous properties of their modular Terwilliger algebras. We conclude our investigation with a few corollaries and questions on the modular Terwilliger algebras of association schemes.

MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

AbstractIt is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

(G,χϕ)-equivariant ϕ-coordinated quasi modules for vertex algebras

To give a unified treatment on the association of Lie algebras and vertex algebras, we study (G,χϕ)-equivariant ϕ-coordinated quasi modules for vertex algebras, where G is a group with χϕ a linear character of G and ϕ is an associate of the one-dimensional additive formal group. The theory of (G,χϕ)-equivariant ϕ-coordinated quasi modules for nonlocal vertex algebra is established in [10]. In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their (G,χϕ)-equivariant ϕ-coordinated quasi modules. Furthermore, for any conformal algebra C, we construct a class of Lie algebras Cˆϕ[G] and prove that restricted Cˆϕ[G]-modules are exactly (G,χϕ)-equivariant ϕ-coordinated quasi modules for the universal enveloping vertex algebra of C. As an application, we determine the (G,χϕ)-equivariant ϕ-coordinated quasi modules for affine and Virasoro vertex algebras.

Imaginary Verma modules for the twisted affine Nappi-Witten Lie algebra $\widetilde{H}_{4}[\sigma

An irreducible criterion of the imaginary Verma modules for the twisted affine Nappi-Witten Lie algebra $\widetilde{H}_{4}[\sigma]$ is determined. For a (reducible) imaginary Verma module, explicit characterizations of the maximal submodule, imaginary vacuum vectors, and the composition structure are also given.