Modules For Algebras Research Articles

Relational representation for subordination Tarski algebras

ABSTRACT In this work, we study the relational representation of the class of Tarski algebras endowed with a subordination, called subordination Tarski algebras. These structures were introduced in a previous paper as a generalisation of subordination Boolean algebras. We define the subordination Tarski spaces as topological spaces with a fixed basis endowed with a closed relation. We prove that there exist categorical dualities between categories whose objects are subordination Tarski algebras and categories whose objects are subordination Tarski spaces. These results extend the known dualities for modal algebras. Finally, we are going to characterise some algebraic conditions written in the quasi-modal language by means of first-order conditions.

Simple Heisenberg-Virasoro modules from Weyl algebra modules

In this paper, we constructed and studied two classes of modules over the Heisenberg-Virasoro algebra HVir from Weyl algebra modules, parallel to the similar construction for Virasoro algebra in [13,9]. More precisely, given any module P over the degree-1 Weyl algebra, we define the HVir-modules M(P,V), where V is a restricted module over the nonnegative part a of HVir, and L(P,W,ξ,b,c), where W is a restricted module over another subalgebra b⊆HVir and ξ,b,c∈C with ξ≠0. We determined the irreducibility of these modules as well as the isomorphisms within each class respectively. Moreover, any two such modules from different classes are never isomorphic although they might be closely related, as we showed in the last section.

Resolutions in Local Algebra and Singularity Theory

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

Projective relative unification through duality

AbstractUnification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between Heyting or modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for formulas to be projective. A close inspection of this characterization will motivate a generalization of standard unification, which we dub relative unification. Applying this result to a number of different logics, we then obtain new proofs of their projective—or non-projective—character. Aside from reproving known results, we show that the projective extensions of $\textbf{K5}$ are exactly the extensions of $\textbf{K45}$. This resolves the open question of whether $\textbf{K5}$ is projective.

ALGEBRAIC SEMANTICS FOR RELATIVE TRUTH, AWARENESS, AND POSSIBILITY

AbstractThis paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth is relative. In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics.

Distributing Persistent Homology via Spectral Sequences

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation boxplus that leads to a method for obtaining images, kernels and cokernels of tame persistence morphisms. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self-contained presentation of the Mayer–Vietoris spectral sequence. Then we study the Persistent Mayer–Vietoris spectral sequence and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PerMaViss, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method.

Rational Cherednik algebras of G(ℓ,p,n) from the Coulomb perspective

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n). In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg, and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category O and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the first classification of simple Dunkl-Opdam modules for the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n).

Intuitionistic Modal Algebras

Recent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus. Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.

On classifying simplicial modules

In this paper, we prove that the classifying functors W ¯ and Diag ° N are simplicially homotopy equivalent. Using this equivalency, we define classifying simplicial modules and give an alternative definition of (co)homology of crossed modules of algebras.

Irreducible modules for super-Virasoro algebras from algebraic D-modules

In this paper, we introduce a new family of functors from the category of modules for the Weyl algebra to the category of modules for the super-Virasoro algebras. The properties of these functors are investigated, with an emphasis on irreducibility preservation and natural isomorphisms. By utilizing these functors, we recover some old irreducible super-Virasoro modules, including those from the irreducible intermediate series as well as irreducible U(h)-free modules. Additionally, we provide several families of new irreducible super-Virasoro modules via our constructed functors.

Tensor Products and Crossed Differential Graded Lie Algebras in the Category of Crossed Complexes

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories.

New simple [formula omitted]-modules from Weyl algebra modules

We construct a class of modules over the affine Lie algebra slˆ2 from the modules over the degree-2 Weyl algebra K2 and modules over the 2-dimensional solvable Lie algebra b. We determine the irreducibility and isomorphisms of these modules with examples given by Bavula's construction of modules for generalized Weyl algebras in [5]. Finally, we show these irreducible slˆ2-modules are generally new.

Crossed modules and non-abelian extensions of Rota-Baxter Leibniz algebras

We introduce the notions of crossed modules of Rota-Baxter Leibniz algebras, 2-term Rota-Baxter sh Leibniz algebras, and discuss the relationship between them and the 3-th cohomology group of Rota-Baxter Leibniz algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. Based on this classification, we study the inducibility of a pair of automorphisms about a non-abelian extension of Rota-Baxter Leibniz algebras, and give the fundamental sequence of Wells in the context of Rota-Baxter Leibniz algebras.

THE IMPLEMENTATION OF A ROUGH SET OF PROJECTIVE MODULE

In ring and module theory, one concept is the projective module. A module is said to be projective if it is a direct sum of independent modules. (U, R) is an approximation space with non-empty set and equivalence relation If X subset U, we can form upper approximation and lower approximation. X is rough set if upper Apr(X) is not equal to under Apr(X). The rough set theory applies to algebraic structures, including groups, rings, modules, and module homomorphisms. In this study, we will investigate the properties of the rough projective module.

Components of AR-quivers for string algebras of type [formula omitted] and a conjecture by Geiss-Leclerc-Schröer

We study modules of certain string algebras, which are referred to as of affine type C˜. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and τ-locally free modules defined by Geiss-Lerclerc-Schröer. In particular, we show that an indecomposable module is τ-locally free if and only if it is preprojective, or preinjective or regular in a tube. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type C˜ and τ-locally free modules of the corresponding string algebras. Furthermore, given a positive root α, we show that if α is real, then there is a unique τ-locally free module M (up to isomorphism) with rank_M=α; otherwise there are families of τ-locally free modules with rank_M=α.

Ideal and factor conditions for crossed modules and algebra-algebroids

In this paper, using the equivalence between the category of crossed modules of algebras and the category of algebra-algebroids, we will explore the notions of ideality and factors for these algebraic structures. We give the structure of a two sided ideal of an algebra-algebroid and the notion of quotient algebra-algebroid. By considering a two sided ideal of an algebra-algebroid, we show that the crossed module corresponding to this ideal is a crossed ideal of the crossed module corresponding to the algebra-algebroid. Conversely, by taking a crossed ideal of a crossed module, we also show that the corresponding algebra-algebroid to this crossed ideal is an ideal of the algebra-algebroid corresponding to the crossed module.

The Grothendieck ring of Yetter-Drinfeld modules over a class of 2n 2-dimensional Kac-Paljutkin Hopf algebras

As is well known, a class of -dimensional Kac-Paljutkin Hopf algebras was introduced by Pansera. It is the generalization of the 8-dimensional Kac-Paljutkin Hopf algebra H 8. In this paper, the classification of Yetter-Drinfeld modules over is given by Radford’s method of constructing Yetter-Drinfeld modules for a Hopf algebra. Furthermore, the tensor product of Yetter-Drinfeld modules over is established, and the Grothendieck ring is described explicitly by generators and relations. Communicated by Julia Plavnik

Classification of simple locally finite modules over the affine-Virasoro algebras

In [12], the authors classified a class of simple modules over untwised affine Kac-Moody Lie algebras, on these modules each weight vector of the positive parts of affine Kac-Moody Lie algebras acts locally finitely. In this paper, for all affine-Virasoro algebras we also classify all simple modules with the similar property. We determine that there are precisely three classes of simple modules on which each weight vector of the positive part of any affine-Virasoro algebra acts locally finitely: simple highest weight or Whittaker Virasoro algebra modules, simple highest weight or Whittaker affine Lie algebra modules, and simple highest weight or Whittaker affine-Virasoro algebra modules which are neither simple Virasoro algebra modules nor simple affine Lie algebra modules. We also obtain three equivalent conditions to characterize such simple modules over affine-Virasoro algebras.

On Lie algebra modules which are modules over semisimple group schemes

Let p be a prime. Given a split semisimple group scheme G over a normal integral domain R which is a faithfully flat $${\mathbb {Z}}_{(p)}$$ -algebra, we classify all finite dimensional representations V of the fiber $$G_K$$ of G over $$K:=Frac (R)$$ with the property that the set of lattices of V with respect to R which are G-modules is as well the set of lattices of V with respect to R which are $$Lie (G)$$ -modules. We apply this classification to get a general criterion of extensions of homomorphisms between reductive group schemes over $$Spec \,K$$ to homomorphisms between reductive group schemes over $$Spec \,R$$ . We also show that for a simply connected semisimple group scheme over a reduced $${\mathbb {Q}}$$ –algebra, the category of its representations is equivalent to the category of representations of its Lie algebra.

The Derived Pure Spinor Formalism as an Equivalence of Categories

We construct a derived generalization of the pure spinor superfield formalism and prove that it exhibits an equivalence of dg-categories between multiplets for a supertranslation algebra and equivariant modules over its Chevalley-Eilenberg cochains. This equivalence is closely linked to Koszul duality for the supertranslation algebra. After introducing and describing the category of supermultiplets, we define the derived pure spinor construction explicitly as a dg-functor. We then show that the functor that takes the derived supertranslation invariants of any supermultiplet is a quasi-inverse to the pure spinor construction, using an explicit calculation. Finally, we illustrate our findings with examples and use insights from the derived formalism to answer some questions regarding the ordinary (underived) pure spinor superfield formalism.