Modules For Algebras Research Articles

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.

Weyl Modules for Toroidal Lie Algebras

In this paper, we study Weyl modules for a toroidal Lie algebra \(\mathcal {T}\) with arbitrary n variables. Using the work of Rao (Pac. J. Math. 171(2), 511–528 1995), we prove that the level one global Weyl modules of \(\mathcal {T}\) are isomorphic to suitable submodules of a Fock space representation of \(\mathcal {T}\) up to a twist. As an application, we compute the graded character of the level one local Weyl module of \(\mathcal {T}\), thereby generalising the work of Kodera (Lett. Math. Phys. 110(11) 3053–3080 2020).

Applications of Group Theory

Groups play a fundamental role in Algebra: many algebraic structures, including rings, fields, and modules, can be seen as formed by adding new operations and axioms based on groups. Researchers often use group theory to explain many kinds of phenomena. In recent years, group theory has been introduced into crystallography to further explore the macroscopic symmetry of crystals from a mathematical point of view. In this paper, the applications of group theory in crystallography and magic cubic will be discussed. Basic definitions and models of these fields are demonstrated. A finite group is a group with a finite number of elements, which are the important contents of group theory. Besides, this paper proves that n−1 elements in a n order group can completely decide the nth element and gives a method of the nth element in a commutative group of order n. The analysis suggests that the research method of group theory has an important influence on other subjects.

Deforming vertex algebras by vertex bialgebras

This is a continuation of a previous study initiated by the third author on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right [Formula: see text]-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra [Formula: see text] and give a construction of deformations of vertex algebras with a right [Formula: see text]-comodule nonlocal vertex algebra structure and a compatible [Formula: see text]-module nonlocal vertex algebra structure. We also give a construction of [Formula: see text]-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.

On representations of Plesken Lie algebras

Cohen and Taylor [On a certain Lie algebra defined by a finite group, Am. Math. Mon. 114 (2007) 633–639] introduced certain Lie algebras constructed using finite groups called the Plesken Lie algebras. In this paper, we describe the representation of these Plesken Lie algebras, their irreducibility and the Plesken Lie algebra modules.

Deformations and cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights

A cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights is introduced. Formal deformations, abelian extensions, skeletal Rota-Baxter 3-Lie 2-algebras and crossed modules of Rota-Baxter 3-Lie algebras are interpreted by using lower degree cohomology groups.

Cyclic cohomology and multiplier Hopf algebras

In this paper, we study cyclic cohomology theory on a regular multiplier Hopf algebra. Given a modular pair consisting of a group-like element, a character and in involution for a regular multiplier Hopf algebra, we associate a cyclic module. The approach is based on the Connes-Moscovici cyclic module of Hopf algebras and we obtain the same cyclic modules when our algebra has an identity.

Faithfulness of generalised Verma modules for Iwasawa algebras

We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type A2. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.

On Quasi 2-Crossed Modules for Lie Algebras and Functorial Relations

In this paper, we have introduced the category of 2-quasi crossed modules for Lie algebras and we have constructed a pair of adjoint functors between this category and that of 2-crossed modules Lie algebras.

Algebraic tools for default modal systems

Abstract Default Logics are a family of non-monotonic formalisms having so-called defaults and extensions as their common foundation. Traditionally, default logics have been defined and dealt with via syntactic notions of consequence in propositional or first-order logic. Here, we build default logics on modal logics. First, we present these default logics syntactically. Then, we elaborate on an algebraic counterpart. More precisely, we extend the notion of a modal algebra to accommodate for defaults and extensions. Our algebraic view of default logics concludes with an algebraic completeness result and a way of comparing default logics borrowing ideas from the concept of bisimulation in modal logic. To our knowledge, this take on default logics approach is novel. Interestingly, it also lays the groundwork for studying default logics from a dynamic logic perspective.

Irreducible representations of Leavitt algebras

For a weighted graph E, we construct representation graphs F, and consequently, LK(E)-modules VF, where LK(E) is the Leavitt path algebra associated to E, with coefficients in a field K. We characterise representation graphs F such that VF are simple LK(E)-modules. We show that the category of representation graphs of E, RG(E), is a disjoint union of subcategories, each of which contains a unique universal object T which gives an indecomposable LK(E)-module VT and a unique irreducible representation graph S, which gives a simple LK(E)-module VS.Specialising to graphs with one vertex and m loops of weight n, we construct irreducible representations for the celebrated Leavitt algebras LK(n,m). On the other hand, specialising to graphs E with weight one, we recover the simple modules of Leavitt path algebras LK(E) constructed by Chen via infinite paths or sinks and give a large class of non-simple indecomposable LK(E)-modules.Our approach gives a completely new way to construct indecomposable and simple modules for Leavitt path algebras of (weighted) graphs. Besides being more visual, this approach allows for carrying calculus on these modules with ease. On the other hand, the approach also allows us, to the best of our knowledge, for the first time, to produce systematically many examples of non-simple indecomposable modules, for these algebras, including Leavitt algebras LK(n,m).

THE PROPERTIES OF ROUGH V-COEXACT SEQUENCE IN ROUGH GROUP

In ring and module theory, the concept of an exact sequence is commonly employed. The exact sequence is generalized into the U-exact sequence and the V-coexact sequence. Rough set theory has also been applied to a variety of algebraic structures, including groups, rings, modules, and others. In this study, we investigated characteristics of a rough V-coexact sequence in rough groups

Wreath Macdonald polynomials at 𝑞=𝑡 as characters of rational Cherednik algebras

Using the theory of Macdonald [Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group S n \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q = t q=t . We generalize this to restricted rational Cherednik algebras of wreath product groups C ℓ ≀ S n C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [Combinatorics, symmetric functions, and Hilbert schemes, Int. Press, Somerville, MA, 2003, pp. 39–111].

On the h-adic Quantum Vertex Algebras Associated with Hecke Symmetries

We study the quantum vertex algebraic framework for the Yangians of RTT-type and the braided Yangians associated with Hecke symmetries, introduced by Gurevich and Saponov. First, we construct several families of modules for the aforementioned Yangian-like algebras which, in the RTT-type case, lead to a certain h-adic quantum vertex algebra $${\mathcal {V}}_c (R)$$ via the Etingof–Kazhdan construction, while, in the braided case, they produce ( $$\phi $$ -coordinated) $${\mathcal {V}}_c (R)$$ -modules. Next, we show that the coefficients of suitably defined quantum determinant can be used to obtain central elements of $${\mathcal {V}}_c (R)$$ , as well as the invariants of such ( $$\phi $$ -coordinated) $${\mathcal {V}}_c (R)$$ -modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of $${\mathcal {V}}_c (R)$$ .

Irreducible modules of toroidal Lie algebras arising from ϕϵ-coordinated modules of vertex algebras

In this paper, for every ϵ∈Z, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the ϕϵ-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When ϵ=1, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.

Higher depth false modular forms

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra [Formula: see text], [Formula: see text], and from [Formula: see text]-invariants of three-manifolds associated with gauge group SU(3).

Graded Specht Modules as Bernstein–Zelevinsky Derivatives of the RSK Model

AbstractWe clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the Robinson-Schensted-Knuth (RSK) construction for quiver Hecke algebras of type $A$, which was recently imported from the setting of representations of $p$-adic groups. For that goal we develop a theory of crystal derivative operators on quiver Hecke algebra modules that categorifies the Berenstein–Zelevinsky strings framework on quantum groups and generalizes a graded variant of the classical Bernstein–Zelevinsky derivatives from the $p$-adic setting. Graded cyclotomic decomposition numbers are shown to be a special subfamily of the wider concept of RSK decomposition numbers.

Irreducible integrable modules for the full toroidal Lie algebras coordinated by rational quantum torus

Let Cq be the non-commutative Laurent polynomial ring associated with a (n+1)×(n+1) rational quantum matrix q. Let sld(Cq)⊕HC1(Cq) be the universal central extension of Lie subalgebra sld(Cq) of gld(Cq). We take the Lie algebra τ=gld(Cq)⊕HC1(Cq). Let Der(Cq) be the Lie algebra of all derivations of Cq and consider the Lie algebra τ˜=τ⋊Der(Cq), called the full toroidal Lie algebra coordinated by rational quantum tori. In this paper we get a classification of irreducible integrable modules with finite dimensional weight spaces for τ˜ with nonzero central action on the modules.

Universal central extensions of braided crossed modules of Lie algebras

In this paper, we give a natural braiding on the universal central extension of a Lie crossed module with a given braiding in the category of Lie crossed modules. We also construct the universal central extension of a braided Lie crossed module in the category of braided Lie crossed modules, showing that, when one of these constructions exists, both of them exist and coincide.