Extension Of Algebra 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.

T-rough symmetric Heyting algebras with tense operators

T-rough Heyting algebras were introduced by Eric SanJuan in 2008 as an algebraic formalism for reasoning on finite increasing sequences over Boolean algebras in general and on generalizations of rough set concepts in particular. In this paper, we introduce the variety of algebras, which we call T-rough symmetric Heyting algebras. These algebras constitute an extension of T-rough Heyting algebras and a generalization of symmetric Heyting algebras of order n. Our main interest is the representation theory of tense operators on T-rough symmetric Heyting algebras. In order to do this, a discrete-style duality for these algebras is developed.

Amplitudes, Hopf algebras and the colour-kinematics duality

It was recently proposed that the kinematic algebra featuring in the colour-kinematics duality for scattering amplitudes in heavy-mass effective field theory (HEFT) and Yang-Mills theory is a quasi-shuffle Hopf algebra. The associated fusion product determines the structure of the Bern-Carrasco-Johansson (BCJ) numerators, which are manifestly gauge invariant and with poles corresponding to heavy-particle exchange. In this work we explore the deep connections between the quasi-shuffle algebra and general physical properties of the scattering amplitudes. First, after proving the double-copy form for gravitational HEFT amplitudes, we show that the coproducts of the kinematic algebra are in correspondence with factorisations of BCJ numerators on massive poles. We then study an extension of the standard quasi-shuffle Hopf algebra to a non-abelian version describing BCJ numerators with all possible gluon orderings. This is achieved by tensoring the original algebra with a particular Hopf algebra of orderings. In this extended version, a specific choice of the coproduct in the algebra of orderings leads to an antipode in the resulting Hopf algebra that has the interpretation of reversing the gluons’ order within each BCJ numerator.

Extensions and tangent prolongations of differentiable loops

The tangent Akivis algebra and Sabinin algebra of degree 3 of a differentiable loop is the tangential object determined by the third-order Taylor polynomial of the multiplication function of the loop. It is endowed with a bilinear skew-symmetric and a trilinear operation defined by the infinitesimal commutator and associator of the loop. The aim of our work is to study tangent algebras of degree 3 of abelian extensions of differentiable loops, which are affine extensions of the tangent algebras of the loop by abelian algebras. This class of loop extensions has previously been studied in terms of computational complexity and in terms of universal algebra. We apply the obtained results to the determination of tangent algebras of degree 3 of tangent prolongation of differentiable loops.

A NOTE ON THE SOLUTION OF THE CHARACTERISTIC EQUATION OVER THE SYMMETRIZED MAX-PLUS ALGEBRA

The symmetrized max-plus algebra is an extension of max-plus algebra. One of the problems in the symmetrized max-plus algebra is determining the eigenvalues of a matrix. If the determinant can be defined, the characteristic equation can be formulated as a max-plus algebraic multivariate polynomial equation system. A mathematical tool for solving the problem using operations as in conventional algebra, known as the extended linear complementary problem (ELCP), to determine the solution to the characteristic equation. In this paper, we will describe the use of ELCP in determining the solution to the characteristic equations of matrices over the symmetrized max-plus algebra.

Automorphism Groups and Uniqueness of Holomorphic Vertex Operator Algebras of Central Charge 24

We describe the automorphism groups of all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras by using their constructions as simple current extensions. We also confirm a conjecture of G. H\"ohn on the numbers of holomorphic vertex operator algebras of central charge $24$ obtained as inequivalent simple current extensions of certain vertex operator algebras, which gives another proof of the uniqueness of holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras.

On the implicit constant fields and key polynomials for valuation algebraic extensions

This article is a natural continuation of our previous works [Dutta 2021] and [Dutta 2022]. In this article, we employ similar ideas as in [Alexandru, Popescu and Zaharescu 1990] to provide an estimate of IC(K(X)∣K,v) when (K(X)∣K,v) is a valuation algebraic extension. Our central result is an analogue of [Dutta 2022, Theorem 1.3]. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.

Existence and Uniqueness of Algebraic Closures

Summary This is the second part of a two-part article formalizing existence and uniqueness of algebraic closures, using the Mizar [2], [1] formalism. Our proof follows Artin’s classical one as presented by Lang in [3]. In the first part we proved that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin’s proof applies Kronecker’s construction to each polynomial p ∈ F [X]\F simultaneously. To do so we needed the polynomial ring F [X 1, X 2, ...] with infinitely many variables, one for each polynomal p ∈ F [X]\F. The desired field extension E then is F [X 1, X 2, …]\I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn’s lemma has to be applied. In this second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F. To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F. In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn’s lemma.

Provenance Framework for Multi-Depth Querying Using Zero-Information Loss Database

Data provenance is a kind of metadata that describes the origin and derivation history of data. It provides the information about various direct and indirect sources of data and different transformations applied on it. Provenance information are beneficial in determining the quality, truthfulness, and authenticity of data. It also explains how, when, why, and by whom this data are created. In a relational database, fine-grained provenance captured at different stages (i.e., multi-layer provenance) is more significant and explanatory as it provides various remarkable information such as immediate and intermediate sources and origin of data. In this paper, we propose a novel multi-layer data provenance framework for Zero-Information Loss Relational Database (ZILRDB). The proposed framework is implemented on top of the relational database using the object relational database concepts to maintain all insert, delete, and update operations efficiently. It has the capability to capture multi-layer provenance for different query sets including historical queries. We also propose Provenance Relational Algebra (PRA) as an extension of traditional relational algebra to capture the provenance for ASPJU (Aggregate, Select, Project, Join, Union) queries in relational database. The framework provides a detailed provenance analysis through multi-depth provenance querying. We store the provenance data in both relational and graph database, and further evaluate the performance of the framework in terms of provenance storage overhead and average execution time for provenance querying. We observe that the graph database offers significant performance gains over relational database for executing multi-depth queries on provenance. We present two use case studies to explain the usefulness of proposed framework in various data-driven systems to increase the understandability of system’s behavior and functionalities.

Non-generators in extensions of infinitary algebras

Contrary to the finitary case, the set Γ(A) of all the non-generators of an infinitary algebra A is not necessarily a subalgebra of A. We show that the phenomenon is ubiquitous: every algebra with at least one infinitary operation can be embedded into some algebra B such that Γ(B) is not a subalgebra of B. As far as expansions are concerned, there are examples of infinite algebras A such that in every expansion B of A the set Γ(B) is a subalgebra of B. However, under relatively weak assumptions on A, it is possible to get some expansion B of A such that Γ(B) fails to be a subalgebra of B.

Notes on the decidability of addition and the Frobenius map for polynomials and rational functions

Let pbe a prime number, Fp a finite field with pelements, Fan algebraic extension of Fp and z a variable. We consider the structure of addition and the Frobenius map (i.e., x →xp) in the polynomial rings F[z] and in fields F(z) of rational functions. We prove that any question about F[z] in the structure of addition and Frobenius map may be effectively reduced to questions about the similar structure of the field F. Furthermore, we provide an example which shows that a fact which is true for addition and the Frobenius map in the polynomial rings F[z] fails to be true in F(z). As a consequence, certain methods used to prove model completeness for polynomials do not suffice to prove model completeness for similar structures for fields of rational functions F(z), a problem that remains open even for F= Fp.

Asymptotic structure of Carrollian limits of Einstein-Yang-Mills theory in four spacetime dimensions

In this paper, three things are done. First, we study from an algebraic point of view the infinite-dimensional Bondi-Metzner-Sachs (BMS)-like extensions of the Carroll algebra relevant to the asymptotic structure of the electric and magnetic Carrollian limits of Einstein gravity. In the course of this study we exhibit by the ``Carroll-Galileo duality'' a new infinite-dimensional BMS-like extension of the Galilean algebra and of its centrally extended Bargmann algebra. Second, we consider the electric Carrollian limit of the pure Einstein theory and indicate that more flexible boundary conditions than the ones that follow from just taking the limit of the Einsteinian boundary conditions are actually consistent. These boundary conditions lead to a bigger asymptotic symmetry algebra that involves spatial supertranslations depending on three functions of the angles (instead of one). Third, we turn to the Carrollian limit of the coupled Einstein-Yang-Mills system. An infinite-dimensional color enhancement of the gauge algebra is found in the electric Carrollian limit of the Yang-Mills field, which allows angle-dependent Yang-Mills transformations at spatial infinity, not available in the Einstein-Yang-Mills case prior to taking the Carrollian electric limit. This enhancement does not occur in the magnetic limit.

Central extensions of 2-filiform Leibniz algebras

In this paper we describe the central extensions of some nilpotent Leibniz algebras and one-dimensional central extensions (up to isomorphism) of all naturally graded 2-filiform Leibniz algebras over the field of complex numbers. It is well-know that 2-filiform Leibniz algebras which are not naturally graded have not yet studied. There are such kind of algebras in the algebras obtained along this article. Also in this work we describe some subclasses of 6-dimensional nilpotent Leibniz algebras with condition quotient algebra by two-dimensional center of the algebra is isomorphic to the four-dimensional naturally graded 2-filiform nonsplit Leibniz algebra.

Non-relativistic and ultra-relativistic expansions of three-dimensional spin-3 gravity theories

In this paper, we present novel and known non-relativistic and ultra-relativistic spin-3 algebras, by considering the Lie algebra expansion method. We start by applying the expansion procedure using different semigroups to the spin-3 extension of the AdS algebra, leading to spin-3 extensions of known non-relativistic and ultra-relativistic algebras. We then generalize the procedure considering an infinite-dimensional semigroup, which allows to obtain a spin-3 extension of two new infinite families of the Newton-Hooke type and AdS Carroll type. We also present the construction of the gravity theories based on the aforementioned algebras. In particular, the expansion method based on semigroups also allows to derive the (non-degenerate) invariant bilinear forms, ensuring the proper construction of the Chern-Simons gravity actions. Interestingly, in the vanishing cosmological constant limit we recover the spin-3 extensions of the infinite-dimensional Galilean and infinite-dimensional Carroll gravity theories.

Non-abelian extensions of pre-Lie algebras

In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra.

Deformations, cohomologies and integrations of relative difference Lie algebras

In this paper, first using the higher derived brackets, we give the controlling algebra of relative difference Lie algebras, which are also called crossed homomorphisms or differential Lie algebras of weight 1 when the action is the adjoint action. Then using Getzler's twisted L∞-algebra, we define the cohomology of relative difference Lie algebras. In particular, we define the regular cohomology of difference Lie algebras by which infinitesimal deformations of difference Lie algebras are classified. We also define the cohomology of difference Lie algebras with coefficients in arbitrary representations, and using the second cohomology group to classify abelian extensions of difference Lie algebras. Finally, we show that any relative difference Lie algebra can be integrated to a relative difference Lie group globally in a functorial way.

Éz fields

Let K be a field. The étale open topology on the K-points V(K) of a K-variety V was introduced in [23]. The étale open topology is non-discrete if and only if K is large. If K is separably, real, p-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large fields enjoy some of the well-known topological properties of definable sets in algebraically, real, and p-adically closed fields. We introduce and study the class of éz fields: K is éz if K is large and every definable set is a finite union of étale open subsets of Zariski closed sets. This should be seen as a generalized notion of model completeness for large fields. Algebraically closed, real closed, p-adically closed, and bounded PAC fields are éz. (In particular pseudofinite fields and infinite algebraic extensions of finite fields are éz.) We develop the basics of a theory of definable sets in éz fields. This gives a uniform approach to the theory of definable sets across all characteristic zero local fields and a new topological theory of definable sets in bounded PAC fields. We also show that some prominent examples of possibly non-model complete model-theoretically tame fields (characteristic zero t-Henselian fields and Frobenius fields) are éz.

Points on x4 + y4 + z4 = 0 over algebraic extensions of ℚ(i)

It is shown that the twisted Fermat quartic [Formula: see text] has no points over any odd degree extensions of [Formula: see text]. All points over quadratic extensions of [Formula: see text] are found.

Rigidity dimensions of Hochschild extensions of hereditary algebras of type D

Rigidity dimension of algebras is a new homological dimension which measures the quality of resolutions of algebras by algebras of finite global dimension and large dominant dimension. In this paper, we calculate the rigidity dimension of the Hochschild extension of a hereditary algebra H of Dynkin type D n ( n ≥ 4 ) by the standard duality H -bimodule. It turns out that this dimension is independent of n and the orientation of D n .

Artin’s Theorem Towards the Existence of Algebraic Closures

Summary This is the first part of a two-part article formalizing existence and uniqueness of algebraic closures using the Mizar system [1], [2]. Our proof follows Artin’s classical one as presented by Lang in [3]. In this first part we prove that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin’s proof applies Kronecker’s construction to each polynomial p ∈ F [X]\F simultaneously. To do so we need the polynomial ring F [X 1, X 2, ...] with infinitely many variables, one for each polynomal p ∈ F [X]\F . The desired field extension E then is F [X 1, X 2, ...]\I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn’s lemma has to be applied. In the second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F . To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F . In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn’s lemma.