Universal Enveloping Algebras of Lie Antialgebras

Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically related to a Lie superalgebra and prove that its enveloping algebra is a quotient of the enveloping algebra of the corresponding Lie superalgebra.     

Cohomology and deformations for the Heisenberg Hom-Lie algebras

ABSTRACT

In this work, we consider the Heisenberg Lie algebra with all its Hom-Lie structures. We completely characterize the cohomology and deformations of any order of all Heisenberg Hom-Lie algebras of dimension 3.     

Low-dimensional cohomology of <Emphasis Type="Italic">q</Emphasis>-deformed Heisenberg-Virasoro algebra of Hom-type

Hom-Lie algebras were introduced by J. Hartwig, D. Larsson, and S. Silvestrov as a generalized Lie algebra. When studying the homology and cohomology theory of Hom-Lie algebras, the authors find that the low-dimensional cohomology theory of Hom-Lie algebras is not well studied because of the Hom-Jacobi identity. In this paper, the authors compute the first and second cohomology groups of the q-deformed Heisenberg-Virasoro algebra of Hom-type, which will be useful to build the low-dimensional cohomology theory of Hom-Lie algebras.     

Central extensions and deformations of Hom-Lie triple systems

In this paper, we study the cohomology theory of Hom-Lie triple systems generalizing the Yamaguti cohomology theory of Lie triple systems. We introduce the central extension theory for Hom-Lie triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Hom-Lie triple systems and the third cohomology group. We develop the 1-parameter formal deformation theory of Hom-Lie triple systems and prove that it is governed by the cohomology group.     

Deformations and generalized derivations of Hom-Lie conformal algebras

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.     

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W（2, 2） Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.     

Track Extensions of Categories and Cohomology

We introduce the notion of n-fold track extensions of a category C by a natural system D and prove that such extensions represent classes in the cohomology of C with coefficients in D introduced by Baues–Wirsching. This generalizes a result of Huebschmann on the cohomology of groups.     

Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra

Hom-Lie algebra (superalgebra) structure appeared naturally in q-deformations, based on σ-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of α ^k -derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted osp(1, 2) superalgebra and q-deformed Witt superalgebra.     

FIXED POINT THEOREMS IN A LOCALLY CONVEX SPACE

Abstract

The notion of diminishing orbital diametral sum for a pair of commuting mappings has been introduced into a Hausdorff locally convex space whose topology is generated by a family of seminorms. Subsequently, this notion is utilized to prove common fixed point theorems which generalize certain theorems of Kirk and Tan.     

A Tiled Order of Finite Global Dimension with No Neat Primitive Idempotent

The aim of this article is to introduce the notion of Hom-Lie color algebras. This class of algebras is a natural generalization of the Hom-Lie algebras as well as a special case of the quasi-hom-Lie algebras. In the article, homomorphism relations between Hom-Lie color algebras are defined and studied. We present a way to obtain Hom-Lie color algebras from the classical Lie color algebras along with algebra endomorphisms and offer some applications. Also, we introduce a multiplier σ on the abelian group Γ and provide constructions of new Hom-Lie color algebras from old ones by the σ-twists. Finally, we explore some general classes of Hom-Lie color admissible algebras and describe all these classes via G–Hom-associative color algebras, where G is a subgroup of the symmetric group S ₃.     

Leibniz algebras constructed by Witt algebras

ABSTRACT

We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.     

Almost minimizers of the one-phase free boundary problem

Abstract

We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein, and Toro. Our proofs provide a different method based on a non-infinitesimal notion of viscosity solutions that we introduced.     

Cohomological approach to asymptotic dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim _Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension.      

Tate–Hochschild cohomology for periodic algebras

This paper is devoted to studying the Tate–Hochschild cohomology for periodic algebras. We will prove that the Tate–Hochschild cohomology ring of a periodic algebra can be written as the localization of the non-negative part of the Tate–Hochschild cohomology ring.

     

Rigidity for orientable functors

In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.     

Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results

Abstract

The idea of statistical relative convergence on modular spaces has been introduced by Orhan and Demirci. The notion of σ-statistical convergence was introduced by Mursaleen and Edely and further extended based on a fractional order difference operator by Kadak. The concern of this paper is to define two new summability methods for double sequences by combining the concepts of statistical relative convergence and σ-statistical convergence in modular spaces. Furthermore, we give some inclusion relations involving the newly proposed methods and present an illustrative example to show that our methods are nontrivial generalizations of the existing results in the literature. We also prove a Korovkin-type approximation theorem and estimate the rate of convergence by means of the modulus of continuity. Finally, using the bivariate type of Stancu-Schurer-Kantorovich operators, we display an example such that our approximation results are more powerful than the classical, statistical, and relative modular cases of Korovkin-type approximation theorems.     

Cohomology theories of Hopf bimodules and cup-product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by Gerstenhaber and Schack, and by Ospel. We prove, when all the spaces involved are finite dimensional, that they are all equal to the Ext functor on the module category of an associative algebra X associated to A, as described by Cibils and Rosso. We also give an expression for a cup-product in the cohomology defined by Ospel, and prove that it corresponds to the Yoneda product of extensions.     

UNIQUE FACTORIZATION AND BIRTH OF ALMOST PRIMES

ABSTRACT

In studying unique factorization of domains we encountered a property of ideals. Using that we define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideals are primary. We also prove that in a regular domain almost primes are precisely primes. Further, we define strictly nonprime ideals and study some inter relations between almost prime ideals, strictly nonprime ideals and factorization of ideals.     

Cohomology Theories of Hopf Bimodules and Cup-Product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.