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.     

Hom-Gel'fand–Dorfman Super-bialgebras and Hom-Lie Conformal Superalgebras

The aim of this paper is to introduce and study Hom-Gel'fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing HomGel'fand–Dorfman super-bialgebras. Also, we obtain some infinite-dimensional Hom-Lie superalgebras from affinization of Hom-Gel'fand–Dorfman super-bialgebras. Finally, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish the equivalence between quadratic Hom-Lie conformal superalgebras and Hom-Gel'fand–Dorfman super-bialgebras.     

Structures of Not-finitely Graded Lie Algebras Related to Generalized Heisenberg–Virasoro Algebras

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg–Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.     

One Parameter Deformation of Symmetric Toda Lattice Hierarchy

In this paper,we study one parameter deformation of full symmetric Toda hierarchy. This deformation is induced by Hom-Lie algebras,or is the applications of Hom-Lie algebras. We mainly consider three kinds of deformation,and give solutions to deformations respectively under some conditions.     

Purely Hom-Lie bialgebras

In this paper, we first show that there is a Hom-Lie algebra structure on the set of(σ, σ)-derivations of an associative algebra. Then we construct the dual representation of a representation of a Hom-Lie algebra.We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-O-operators.     

Lie-Yamaguti着色代数的导子与形变

In this paper,we introduce the representation and cohomology theory of Lie-Yamaguti color algebras.Furthermore,we introduce the notions of generalized derivations of Lie-Yamaguti color algebras and present some properties.Finally,we study linear deformations of LieYamaguti color algebras,and introduce the notion of a Nijenhuis operator on a Lie-Yamaguti color algebra,which can generate a trivial deformation.     

RESEARCH ANNOUNCEMENTS——Structure of Solvable Quadratic Lie Algebras

Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics^[10，12，13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras. In this paper, we study solvable quadratic Lie algebras. In Section 1, we study quadratic solvable Lie algebras whose Cartan subalgebras consist of semi-simple elements. In Section 2,we present a procedure to construct a class of quadratic Lie algebras, and we can exhaust all solvable quadratic Lie algebras in such a way. All Lie algebras mentioned in this paper are finite dimensional Lie algebras over a field F of characteristic 0.     

Riemannian Geometry on Hom-ρ-commutative Algebras*

Recently, some concepts such as Hom-algebras, Hom-Lie algebras, Hom-Lie admissible algebras, Hom-coalgebras are studied and some classical properties of algebras and some geometric objects are extended on them. In this paper by recalling the concept of Hom-ρ-commutative algebras, the authurs intend to develop some of the most classical results in Riemannian geometry such as metric, connection, torsion tensor, curvature tensor on it and also they discuss about differential operators and get some ...     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.     

On Some Varieties of Soluble Lie Algebras

In this paper,we study a class of soluble Lie algebras with variety relations that the commutator of m and n is zero.The aim of the paper is to consider the relationship between the Lie algebra L with ...     

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.     

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.     

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.     

On Hom-Lie antialgebra

Abstract

In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.

Communicated by Dr. Pavel Kolesnikov     

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.     

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.