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.     

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.     

L-o cto-algebras

L-octo-algebra with 8 operations as the Lie algebraic analogue of octo-algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo-algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri-algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.     

A Class of Solvable Lie Algebras and Their Hom-Lie Algebra Structures

The finite-dimensional indecomposable solvable Lie algebras s with Q2n＋1 as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n＋1＋1. Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.     

Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds

The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich's type theorem and the Smale's γ-theory are extended.     

Linear Maps Preserving Invertibility or Related Spectral Properties

We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various parts of spectrum, spectral radius, quasinilpotents, etc. We present some results on elementary operators and additive operators preserving invertibility or related properties. In particular, we give a negative answer to a problem posed by Gao and Hou on characterizing spectrumpreserving elementary operators. Several open problems are also mentioned.     

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.     

Piecewise-Koszul algebras

It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos.     

Generalized Einstein tensor for a Weyl manifold and its applications

It is well known that the Einstein tensor G for a Riemannian manifold defined by G βα = R βα 1/2 Rδβα , R βα = g βγ R γα where R γα and R are respectively the Ricci tensor and the scalar curvature of the manifold, plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work, we first obtain the generalized Einstein tensor for a Weyl manifold. Then, after studying some properties of generalized Einstein tensor, we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover, we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally, a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.     

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive(generalized)ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive(generalized)ξ-Lie derivation with ξ = 1 if and only if it is an additive(generalized)derivation satisfying L(ξA)= ξL(A)for all A. These results are then used to characterize additive(generalized)ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.     

分裂的正则Hom-Poisson 着色代数结构

We introduce the class of split regular Hom-Poisson color algebras as the natural generalization of split regular Hom-Poisson algebras and the one of split regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show∑that such a split regular Hom-Poisson color algebras A is of the form A = U +αIα with U a subspace of a maximal abelian subalgebra H and any Iα, a well described ideal of A, satisfying[Iα, Iβ] + IαIβ = 0 if [α]≠[β]. Under certain conditions, in the case of A being of maximal length, the simplicity of the algebra is characterized.     

Representations of four-derivation Lie algebras of Block type

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on ＂polynomials＂ for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.     

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.     

Admissible solutions of systems of complex partial difference equations on Cn

In a recent papers,some authors applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. We will mainly investigate Malmquist theorem of a type of systems of complex partial difference equations on Cn,improvements and extensions of such results are presented in this paper.     

On Marcinkiewicz integrals associated to compound mappings with rough kernels

The Lp bounds for the parametric Marcinkiewicz integrals associated to compound mapq pings, which contain many classical surfaces as model examples, are given, where the kernels of our operators are rather rough on the unit sphere as well as in the radial direction. These results substan- tially improve and extend some known results.     

Homological dimensions of the extension algebras of monomial algebras

The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras — dual and trivially twisted extensions — with a unified combinatorial approach using the two combinatorial algorithms — Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden subtle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more efficient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras.     

Derivations and extensions of lie color algebra

In this article,the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer（L）and central extension H^2（L,F）on some Lie color algebras.Meanwhile,they generalize the notion of double extension to quadratic Lie color algebras,a sufficient condition for a quadratic Lie color algebra to be a double extension and further properties are given.     

On Completeness and Minimality of Random Exponential System in a Weighted Banach Space of Functions Continuous on the Real Line

In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin's classical results.     

HOLOMORPHIC HARMONIC ANALYSIS ON COMPLEX REDUCTIVE GROUPS

The authors define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties such as the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of K-admissible measures. The authors prove that K-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of K-admissible measures.     

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.