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     

Cohomology of oriented algebras

In this paper, our goal is to develop the equivariant version of Hochschild cohomology. In particular, we develop a cohomology theory for oriented algebras.     

3-Lie bialgebras

3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of characteristic zero are provided.     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.     

Nijenhuis algebras, NS algebras, and N-dendriform algebras

In this paper, we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras and related algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.     

对合导子构造的3-李双代数与3-Pre-李代数

研究具有对合导子的3-李代数的结构,证明了具有对合导子的m-维3-李代数A存在相容的3-Pre-李代数,且在2m-维半直积3-李代数A■_(ad~*)A~*上存在局部上循环3-李双代数结构.利用对合导子构造了3-李代数A■_(ad~*)A~*上的3-李Yang-Baxter方程的解和一类3-Pre-李代数,并构造了8-维和10-维局部上循环3-李双代数.     

3-Lie代数的次理想

主要研究3-Lie代数的子代数及次理想的结构.证明了由次理想生成的子代数不一定是次理想.给出了由次理想生成的子代数是次理想的充要条件.最后研究了次理想和子代数的G_n-对之间的关系.     

Cohomology of a Poisson Algebra

We propose a technique for calculating the cohomology of a Poisson algebra using the Laplace transformation of distributions with compact support. We find the lowest-order cohomologies of this algebra with coefficients in two natural representations: the trivial and the adjoint representations.     

T*-extension of a 3-Lie algebra

We introduce a new technique called T*-extension, of constructing a metric 3-Lie algebra out of an arbitrary 3-Lie algebra, and explore all possible metrics and corresponding signatures on this resulting metric 3-Lie algebra.     

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.     

3-Lie Algebras Realized by Cubic Matrices

Associative multiplications of cubic matrices are provided. The N3-dimensional 3-Lie algebras are realized by cubic matrices, and structures of the 3-Lie algebras are studied.     

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension 16

Abstract

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.     

Hochschild Cohomology and Representation-Finite Algebras

Using Grothendieck's semicontinuity theorem for half-exact functors,we derive two semicontinuity results on Hochschild cohomology.We apply these to show that the first Hochschild cohomogy groupof the mesh algebra of a translation quiver over a domain vanishesif and only if the translation quiver is simply connected. Wethen establish an exact sequence relating the first Hochschildcohomology group of an algebra to that of the endomorphism algebraof a module and apply it to study the first Hochschild cohomologygroup of an Auslander algebra. Our main result shows that fora finite-dimensional and representation-finite algebra algebraA over an algebraically closed field with Auslander algebra the following conditions are equivalent:

1. (1)A admits no outer derivation;
2. (2) admits no outer derivations;
3. (3) A is simply connected;
4. (4) is strongly simply connected.

. 2000 Mathematics Subject Classification 16E30, 16G30.     

3-李代数的广义导子

给出了3-李代数的广义导子、拟导子、拟型心的定义,研究了他们之间的结构关系,并对具有极大对角环面的3-李代数的拟导子和拟型心结构进行了系统的研究.证明了(1)广义导代数GDer(A)可以分解成拟导子代数QDer(A)和拟型心QΓ(A)的直和;(2)3-李代数A的拟导子可以扩张成一个具有较大维数的3-李代数的导子;(3)拟导子代数QDer(A)包含在拟型心的正规化子中,表示为[QDer(A),QΓ(A)]?QΓ(A);(4)如果A包含极大对角环面T,那么QDer(A)和Qr(A)是T的对角模,也就是(T,T)半单地作用在QDer(A)和QΓ(A)上.     

Homological properties of sklyanin algebras

In their recent paper [13], Tate and Van den Bergh studied certain quadratic algebras, called the “Sklyanin algebras”. They proved that these algebras have the Hilbert series of a polynomial algebra, are Noetherian and Koszul, and satisfy the Auslander-Gorenstein and Cohen-Macaulay conditions. This paper gives an alternative proof of these results, as suggested in [13], and thereby answering a question in their paper.     

The Realization of 4-dimensional 3-Lie Algebras

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.