Derivations of Tensor Product Algebras

It is known that under certain finite dimensionality condition the derivation algebra of tensor product of two algebras can be obtained in terms of the derivation algebras and the centroids of the involved algebras. We extend this theorem to infinite dimensional case and as an application, we determine the derivation algebra of the fixed point algebra of the tensor product of two algebras, with respect to the tensor product of two finite order automorphisms. These provide the framework for calculating the derivations of some infinite dimensional Lie algebras.     

Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

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.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Galois Theory for Multiplier Hopf Algebras with Integrals

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.     

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.     

Relationship between genetic algebras and semicommutative matrices

Three kinds of noncommutative Gonshor genetic algebras are defined and characterized in terms of matrices. A necessary condition for an algebra to have one of these properties is the semicommutativity of a set of matrices representing the left (and the right) transformations induced by basis elements. For Gonshor genetic algebras which are interpretable, bounds for the train roots of the algebraare given. In terms of matrices this result yields bounds for the eigenvalues of a set ofcertain stochastic semicommutative matrices.     

Bicrossed Products for Finite Groups

We investigate one question regarding bicrossed products of finite groups which we believe has the potential of being approachable for other classes of algebraic objects (algebras, Hopf algebras). The problem is to classify the groups that can be written as bicrossed products between groups of fixed isomorphism types. The groups obtained as bicrossed products of two finite cyclic groups, one being of prime order, are described.     

On Primitive Cellular Algebras

First we define and study the exponentiation of a cellular algebra by a permutation group that is similar to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study, for cellular algebras, the notion of a base, which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally, we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential. Bibliography: 16 titles.     

Depth One Extensions of Semisimple Algebras and Hopf Subalgebras

Depth one extensions of finite dimensional semisimple algebras are completely characterized in terms of their algebra centers. For extensions of semisimple Hopf algebras this characterization translates into a trivial monoidal action of the dual fusion category Rep(A ^*) on Rep(B).     

Zur konstruktion von Lie-Algebren aus Jordan-Tripelsystemen

In this paper we study certain Lie algebras which are constructed from the (-1)-eigenspaees of an involution of a Jordan algebra. The construction is a generalisation of the Koecher-Tits-construction. We give necessary conditions in terms of the Jordan algebras for the Lie algebras being simple. If the (-1)-spaces are Peirce-1/2-components then we obtain a close relation between the Lie algebras under consideration and the structure algebras of Jordan algebras. We finally give a list of those types of simple Lie algebras which can be formed by this construction; among them are Lie algebras of type E₆ and E₇.Of fundamental importance for our considerations is a close connection between the constructed Lie algebras and the standard imbeddings of Lie triple systems.     

Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.     

代数表示论的某些新进展

代数表示理论是代数学的一个新的重要分支，在近二十五年的时间里，这一理论有很大的发展，关于代数表示的基础理论的介绍可参见文献（１０１），本文主要从Ｈａｌｌ代数和拟遗传代数两个方面介绍代数表示论的一些最新进展，第一章给出了Ｈａｌｌ代数的基本理论及其方法，并且着重指出了利用这一理论和方法通过代数表示论去实现Ｋａｃ－Ｍｏｏｄｙ李代数及相应的量子包络代数，第二章介绍了拟遗传代数及其表示理论，以及这一理论与复     

Flat Nonunimodular Lorentzian Lie Algebras

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (?, +,…, +) such that left multiplications are skew-symmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4.     

On the predecessor relation in abstract algebras

In this paper we generalize the Dedekind theory of order for the natural numbers N to abstract algebras with arbitrarily many finitary or infinitary operations. For any algebra ??, we introduce an algebraic predecessor relation P?? and its transitive hull P*?? coinciding in N with the unary injective successor function' resp. the >-relation. For some important classes of algebras ??, including Peano algebras (absolutely free algebras, word algebras), the algebraic predecessor relation is well-founded. Hence, its transitive hull, the natural ordering >?? of ??, is a well-founded partial order, which turns out to be a convenient device for classifying Peano algebras with respect to the number of operations and their arities. Moreover, the property of well-foundedness is an efficient tool for giving simple proofs of structure theorems as, e. g., that the class of all Peano algebras is closed under subalgebras and non-void direct products. - Finally, we will show how in the case of a formal language ??, i. e., the Peano algebra ?? of expressions (= terms & formulas), relations P??, resp. P*_?? can be used to define basic syntactical notions as occurences of free and bound variables etc. without any reference to a particular representation (“coding”) of the formal language. MSC: 03B22, 03E30, 03E75, 03F35, 08A55, 08B20.