Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Universal quantum semigroupoids

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) graded algebra A universally while preserving grading. We restrict our attention to algebraic structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that coact on a connected graded algebra universally while preserving grading. Our main result is that when A is the path algebra $\mathbb{k}Q$ of a finite quiver Q, each of the various UQSGds introduced here is isomorphic to the face algebra attached to Q. The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated.     

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G ⁿ , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.     

Constructing affine Lie algebras

The ?-grading determined by a long simple root of a rank n+1 a?ne Lie algebra over ? arises from a representation of a rank n semi-simple complex Lie algebra. Analysis of the relationship between the grading and the representation yields constructions that generalize the minuscule and adjoint algorithms as well as Kac’s construction of nontwisted a?ne Lie algebras.     

Standardly Stratified Extension Algebras

Abstract

The paper generalizes some of our previous results on quasi-hereditary Koszul algebras to graded standardly stratified Koszul algebras. The main result states that the class of standardly stratified algebras for which the left standard modules as well as the right proper standard modules possess a linear projective resolution – the so called linearly stratified algebras – is closed under forming their Yoneda extension algebras. This is proved using the technique of Hilbert and Poincaré series of the corresponding modules.

     

Generalizing the notion of Koszul algebra

We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.     

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     

关于局部顶点李代数的一些注记

本文研究局部顶点李代数与顶点代数之间的关系.利用由局部顶点李代数构造顶点代数的方法,定义局部顶点李代数之间的同态,证明了同态可以唯一诱导出由局部顶点李代数构造所得到的顶点代数之间的同态.     

关于格上蕴涵代数及其对偶代数

给出了格蕴涵代数、MV代数、R₀代数等一些格上蕴涵代数之间的关系,并建立了它们的对偶代数.其结果描述了这些代数内部结构的特征,同时也为从语义的角度进一步研究格值逻辑系统提供了一个新的途径.     

Equations des algèbres Lie triple qui sont des algèbres train

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank $\le 4$ with an idempotent are Jordan algebras.Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of $e$-stable power-associative algebras.We also prove that the algebra obtained by $2$-gametization process of a Lie triple algebra is a Lie triple one.     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.     

Graded extended Lie-type algebras

The class of extended Lie-type algebras contains the ones of associative algebras, Lie algebras, Leibniz algebras, dual Leibniz algebras, pre-Lie algebras, and Lie-type algebras, etc. We focus on the class of extended Lie-type algebras graded by an Abelian group G and study its structure, by stating, under certain conditions, a second Wedderburn-type theorem for this class of algebras.     

Polyadic and cylindric algebras of sentences

In this note we give an interpretation of cylindric algebras as algebras of sentences (rather than formulas) of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two Kinds of Solvable Complete Lie Algebras

In this paper,we will give the definition of completable nilpotent Lie algebras,discuss its decomposition and prove that the heisenberg algebras and extensions of abelian quadratic Lie algebras are all completable nilpotent Lie algebras.     

Binary Lie algebras satisfying the third Engel condition

Let Φ be a unital associative commutative ring with 1/2. The local nilpotency is proved of binary Lie Φ-algebras satisfying the third Engel condition. Moreover, it is proved that this class of algebras does not contain semiprime algebras.     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

Monoidal Hom–Hopf Algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras, and Lie algebras.     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.     

Regular Unital Finite-Dimensional Lattice-Ordered Algebras

The main result in this article is to show that a regular unital finite-dimensional lattice-ordered algebra over ? with zero ?-radical is isomorphic to a finite direct sum of lattice-ordered matrix algebras of lattice-ordered group algebras of finite groups over ?.