Notes on δ-Koszul Algebras

This paper is a continuous work of δ-Koszul algebras, which were first introduced by Green and Marcos in 2005 (see Green and Marcos, Commun Algebra 33(6):1753–1764, 2005). Let K^d(A)\mathcal{K}^{\delta}(A) be the category of δ-Koszul modules. It is proved that K^d(A)\mathcal{K}^{\delta}(A) preserves kernels of epimorphisms if and only if the “minimal Horseshoe Lemma” (“MHL” for short) holds. Further, a special class of δ-Koszul algebras named periodic δ -algebras are introduced, which have close connection with Koszul algebras and provide answers to the questions raised by Green and Marcos (Commun Algebra 33(6):1753–1764, 2005). Finally, we construct new periodic δ-algebras from the given ones in terms of one-point extension and sum-extension.     

δ-Koszul对象的若干注记

本文首先给出了δ-分解决定的代数的存在性的一个充分条件和一个必要条件,它与2005年Green和Marcos提出的关于δ-Koszul代数的三个问题有关.2007年,对有限生成分次模范畴中的短正合列ζ:0→K→M→N→0,Cheng和Ye证明了:若K,M是δ-Koszul模,则N也是δ-Koszul模;若K,N是δ-Koszul模,则M也是δ-Koszul模,并且给出反例说明即使M,N是δ-Koszul模,K也未必是δ-Koszul模.本文的另一主要内容是讨论了当M,N是δ-Koszul模时,K是δ-Koszul模的一些条件.     

Ringel–Hall Algebras of Quotient Algebras of Path Algebras

We determine the generating relations for Ringel–Hall algebras associated with quotient algebras of path algebras of Dynkin and tame quivers, and investigate their connection with composition subalgebras.     

Simple Lie Algebras and Complete Lie Algebras

ＳｉｍｐｌｅＬｉｅＡｌｇｅｂｒａｓａｎｄＣｏｍｐｌｅｔｅＬｉｅＡｌｇｅｂｒａｓ＊）ＹａｎｇＱｉ（杨奇）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＴｉａｎｊｉｎＵｎｉｖｅｒｓｉｔｙ，Ｔｉａｎｊｉｎ，３０００７２）ＭｅｎｇＤａｏｊｉ（孟道骥）（Ｄ...     

BCC—Algebras and BCK—Algebras

ＢＣＣ－ＡｌｇｅｂｒａｓａｎｄＢＣＫ－ＡｌｇｅｂｒａｓＺｈａｎｇＸｉａｏｈｏｎｇ（张小红）；ＷｅｉＺｏｎｇｌｉ（卫宗礼）（ＨａｎｚｈｏｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，７２３０００）（ＬｕｏｙａｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，４７１０２２）...     

Algebras of Quotients of Jordan–Lie Algebras

In this article, we introduce the notion of algebra of quotients of a Jordan–Lie algebra. Properties such as semiprimeness or primeness can be lifted from a Jordan–Lie algebra to its algebras of quotients. Finally, we construct a maximal algebra of quotients for every semiprime Jordan–Lie algebra.     

Semi-Heyting Algebras Term-equivalent to Gödel Algebras

In this paper we investigate those subvarieties of the variety $\mathcal {SH}$ of semi-Heyting algebras which are term-equivalent to the variety $\mathcal L_{\mathcal H}$ of Gödel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety $\mathcal L^{\rm Com}$ of commutative semi-Heyting algebras and the variety $\mathcal L^{\vee}$ generated by the chains in which a?<?b implies a → b?=?b. We also study the variety $\mathcal C$ generated within $\mathcal{SH}$ by $\mathcal L_{\mathcal H}$ , $\mathcal L_\vee$ and $\mathcal L_{\rm Com}$ . In particular we prove that $\mathcal C$ is locally finite and we obtain a construction of the finitely generated free algebra in $\mathcal C$ .     

Prime-Dimensional Hopf Algebras

Ｐｒｉｍｅ－ＤｉｍｅｎｓｉｏｎａｌＨｏｐｆＡｌｇｅｂｒａｓＣａｉＣｈｕａｎｒｅｎ（蔡传仁）ａｎｄＣｈｅｎＨｕｉｘｉａｎｇ（陈惠香）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＹａｎｇｚｈｏｕＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，Ｊｉａｎｇｓｕ２２５...     

Simple Poisson–Farkas Algebras and Ternary Filippov Algebras

We establish connection between the differentiably simple associative commutative algebras with unity and the simple Filippov algebras.     

Calabi–Yau Algebras Viewed as Deformations of Poisson Algebras

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w?=?fz is 3-Calabi–Yau. The algebra B can be viewed as a 3-Calabi–Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained.     

On Amenable Banach Algebras

In this paper we extend the concept of amenable algebra given by Jobnsonin[3]to n-amenable algebra.We prove several properties which are extensionof1-amenable algebras appeared in[3]and[4],and we give a sufficientcondition for a Banach algebra being2-amenable.     

∗-Prime Group Algebras

An associative algebra R over a field K is said to be right ?-prime if for every nonzero r ? R, there exists a finitely generated subalgebra S of R such that rSt = 0 implies t = 0. Clearly, strongly prime implies ?-prime and ?-prime implies prime. A large number of examples of group algebras are given which show that the concept of ?-prime lies strictly between prime and strongly prime. A complete characterization of ?-prime group algebras is given. It is proved that a group algebra KG of the group G over the field K is ?-prime if and only if Λ⁺(G) = (1). Intersection theorems play an important role in the study. In the process, a new intersection theorem for ?-prime group algebras is obtained. Elementwise characterization of the ?-prime radical is given and its relation with some well-known radicals is discussed.     

Real Banach ✶ Algebras

In this paper, we study real Banach ✶ algebras systematically. We present the right form of Pták's inequality [1,4] in the real case, and generalize the results of Vukman in [3] to the general case (algebras with or without an identity). Moreover, this paper is a real analogue of Pták's work [1] in the complex case. Received February 22, 2000, Accepted March 14, 2000     

3-Bihom-ρ-Lie Algebras, 3-Pre-Bihom-ρ-Lie Algebras

The purpose is to introduce the notions of 3-Bihom-ρ-Lie algebras and 3-preBihom-ρ-Lie algebras. The authors describe their constructions and express the related lemmas and theorems. Also, they define the 3-Bihom-ρ-Leibniz algebras and show that a3-Bihom-ρ-Lie algebra is a 3-Bihom-ρ-Leibniz algebra with the ρ-Bihom-skew symmetry property. Moreover, a combination of a 3-Bihom-ρ-Lie algebra bracket and a Rota-Baxer operator gives a 3-pre-Bihom-ρ-Lie algebra structure.     

Characteristic Semi-simple Lie Algebras and Completely Semi-simple Lie Algebras over any Field

In this paper, we study some properties of the finite dimensional characteristic semi-simple (C.S.S.) Lie algebras and completely semi-simple Lie algebras over any field. The definitions and some results of these algebras have been given by G.B.Seligman in [1]. We can easily prove the following lemmas: Lemma 1 The centre of any finite dimensional Lie algebra L is a characteristic ideal of L.     

Isomorphisms Between Quasi-Banach Algebras

In this paper, the author proves the Hyers-Ulam-Rassias stability of homo-morphisms in quasi-Banach algebras. This is used to investigate isomorphisms between quasi-Banach algebras.     

One-Point Extensions of t-Koszul Algebras

We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extensions and cokernels of monomorphisms. We study the one-point extension algebras, and a necessary and sufficient condition for such an algebra to be t-Koszul is given. We also consider the conditions such that the category of t-Koszul modules and the category of quadratic modules coincide.     

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.     

Some Results Concerning Hereditary Algebras

ＳｏｍｅＲｅｓｕｌｔｓＣｏｎｃｅｒｎｉｎｇＨｅｒｅｄｉｔａｒｙＡｌｇｅｂｒａｓ￥ＬｕＹｅｇｕａｎｇ（Ｈｅｆｅｉｏｆ，ＩｎｓｔｉｔｕｔｅｏｆＥｃｏｎｏｍｙａｎｄＴｅｃｈｎｏｌｏｇｙ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐｅｐｅｒ，ｔｈｅＡｌｐｒｉｎ＇ｓｃｏｕ...