The Strong Endomorphism Kernel Property in Ockham Algebras

An endomorphism on an algebra 𝒜 is said to be “strong” if it is compatible with every congruence on 𝒜; and 𝒜 is said to have the “strong endomorphism kernel property” if every congruence on 𝒜, different from the universal congruence, is the kernel of a strong endomorphism on 𝒜. Here we consider this property in the context of Ockham algebras. In particular, for those MS-algebras that have this property we describe the structure of their dual space in terms of 1-point compactifications of discrete spaces.     

An Extended Ockham Algebra with Endomorphism Kernel Property

An algebraic structure A is said to have the endomorphism kernel property if every congruence on A, other than the universal congruence, is the kernel of an endomorphism on A. In this paper, we consider the EKP （that is, endomorphism kernel property） for an extended Ockham algebra A. In particular, we describe the structure of the finite symmetric extended de Morgan algebras having EKP.     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.     

The Coradical Filtration for Drinfeld Double Ringel-Hall Algebras

Abstract

Let 𝒟(Λ) be the Drinfeld double Ringel-Hall algebra with Λ being any finite dimensional hereditary algebra over a finite field k. We determine the coradical filtration for 𝒟(Λ). As an application, we describe the group of Hopf algebra automorphisms of the Drinfeld double Ringel composition algebra of Λ.     

A Class of Nongraded Simple Lie Algebras

Abstract

Nongraded simple Lie algebras appear naturally in mathematical physics. In this paper, a new class of nongraded simple Lie algebras are presented based on the pairs (𝒜, 𝒟) consisting of a commutative associative unital algebra 𝒜 and a finite dimensional commutative derivation subalgebra 𝒟 such that 𝒜 is 𝒟-simple. The isomorphism classes of these nongraded Lie algebras are also determined and the structure space of these algebras is given explicitly.     

On factoring an operator using elements of its kernel

A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation ? acting on some ring of functions, this paper considers the more general situation of an endomorphism 𝔇 acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of 𝔇 followed by left multiplication by elements of the algebra. Assume that the set of such operators is closed under multiplication and that a Wronski-like matrix produced from some finite list of elements of the algebra is invertible (analogous to the linear independence condition). Then, it is shown that the set of operators whose kernels contain all of those elements is the left ideal generated by an explicitly given operator. In other words, an operator has those elements in its kernel if and only if it has that generator as a right factor. Three examples demonstrate the application of this result in different contexts, including one in which 𝔇 is an automorphism of finite order.     

Characterizations of Jordan derivations and Jordan homomorphisms

Let 𝒜 be a unital Banach algebra and ? be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ? satisfying δ(ST)?=?δ(S)T?+Sδ(T) for any S, T?∈?𝒜 with ST?=?W, where W is a left or right separating point of ?, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ? which satisfies h(S)h(T)?=?h(ST) for any S,?T?∈?𝒜 with ST?=?W is a Jordan homomorphism if h(W) is a separating point of ?.     

Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let 𝒜 be a commutative unital algebra over an algebraically closed field k of characteristic ≠2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let 𝒬 be a Hopf algebra that coacts on 𝒜 inner-faithfully, while leaving V invariant. We prove that 𝒬 must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) 𝒬 is co-semisimple, finite-dimensional, and char(k) = 0.     

PBW-Pairs of Varieties of Linear Algebras

The notion of a Poincaré–Birkhoff–Witt (PBW)-pair of varieties of linear algebras over a field is under consideration. Examples of PBW-pairs are given. We prove that if (𝒱, 𝒲) is a PBW-pair and the variety 𝒱 is homogeneous and Schreier, then so is 𝒲; the results similar to the Schreier property for PBW-pairs are also true for the Freiheitssatz and Word problem. In particular, it follows that the Freiheitssatz is true for the varieties of Akivis and Sabinin algebras. We give also examples of varieties that do not satisfy the Freiheitssatz. It is shown that an element u of a free algebra 𝒲[X] in a homogeneous Schreier variety of algebras 𝒲 satisfying the Freiheitssatz is a primitive element (a coordinate polynomial) if and only if the factor algebra of 𝒲[X] by the ideal generated by the element u is a free algebra in 𝒲. We consider also properties of primitive elements.     

On the composition subalgebras of Bridgeland’s Hall algebras

Let A be a finite dimensional hereditary algebra over a finite field k and 𝒫 the category consisting of finite dimensional projective (left) A-modules. In this paper, we consider the composition subalgebra of Bridgeland’s Hall algebra of the category 𝒞_m(𝒫) of m-cyclic complexes for any positive integer m≥2 and determine its generating relations.     

The Yoneda Algebra of a Graded Ore Extension

Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext _A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦₂ property, B will also be 𝒦₂. We prove the converse of this result.     

The Lower Socle and Finite Rank Elementary Operators

Abstract

We define the lower socle of a semiprime algebra 𝒜 as the sum of all minimal left ideals 𝒜e where e is a minimal idempotent such that the division algebra e𝒜e is finite dimensional. We study the connection between the condition that the elements a _k , b _k , 1 ≤ k ≤ n, lie in the lower socle of 𝒜 and the condition that the elementary operator x ? a ₁ xb ₁ + ? + a _n xb _n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

Finite Abelian Groups with the Strong Endomorphism Kernel Property

An endomorphism h of a group G is said to be strong whenever for every congruence θ on G, (x, y) ∈ θ implies (h(x), h(y)) ∈ θ for every x, y ∈ G. A group G is said to have the strong endomorphism kernel property if every congruence on G is the kernel of a strong endomorphism. In this note, we study the strong endomorphism kernel property in the class of Abelian groups. In particular, we show that a finite Abelian group has the strong endomorphism kernel property if and only if it is cyclic.     

Generating primitive positive clones

Suppose is a set of operations on a finite set A. Define PPC() to be the smallest primitive positive clone on A containing . For any finite algebra A, let PPC#(A) be the smallest number n for which PPC(CloA) = PPC(Clo _n A). S. Burris and R. Willard [2] conjectured that PPC#(A) ≤|A| when CloA is a primitive positive clone and |A| > 2. In this paper, we look at how large PPC#(A) can be when special conditions are placed on the finite algebra A. We show that PPC#(A) ≤|A| holds when the variety generated by A is congruence distributive, Abelian, or decidable. We also show that PPC#(A) ≤|A| + 2 if A generates a congruence permutable variety and every subalgebra of A is the product of a congruence neutral algebra and an Abelian algebra. Furthermore, we give an example in which PPC#(A) ≥|A| - 1)² so that these results are not vacuous. Received August 30, 1999; accepted in final form April 4, 2000.     

Some remarks on Bridgeland’s Hall algebras

We study the functorial properties of Bridgeland’s Hall algebras. Specifically, let 𝒜 and ? be two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and let F:𝒜→? be a fully faithful exact functor, which preserves projectives, then F induces an embedding of algebras from the Bridgeland’s Hall algebra of 𝒜 to the one of ?. In addition, let A be a finite-dimensional algebra over a finite field and B some special quotient algebra of A, then the Bridgeland’s Hall algebra of B is the quotient algebra of the one of A. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras.     

Jordan generalized derivations on triangular algebras

Let 𝒜 be a unital algebra and let ? be a unitary 𝒜-bimodule. We consider Jordan generalized derivations mapping from 𝒜 into ?. Our results on unitary algebras are applied to triangular algebras. In particular, we prove that any Jordan generalized derivation of a triangular algebra is a generalized derivation.     

Caldero–Keller Approach to the Denominators of Cluster Variables

Buan, Marsh, and Reiten proved that if a cluster-tilting object T in a cluster category 𝒞 associated to an acyclic quiver Q satisfies certain conditions with respect to the exchange pairs in 𝒞, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to Q has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T. In this article, we give an alternative proof of this result using the Caldero–Keller approach to acyclic cluster algebras and the work of Palu on cluster characters.