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.     

Triangulated Quotient Categories

A notion of mutation of subcategories in a right triangulated category is defined in this article. When (𝒵, 𝒵) is a 𝒟-mutation pair in a right triangulated category 𝒞, the quotient category 𝒵/𝒟 carries naturally a right triangulated structure. Moreover, if the right triangulated category satisfies some reasonable conditions, then the right triangulated quotient category 𝒵/𝒟 becomes a triangulated category. When 𝒞 is triangulated, our result unifies the constructions of the quotient triangulated categories by Iyama-Yoshino and by Jørgensen, respectively.     

Calculus of Functorial Morphisms

We prove that, if F, G: 𝒞 → 𝒟 are two right exact functors between two Grothendieck categories such that they commute with coproducts and U is a generator of 𝒞, then there is a bijection between Nat(F, G) and the centralizer of Hom_𝒟(F(U), G(U)) considered as an Hom_𝒞(U, U)-Hom_𝒞(U, U)-bimodule. We also prove a dual of this result and give applications to Frobenius functors between Grothendieck categories.     

The derived category of an algebra with radical square zero

Koszul duality and covering theory are combined to realize the bounded derived category 𝒟 of an algebra with radical square zero as a certain orbit category of the bounded derived category of finitely presented representations of an associated infinite quiver. As a consequence, the possible shapes of the connected components of the Auslander–Reiten quiver of 𝒟 are described.     

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.     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

Mutation pairs and quotient categories of Abelian categories

The notion of 𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a 𝒟-mutation pair in an abelian category 𝒜, the quotient category 𝒵∕𝒟 carries naturally a triangulated structure. Moreover, our result generalize the construction of the quotient triangulated category by Happel [10 Happel, D. (1988). Triangulated Categories in the Representation of Finite Dimensional Algebras. London Mathematical Society, LMN, Vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar], Theorem 2.6]. Finally, we find a one-to-one correspondence between cotorsion pairs in 𝒜 and cotorsion pairs in the quotient category 𝒵∕𝒟, and study homological finiteness of subcategories in a mutation pair.     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.     

Residual Properties of Free Products

Let 𝒞 be a class of groups. We give sufficient conditions ensuring that a free product of residually 𝒞 groups is again residually 𝒞, and analogous conditions are given for LE-𝒞 groups. As a corollary, we obtain that the class of residually amenable groups and the one of locally embeddable into amenable (LEA) groups are closed under taking free products.

Moreover, we consider the pro-𝒞 topology and we characterize special HNN extensions and amalgamated free products that are residually 𝒞, where 𝒞 is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.     

Coherent Continuation on the Category of (𝔤, K)-Modules

Abstract

Let 𝔤 be a complex semisimple Lie algebra. Let K be an algebraic group acting on the flag variety of 𝔤 with finitely many orbits. We give a geometric interpretation of the coherent continuation on the category of finitely generated (𝔤, K )-modules in terms of the intertwining functors on the category of K-equivariant 𝒟-modules.     

The Endomorphism Kernel Property in Finite Distributive Lattices and de Morgan Algebras

Abstract

An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.     

On Identities of a Ternary Quaternion Algebra

This article studies a simple 4-dimensional ternary algebra 𝒜 which appears analogously to the quaternions from the Lie algebra 𝔰𝔩(2). We describe the heights 1 and 2 identities, and the derivations of 𝒜. Based on 𝒜, some ternary enveloping algebras for ternary Filippov algebras are constructed.     

On reducible matrix patterns

A tool to study the inertias of reducible nonzero (resp. sign) patterns is presented. Sumsets are used to obtain a list of inertias attainable by the pattern 𝒜 ⊕ ? dependent upon inertias attainable by patterns 𝒜 and ?. It is shown that if ? is a pattern of order n, and 𝒜 is an inertially arbitrary pattern of order at least 2(n ? 1), then 𝒜 ⊕ ? is inertially arbitrary if and only if ? allows the inertias (0, 0, n), (0, n, 0) and (n, 0, 0). We illustrate how to construct other reducible inertially (resp. spectrally) arbitrary patterns from an inertially (resp. spectrally) arbitrary pattern 𝒜 ⊕ ?, by replacing 𝒜 with an inertially (resp. spectrally) arbitrary pattern 𝒮. We identify reducible inertially (resp. spectrally) arbitrary patterns of the smallest orders that contain some irreducible components that are not inertially (resp. spectrally) arbitrary. It is shown there exist nonzero (resp. sign) patterns 𝒜 and ? of orders 4 and 5 (resp. 4 and 4) such that both 𝒜 and ? are non-inertially-arbitrary, and 𝒜 ⊕ ? is inertially arbitrary.     

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 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.     

On Disjoint Sets of Natural Classes

Abstract

An M-natural class is any subclass of σ[M] which is closed under (1) submodules,(2) isomorphic copies,(3) direct sums and (4) M-injective envelopes. Let 𝒞 be any set of pairwise disjoint M-natural classes. We define the 𝒞-dimension of an R-module and examine how finite 𝒞-dimension is related to certain injectivity conditions in σ[M]. We also define a 𝒞-chain and relate ACC on 𝒞-chains again to certain injectivity conditions.     

Another lattice of varieties of completely regular semigroups

Completely regular semigroups S are taken here with the unary operation of inversion within the maximal subgroups of S. As such they form a variety 𝒞? whose lattice of subvarieties is denoted by ?(𝒞?). The relation on ?(𝒞?) which identifies two varieties if they contain the same bands is denoted by B^∧. The upper ends of B^∧-classes which are neither equal to 𝒞? nor contained in the variety 𝒞𝒮 of completely simple semigroups are generated by two countably infinite ascending chains called canonical varieties. In a previous publication, we constructed the sublattice Σ of ?(𝒞?) generated by 𝒞𝒮 and the first four canonical varieties. Here we extend Σ to the sublattice Ψ of ?(𝒞?) generated by 𝒞𝒮 and the first six canonical varieties. For each of the varieties in Ψ?Σ, we construct the ladder and a basis of its identities.     

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.