Mathematics

In this paper, the definitions of algebras of quotients and Martandale-like qoutients of Leibniz algebras are introduced and the interactions between the two quotients are determined. Firstly, some important properties which not only hold for a Leibniz algebras but also can been lifted to its algebras of quotients are investigated. Secondly, for any semiprime Leibniz algebra, its maximal algebra of quotients is constucted and a Passman-like characterization of the maximal algebra is described. Thirdly, the relationship between a Leibniz algebra and the associative algebra which is generated by left and right multiplication operators of the corresponding Leibniz algebras of quotients are examined. Finally, the definition of dense extensions and some vital properties about Leibnia algebras via dense extensions are introduced.

In this paper, we introduce the notion of algebras of quotients of Hom-Lie algebras and investigate some properties which can be lifted from a Hom-Lie algebra to its algebra of quotients. We also give some necessary and sufficient conditions for Hom-Lie algebras having algebras of quotients. We also examine the relationship between a Hom-Lie algebra and the associative algebra generated by inner derivations of the corresponding Hom-Lie algebra of quotients. Moreover, we introduce the notion of dense extensions and get a proposition about Hom-Lie algebras of quotients via dense extensions.

We study almost inner derivations of 2 -step nilpotent Lie algebras of genus 2 , i.e., having a 2 -dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner derivations of such algebras in terms of minimal indices and elementary divisors over an arbitrary algebraically closed field of characteristic not 2 and over the real numbers.

A submodule W of V is summand absorbing, if x+y∈W implies x∈W,y∈W for any x,y∈V . Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules.

We combine the language of monoids with the language of preorders to formulate an abstract factorization theorem with several applications. In particular, this leads to (i) a generalization of P.M. Cohn's classical theorem on "atomic factorizations" from cancellative to Dedekind-finite monoids (and, hence, to a variety of rings that are not domains); (ii) a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring R is isomorphic to a direct sum of finitely many indecomposable R -modules (in fact, we obtain the result as a special case of a general decomposition theorem for the objects of certain categories with finite products, where the indecomposable R -modules are characterized as the atoms of a certain "monoid of modules"). Also, we recover and extend an existence theorem of D.D. Anderson and S. Valdes-Leon on "irreducible factorizations" in commutative rings [RMJM 1996]; a refinement of Cohn's theorem to "nearly cancellative" monoids due to Y. Fan et al. [JA 2018]; and a characterization theorem of A.A. Antoniou and the author about atomic factorizations in certain monoids of sets [PJM 202?].

The notion of Fourier transformation is described from an algebraic perspective that lends itself to applications in Symbolic Computation. We build the algebraic structures on the basis of a given Heisenberg group (in the general sense of nilquadratic groups enjoying a splitting property); this includes in particular the whole gamut of Pontryagin duality. The free objects in the corresponding categories are determined, and various examples are given. As a first step towards Symbolic Computation, we study two constructive examples in some detail -- the Gaussians (with and without polynomial factors) and the hyperbolic secant algebra.

In this paper, we introduce an element ? - δ -primary to another element in a compactly generated multiplicative lattice L and obtain its characterizations. We prove many of its properties and investigate the relations between these structures. By a counter example, it is shown that if an element b?�L is ? - δ -primary to a proper element p?�L then b need not be δ -primary to p and found conditions under which an element b?�L is δ -primary to a proper element p?�L if b is ? - δ -primary to p .

We construct an integral form for the universal enveloping algebra of the Onsager algebra and an explicit integral basis for this integral form. We also formulate straightening identities among some products of basis elements.

Let Λ be an artin algebra. We give an upper bound for the dimension of the bounded derived category of the category modΛ of finitely generated right Λ -modules in terms of the projective and injective dimensions of certain class of simple right Λ -modules as well as the radical layer length of Λ . In addition, we give an upper bound for the dimension of the singularity category of modΛ in terms of the radical layer length of Λ .

The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications. This paper carries out an algebraic study of general integral operators and equations, and shows that there are rich algebraic structures underlying Volterra integral operators and the corresponding equations. First Volterra integral operators are shown to produce a matching twisted Rota-Baxter algebra satisfying twisted integration-by-parts operator identities. In order to provide a universal space to express general integral equations, free operated algebras are then constructed in terms of bracketed words and rooted trees with decorations on the vertices and edges. Further explicit constructions of the free objects in the category of matching twisted Rota-Baxter algebras are obtained by a twisted and decorated generalization of the shuffle product, providing a universal space for separable Volterra equations. As an application of these algebraic constructions, it is shown that any integral equation with separable Volterra kernels is operator linear in the sense that the equation can be simplified to a linear combination of iterated integrals.