Mathematics

Given a nilpotent Lie algebra L of dimension ≤6 on an arbitrary field of characteristic ≠2 , we show a direct method which allows us to detect the capability of L via computations on the size of its nonabelian exterior square L∧L . For dimensions higher than 6 , we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of L∧L via the size of the Schur multiplier M(L/ Z ∧ (L)) of L/ Z ∧ (L) , where Z ∧ (L) denotes the exterior center of L .

An orthogonality space is a set equipped with a symmetric and irreflexive binary relation. We consider orthogonality spaces with the additional property that any collection of mutually orthogonal elements gives rise to the structure of a Boolean algebra. Together with the maps that preserve the Boolean structures, we are led to the category NOS of normal orthogonality spaces. Moreover, an orthogonality space of finite rank is called linear if for any two distinct elements e and f there is a third one g such that exactly one of f and g is orthogonal to e and the pairs e,f and e,g have the same orthogonal complement. Linear orthogonality spaces arise from finite-dimensional Hermitian spaces. We are led to the full subcategory LOS of NOS and we show that the morphisms are the orthogonality-preserving lineations. Finally, we consider the full subcategory EOS of LOS whose members arise from positive definite Hermitian spaces over Baer ordered ⋆ -fields with a Euclidean fixed field. We establish that the morphisms of EOS are induced by generalised semiunitary mappings.

Let R be a ring and n , k two non-negative integers. In this paper, we introduce the concepts of n -weak injective and n -weak flat modules and via the notion of special super finitely presented modules, we obtain some characterizations of these modules. We also investigate two classes of modules with richer contents, namely WI n k (R) and WF n k ( R op ) which are larger than that of modules with weak injective and weak flat dimensions less than or equal to k . Then on any arbitrary ring, we study the existence of WI n k (R) and WF n k ( R op ) covers and preenvelopes

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an n -dimensional null-filiform Zinbiel algebra is isomorphic to an (n+1) -dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.

Centralizers of rank one in the first Weyl algebra have genus zero.

It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential endomorphism rings are faithful. The paper will appear in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.

In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras.

We show that the isomorphism class of a two-step solvable Lie poset subalgebra of a semisimple Lie algebra is determined by its dimension. We further establish that all such algebras are absolutely rigid.

We develop tools for classification of contraction algebras and apply these to solve the problem on classification up to isomorphism of 8 and 9 dimensional algebras corresponding to 3-fold flops. We prove that there is only one up to isomorphism contraction algebra of dimension 8, and two algebras of dimension 9. The formulae for the dimension of algebra, depending on the type of the potential are obtained. In the second part of the paper we show that associated graded structure to brace and truss with appropriate descending ideal filtration is pre-Lie.

A hypergroup is stringent if a⊞b is a singleton whenever a≠−b . A hyperfield is stringent if the underlying additive hypergroup is. Every doubly distributive skew hyperfield is stringent, but not vice versa. We present a classification of stringent hypergroups, from which a classification of doubly distributive skew hyperfields follows. It follows from our classification that every such hyperfield is a quotient of a skew field.