Degenerations of binary Lie and nilpotent Malcev algebras

We describe degenerations of four-dimensional binary Lie algebras, and five- and six-dimensional nilpotent Malcev algebras over ?. In particular, we describe all irreducible components of these varieties.     

The algebraic and geometric classification of nilpotent terminal algebras

We give algebraic and geometric classifications of 4-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are 41 one-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 18 two-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 2 three-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by 21 additional isomorphism classes (see Theorem 13). The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the Zariski closures of a one-parameter family of algebras, a two-parameter family of algebras and a three-parameter family of algebras (see Theorem 15). In particular, there are no rigid 4-dimensional complex nilpotent terminal algebras.     

The algebraic classification of nilpotent Tortkara algebras

Abstract

We classify all complex 6-dimensional nilpotent Tortkara algebras.

Communicated by Alberto Facchini     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Square-zero derivations of matrix algebras over commutative rings

Let R be a commutative ring with identity, M _n (R) the R-algebra consisting of all n by n matrices over R. In this article, for n ≥ 5 we classify linear maps φ from M _n (R) into itself satisfying φ(x)x + xφ(x) = 0 whenever x ² = 0. We call such maps as square-zero derivations.     

Homotopy stability of orthogonal spaces in commutative banach algebras

We study the concept of real stable rank for a complex commutative Banach algebra A (rsr A). It is shown that this invariant has a behaviour completely analogous to the classical Bass stable rank; in particular, we establish a precise relation between both invariants.Further, we use this machinery to show that the connected components of the orthogonal spaces of A, O _k ⁿ (A)={ A ^k×n:^t=id_k×k}, stabilize when k and n increase in a way that depends on rsr A Finally, we prove an analogous stabilization result for the homotopy classes of maps from the j-sphere into O _k ⁿ (A), where j is an arbitrary positive integer.Research supported by a grant of the CONICET, Argentina.     

Modules and ideals of algebras of associative type

In this paper, we study some properties of algebras of associative type introduced in previous papers of the author. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of algebras of associative type, it is proved that any module over a semisimple algebra is completely reducible. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent.     

The algebraic and geometric classification of Zinbiel algebras

This paper is devoted to the complete algebraic and geometric classification of complex 5-dimensional Zinbiel algebras. In particular, we proved that the variety of complex 5-dimensional Zinbiel algebras has dimension 24, it is defined by 16 irreducible components and it has 11 rigid algebras.     

Solutions for PDEs with constant coefficients and derivability of functions ranged in commutative algebras

Given a PDE with real or complex partial derivatives and with constant coefficients, we propose a method of assigning to it a set of algebra‐valued functions in such a manner that the components of the latter are solutions of the PDE. Copyright © 2013 John Wiley & Sons, Ltd.     

Jordan higher derivable maps on triangular algebras by commutative zero products

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

Structure of finitely generated commutative alternative algebras and special Moufang loops

It is proved that the commutator ideal of the multiplication algebra of a free commutative alternative algebra of rank n is nilpotent of index n ? 1. As a corollary to this fact, the Bruck theorem for special commutative Moufang loops is derived.