We prove that a simple unital Jordan superalgebra of arbitrary dimension belongs to the list of known simple unital superalgebras or lies in a certain proper subvariety.

In this paper, we prove that in a homogeneous variety V one of the properties of local nilpotency and of local finiteness is radical in the sense of Amitsur-Kurosh if and only of so is the second property.

We study an analogue of the Andreadakis-Johnson filtration for automorphism groups of free algebras and introduce the notion of tangent Lie algebras for certain automorphism groups, defined as subalgebras of the Lie algebra of derivations. We show that, for many classical varieties of algebras, the tangent Lie algebra is contained in the Lie algebra of derivations with constant divergence. We also introduce the concepts of approximately tame and absolutely wild automorphisms of free algebras in arbitrary varieties and employ tangent Lie algebras to investigate their properties. It is shown that nearly all known examples of wild automorphisms of free algebras are absolutely wild, with the notable exceptions of the Nagata and Anick automorphisms. We show that the Bergman automorphism of free matrix algebras of order two is absolutely wild. Furthermore, we prove that free algebras in any variety of polynilpotent Lie algebras-except for the abelian and metabelian varieties-also possess absolutely wild automorphisms. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The purpose of this paper is a partial progress towards classification of simple infinite dimensional Jordan superalgebras. First, we prove that the only simple infinite dimensional Jordan superalgebras with finite dimensional even parts are the superalgebras of superforms. Then we consider the superalgebras whose even parts are infinite dimensional algebras of ``Clifford type'', that is, direct sums of algebras of bilinear forms. The results of \cite{RZ} show that the number of summonds in these sums is 1 or 2. We prove that the second case is impossible and that the simple infinite dimensional Jordan superalgebras of the first type are the superalgebras of superforms.

We construct an Anick type wild automorphism in a 3-generated free Poisson algebra which induces a tame automorphism in a 3-generated polynomial algebra. We also show that this automorphism is stably tame.

We describe all automorphisms of a free metabelian anticommutative algebra of rank n≥ 3 over a field K that move only one variable while fixing the others. Such automorphisms are called Chein automorphisms in the cases of free metabelian groups and free metabelian Lie algebras. We show that all automorphisms of a free metabelian anticommutative algebra of rank n=2 are linear, and that the simplest non elementary Chein automorphism of degree 3 is absolutely wild for all n≥ 3.

We prove that every irreducible Poisson supermodule over the Grassmann Poisson superalgebra G_n over a field of characteristic different from 2 is isomorphic to the regular Poisson supermodule Reg G_n or to its opposite supermodule. Moreover, every unital Poisson supermodule over G_n is completely reducible. If P is a unital Poisson superalgebra which contains G_n with the same unit then P≅ Q⊗ G_n for some Poisson superalgebra Q. Furthermore, we classify the supermodules over G_n in the category of dot-bracket superalgebras with Jordan brackets, and we prove that every irreducible Jordan supermodule over the Kantor double Kan G_n is isomorphic to the supermodule Kan V, where V is an irreducible dot-bracket supermodule with a Jordan bracket over G_n.

A new series of central elements is found in the free alternative algebra. More exactly, let Alt[X] and SMalc[X] ⊂ Alt[X] be the free alternative algebra and the free special Malcev algebra over a field of characteristic 0 on a set of free generators X, and let f (x, y, x1,…, xn) ∈ SMalc[X] be a multilinear element which is trivial in the free associative algebra. Then the element un = un (x, x1,…,xn) = f (x2, x, x1,…,xn) − f (x, x2,x1,…,xn) lies in the center of the algebra Alt[X]. The elements un(x, x1,…, xn) are uniquely defined up to a scalar for a given n (that is, they do not depend on f but only on deg f), and they are skew-symmetric on the variables x1,…,xn. Moreover, un = 0 for n = 4m + 2, 4m + 3 and un ≠ 0 for n = 4m, 4m + 1. The ideals generated by the elements u4m, u4m+1 lie in the associative center of the algebra Alt[X] and have trivial multiplication.

We prove that for every natural number n, there exists a natural number N (n) such that every multilinear skew-symmetric polynomial in N (n) or more variables which vanishes in the free associative algebra also vanishes in any n-generated alternative algebra over a field of characteristic 0. Previously, a similar result was proved only for a series of skew-symmetric polynomials constructed by I. P. Shestakov in [Algebra and Logic, 16, No. 2, 153-166 (1977)].

We construct linear bases for free commutative two-step-associative algebras and study their automorphisms. It turns out that every automorphism of a polynomial algebra without unit can be lifted to an automorphism of a free commutative two-step-associative algebra. Moreover, for any n >= 2, a wild automorphism is constructed for the n-generated free commutative two-step-associative algebra which is not stably tame and cannot be lifted to an automorphism of the n-generated free commutative nonassociative algebra.

We prove that every simple finite dimensional binary Lie superalgebra over the complex numbers field C with non-zero odd part is either a Lie superalgebra or has a solvable even part.

A variety of associative algebras is called nonmatrix if it does not contain the algebra of 2 x 2 matrices over the given field. Nonmatrix varieties were introduced and studied by V.N.Latyshev in relation with the Specht problem. Some characterizations of nonmatrix varieties were obtained in the paper [10]. In the given paper the notion of nonmatrix variety is extended for nonassociative algebras, and the characterization from [10] is generalized for alternative, Jordan, and some other varieties of algebras.

It is proved that, for any natural n, the subalgebra generated by words of length divisible by n on generators (the Veronese n-subalgebra) in a free finitely generated alternative algebra is finitely generated.

The Jacobson Coordinatization Theorem describes the structure of unitary Jordan algebras containing the algebra H_n(F) of symmetric nxn matrices over a field F with the same identity element, for n≥ 3. In this paper we extend the Jacobson Coordinatization Theorem for n=2. Specifically, we prove that if J is a unitary Jordan algebra containing the Jordan matrix algebra H_2(F) with the same identity element, then J has a form J=H_2(F)⊗ A_0+k⊗ A_1, where A=A_0+A_1 is a Z_2-graded Jordan algebra with a partial odd Leibniz bracket , an k=e_12-e_21∈ M_2(F) with the multiplication given by (a⊗ b)(c⊗ d)=ac⊗ bd + [a,c]⊗b,d, the commutator [a,c] is taken in M_2(F).

The classification of irreducible unital commutative power-associative modules for Hn(F), the algebra of symmetric matrices with the Jordan product, over a field F of characteristic not 2, 3 and 5 are given, for n≥3. It is proved that there exists, up to isomorphisms, only one irreducible module which is not Jordan. It is also shown that every finite dimensional unital commutative power associative module for this algebra is completely reducible.

Recently V. H. López Solís and I. Shestakov [9] solved an old problem by N.Jacobson [2] on describing of unital alternative algebras containing the matrix 2× 2 algebra M_2 as a unital subalgebra. Here we give another description of M_2 -algebras via the 6-dimensional alternative superalgebra B(4, 2) and an auxiliar Z_2 -graded algebra . It occurs that the category of alternative M_2 -algebras is isomorphic to the category of -algebras. We describe also the free -algebras and construct their bases.

It is proved that for any natural number $n$ the subalgebra of a free finitely generated alternative algebra generated by all the words on generators whose length is a multiple of $n$ (the Veronese $n$-subalgebra), is finitely generated.

Yu. A. Medvedev [Algebra and Logic, 19, No. 3, 191-201 (1980)] constructed an example of alternative algebra that he used to prove that a certain variety of alternative algebras possess the non-Specht property over a field of characteristic 2. Though his result concerned the characteristic 2 case, the example was claimed to be alternative over an arbitrary field, and it was later used by V. T. Filippov in a series of papers. Unfortunately, Medvedev's example is in fact not alternative in any characteristic. Therefore, whether the variety considered by Medvedev has the non-Specht property is still not clear. Moreover, the results of Filippov's papers, in which Medvedev's example was used, also become questionable. We construct new examples and employ them to prove that the results of Filippov remain true.