Communications in Mathematics |

Editors: Cristina Draper & Ivan Kaygorodov

In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.

A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in An− 1 with respect to a slightly different order and prove that this poset is graded.

We describe the images of multilinear polynomials of arbitrary degree evaluated on the 3×3 upper triangular matrix algebra over an infinite field.

In this paper we give the description of some non-strongly nilpotent Leibniz algebras. We pay our attention to the subclass of nilpotent Leibniz algebras, which is called filiform. Note that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three disjoint families. We describe the pre-derivations of filiform Leibniz algebras for the first and second families and determine those algebras in the first two classes of filiform Leibniz algebras that are non-strongly nilpotent.

In this article, we provide an algorithm with Wolfram Mathematica code that gives a unified computational power in classification of finite dimensional nilpotent algebras using Skjelbred-Sund method. To illustrate the code, we obtain new finite dimensional Moufang algebras.

We give the description of Rota–Baxter operators, Reynolds operators, Nijenhuis operators and average operators on 3-dimensional nilpotent associative algebras over ℂ.

Let (A, u) and (B, b) be two pointed sets. Given a family ofthree maps F = {f1 : A → A; f2 : A × A → A; f3 : A × A → B}, thisfamily provides an adequate decomposition of A \ {u} as the orthogonaldisjoint union of well-described F-invariant subsets. This decompositionis applied to the structure theory of graded involutive algebras, gradedquadratic algebras and graded weak H∗-algebras.

In the present paper we prove that every local and 2-local derivation on conservative algebras of 2-dimensional algebras are derivations. Also, we prove that every local and 2-local automorphism on conservative algebras of 2-dimensional algebras are automorphisms.

We connect the theorems of Rentschler [18] and Dixmier [10] onlocally nilpotent derivations and automorphisms of the polynomial ring A0and of the Weyl algebra A1, both over a field of characteristic zero, byestablishing the same type of results for the family of algebrasAh = hx, y | yx − xy = h(x)i,where h is an arbitrary polynomial in x. In the second part of the paper weconsider a field F of prime characteristic and study F[t]-comodule algebrastructures on Ah. We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show howthis subalgebra determines the automorphism group of Ah.

We find examples of polynomials f ∈ D [t; σ, δ] whose eigenring ℰ(f) is a central simple algebra over the field F = C ∩ Fix(σ) ∩ Const(δ).

Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.