Related papers: Relatively free algebras with the identity x^3=0

Theorem 7 in Ref. [Linear Algebra Appl., 430, 1-6, (2009)] states sufficient conditions to determine whether a pair generates the algebra of 3x3 matrices over an algebraically closed field of characteristic zero. In that case, an explicit…

The algebra of invariants of several 3 x 3 matrices under the action of the orthogonal group by simultaneous conjugation is considered over a field of characteristic different from two. The maximal degree of elements of minimal system of…

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent…

There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group on three elements. The first…

We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here,…

The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are…

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric.…

It is well known that a finite-dimensional Lie algebra over a field of characteristic zero is simple exactly when its derivation algebra is simple. In this paper we characterize those Lie algebras of arbitrary dimension over any field that…

We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n+k generators and k relations and has an n-element system of generators, then this algebra is a free…

Classification, up to isomorphism, of algebras from a non-empty subset of the variety of $n$- dimensional algebras is presented. It is shown that these algebras have only trivial automorphism and if the basic field is algebraically closed…

Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = \{x_1, x_2, \ldots\}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle…

We classify (possibly non commutative) algebras of low rank over a domain R. We first review results for algebras of rank 2 and for finite-dimensional division algebras over the real numbers. These results motivate us to consider which…

We show that an arbitrary algebra ${ A}$, (of arbitrary dimension, over an arbitrary base field and any identity is not suppose for the product), is semisimple if and only if it has zero annihilator and admits a semi-division linear basis.…

We study axial algebras, that is, commutative non-associative algebras generated by idempotents whose adjoint actions are semisimple and obey a fusion law. Considering the case, when said adjoint actions having $3$ eigenvalues and the…

Any variety of classical algebras has a so-called conformal counterpart. For example one can consider Lie conformal or associative conformal algebras. Lie conformal algebras are closely related to vertex algebras. We define free objects in…

Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. The following result is proved: If K is a locally maximal compact set of zeroes of X and the…

This note is to show the effectiveness of the notion of pseudoalgebra in the theory of conformal algebras. We adduce very simple construction of free associative conformal algebra and find its linear basis. There is no any new result but we…

We compute the orders of free commutative Moufand loops of exponent 3 with $n\leq 7$ free generators and find embeddings of such loops into a loop of invertible elements of the free commutative alternative algebra with identity $x^3=0$.

We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such…

By definition the identities $[x_1,x_2]+[x_2,x_1]=0$ and $[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0$ hold in any Lie algebra. It is easy to check that the identity $[x_1,x_2,x_3,x_4]+[x_2,x_1,x_4,x_3]+[x_3,x_4,x_1,x_2]+[x_4,x_3,x_2,x_1] =…