Related papers: Algebras, dialgebras, and polynomial identities

Many years ago, Rota proposed a program on determining algebraic identities that can be satisfied by linear operators. After an extended period of dormant, progress on this program picked up speed in recent years, thanks to perspectives…

The aim of this paper is to extend the notion of bialgebra for Leibniz algebras (and Lie algebras) to $3$-Leibniz algebras (and $3$-Lie algebras) by use of the cohomology complex of $3$-Leibniz algebras. Also, some theorems about Leibniz…

Based on the relation between the notions of Lie triple systems and Jordan algebras, we introduce the $n$-ary Jordan algebras,an $n$-ary generalization of Jordan algebras obtained via the generalization of the following property $\left[…

We set out the general theory of ``Beck modules'' in a variety of algebras and describe them as modules over suitable ``universal enveloping'' unital associative algebras. We develop a theory of ``noncommutative partial differentiation'' to…

For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional…

We put forward a definition for spectral triples and algebraic backgrounds based on Jordan coordinate algebras. We also propose natural and gauge-invariant bosonic configuration spaces of fluctuated Dirac operators and compute them for…

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 design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong…

The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg algebras and pre-Lie algebras. We determine the associated quadratic operads and their dual…

We study a noncommutative generalization of Jordan algebras called Jordan dialgebras. These are algebras that satisfy the identities $[x_1 x_2]x_3= 0$, $(x_1^2,x_2,x_3)=2(x_1,x_2,x_1x_3)$, $x_1(x_1^2 x_2)=x_1^2(x_1 x_2)$; they are related…

Let $(A,\mu)$ be a nonassociative algebra over a field of characteristic zero. The polarization process allows us to associate two other algebras, and this correspondence is one-one, one commutative, the other anti-commutative. Assume that…

In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.

This paper's central theme is to prove the existence of an n-algebra whose multiplication cannot be expressed employing any binary operation. Furthermore, to prove if two algebras are not isomorphic, this property does not hold for…

In this article we shows some results about algebra with the group of units having special polynomial identity.

Polylogrithmic functions, such as the logarithm or dilogarithm, satisfy a number of algebraic identities. For the logarithm, all the identities follow from the product rule. For the dilogarithm and higher-weight classical polylogarithms,…

We present an elementary approach in characterizing Lie polynomials in the generators $A,B$ of an algebra with a defining relation that is in the form of a deformed or twisted commutation relation $AB=\sigma(BA)$ where the deformation or…

We consider finite-dimensional complex Lie algebras. We generalize the concept of Lie derivations via certain complex parameters and obtain various Lie and Jordan operator algebras as well as two one-parametric sets of linear operators.…

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 introduce the notion of a generalized representation of a Jordan algebra with unit. The greneralized representation has the following properties: (1) Usual representations and Jacobson representations correspond to special cases of…

We give a review of recent works for non-associative algebras, especially Lie algebras satisfying the triality relation. They are also intimately related to S_4 (symmetric group of 4-objects) symmetry of the Lie algebras.