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Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (α , β) , over a field of characteristic other than 2, has dimension at most 8 if α ∉ { 2 β , 4 β } . In this note, we show that Rehren's bound does not hold in the case α = 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2 , 1 2) over an arbitrary field F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features. [ABSTRACT FROM AUTHOR]

We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras. [ABSTRACT FROM AUTHOR]

Through this paper, we show that the criteria for real evolution algebra to be a baric algebra can be extended to any evolution algebra over a commutative field of characteristic 6 ≠ = 2. Then we prove that an evolution algebra E is a train algebra of rank r + 1 if and only if the kernel of its weight function is nil of nil-index r > 1. We also study special train evolution algebra and characterize idempotents, power-associativity and automorphism in evolution train algebra. Finally we classify up to dimension 5, indecomposable evolution nil-algebra of nil-index 4 that are not power-associative. [ABSTRACT FROM AUTHOR]

We study the ideal of polynomial identities of a single indeterminate satisfied by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and sufficient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities satified by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8. [ABSTRACT FROM AUTHOR]

We study the relationship of backcrossing algebras with mutation algebras and algebras satisfyingω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identityfthere is a backcrossing algebra satisfyingf. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results. [ABSTRACT FROM AUTHOR]

In this paper we deal with the Wedderburn b-decomposition for alternative baric algebras. [ABSTRACT FROM AUTHOR]

In this paper we work with the variety of commutative algebras satisfying the identity β((x²y)x - ((yx)x)x)+γ(x³y - ((yx)x)x) = 0, where β, γ are scalars. They are called generalized almost-Jordan algebras. We prove that this variety is equivalent to the variety of commutative algebras satisfying (3β + γ)(Gy(x, z, t) - Gx(y, z, t)) + (β + 3γ)(J(x, z, t)y - J(y, z, t)x) = 0, for all x, y, z, t ∈ A, where J(x, y, z) = (xy)z+(yz)x+(zx)y and Gx(y, z, t) = (yz, x, t)+(yt, x, z)+ (zt, x, y). Moreover, we prove that if A is a commutative algebra, then J(x, z, t)y = J (y, z, t)x, for all x, y, z, t ∈ A, if and only if A is a generalized almost-Jordan algebra for β = 1 and γ = -3, that is, A satisfies the identity (x²y)x + 2 ((yx)x) x - 3x³y = 0 and we study this identity. We also prove that if A is a commutative algebra, then Gy(x, z, t) = Gx(y, z, t), for all x, y, z, t ∈ A, if and only if A is an almost-Jordan or a Lie Triple algebra. [ABSTRACT FROM AUTHOR]

We define and study the properties of baric algebras defined by ω-polynomial identities, called ω-PI algebras. We show that every finite dimensional baric algebra is ω-PI. Next we introduce the study of ω-PI algebras of degree 4 with one indeterminate. By gametization we reduce their study to four types. We study the first type corresponding to algebras that are neither barycentric nor invariant by gametization. We show that the variety of these algebras is partitioned into only two subvarieties admiting an unique ponderation, an idempotent, and verifying ω-monomial identities of degrees > 4. [ABSTRACT FROM AUTHOR]

In this article we study the algebras satisfying the ω-polynomial identity x2x2 - x4 = δ(x2 - x) with δ ≠ 0 but do not satisfy any monomial identities of degree ≤4. We show that there exist such algebras for all δ ≠ 0 and they have a unique baric function. We give conditions for the existence of idempotents of weight 0 or 1, and we construct the three Peirce decompositions associated to these idempotent elements. [ABSTRACT FROM AUTHOR]

In this article we prove that, if (U, ω) is a finite dimensional baric algebra of (γ, δ) type over a field F of characteristic ≠ 2,3,5 such that γ2 - δ2 + δ = 1 and δ ≠ 0,1, then rad(U) = R(U) ∩ (bar(U))2, where R(U) is the nilradical (maximal nil ideal) of U. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if ( U, w) is a finite dimensional Jordan baric algebra such that $$\text{rad}(U)\subseteq(\text{bar}(U))^3$$ then, $$\text{rad}(U)=R(U)\cap(\text{bar}(U))^3$$ , where R( U) is the nilradical (maximal nil ideal) of U. We also give conditions so that $$\text{rad}(U)\subseteq (\text{bar}(U))^3$$ and an example showing that such conditions are necessary. [ABSTRACT FROM AUTHOR]