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Primitive axial algebras of Monster type are a class of non-associative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2-generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra H , or its characteristic 5 cover H ˆ. In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of H and H ˆ. We proceed in a unified way, by defining a cover of H in all characteristics. This cover has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic 5. [ABSTRACT FROM AUTHOR]

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]

In [15] , Yabe gives an almost complete classification of primitive symmetric 2-generated axial algebras of Monster type, leaving open the case of algebras in characteristic 5 and axial dimension larger than 5. In this note, we complete the classification of these algebras, by constructing a new infinite-dimensional primitive 2-generated symmetric axial algebra H ˆ of Monster type (2 , 1 2) over a field of characteristic 5, and showing all the cases not falling into Yabe's classification are factors of H ˆ. [ABSTRACT FROM AUTHOR]

In this paper we will explain an interesting phenomenon which occurs in general nonassociative algebras. More precisely, we establish that any finite-dimensional commutative nonassociative algebra over a field satisfying an identity always contains 1 2 in its Peirce spectrum. We also show that the corresponding 1 2 -Peirce eigenspace satisfies the Jordan type fusion laws. The present approach is based on an explicit representation of the Peirce polynomial for an arbitrary algebra identity. To work with fusion rules, we develop the concept of the Peirce symbol and show that it can be explicitly determined for a wide class of algebras. We also illustrate our approach by further applications to genetic algebras and algebra of minimal cones (the so-called Hsiang 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]

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]