Avelino Suazo

Commutative Power-Associative Nilalgebras of Nilindex 5

In this paper we study the structure of commutative power-associative nilalgebras of dimension 8 and nilindex ≤ 5 over a field of characteristic different from 2, 3 and 5. We prove that every algebra in this class verifies the identities x4y = 0 and x(x(x(x(xy)))) = 0. In particular, we finish the proof of the Albert’s problem [0] in the following case: every commutative power-associative nilalgebra of dimension ≤ 8 over a field of characteristic ≠ 2, 3 and 5 is solvable. The solvability of these algebras for dimension 4, 5 and 6 were proved in [0], [0] and [0] respectively.

SOLVABILITY OF COMMUTATIVE POWER-ASSOCIATIVE NILALGEBRAS OF NILINDEX 4 AND DIMENSION

∈A.In the following a greek letter indicates anelement of the field K.In [8],D. Suttles constructs (as a counterexample to a conjecture due toA. A. Albert) a commutative power-associative nilalgebra of nilindex 4 anddimension 5,which is solvable and is not nilpotent. In [3] (Theorem 3.3), weprove that this algebra is the unique commutative power-associative nilalge-bra of nilindex 4 and dimension 5,which is not Jordan algebra. At presentthere exists the following conjecture: Any finite-dimensional commutativepower-associative algebra is solvable. The solvability of these algebras fordimension 4 , 5 and 6 , are proved in [5],[3] and [2] respectively.From Theorem 2 of [4] and [6] we obtain the following result:Theorem 1.1 : If Ais a commutative power-associative nilalgebra ofnilindex nwith dimension ≤n+ 2 and the characteristic is zero or ≥n,

JORDAN NILALGEBRAS OF NILINDEX N AND DIMENSION N+1

ABSTRACT It is known the classification of Jordan nilalgebras of nilindex n and dimension n with . In this work we describe Jordan nilalgebras of nilindex n and dimension . We also give a description of commutative power-associative nilalgebras of dimension 5.

The multiplication algebra of a bernstein algebra: basic results

We introduce the multiplication algebra of a Bernstein algebra, establish its Peirce decomposition relative to an idempotent of A and state some basic properties of this algebra of endomorphtsms

On Plenary Train Algebras of Rank 4

The existence of idempotent elements in plenary train algebras of rank greater than 3, is an open problem to be solved. J. Carlos Gutierrezs results on plenary train algebras in Gutierrez (2000) are based on the underlying assumption of the existence of an idempotent. In this article we study conditions on the scalars defining a plenary train algebra of rank 4 to assure the existence of such an idempotent.

On the multiplication algebra of a bernstein algebra

This paper deals with the way in which some restrictions on the structure of Bernstein algebras and on the multiplication algebra of a Bernstein algebra interrelate, the first class of restrictions being expressed as the vanishing of some products between Peirce subspaces, and the second by bounds on the dimension of the multiplication algebra (or some of its key subspaces). The course of this enquiry leads to the introduction in Section 4 of some new numerical invariants of Bernstein algebras, namely, the dimensions of the subspaces .

THE MULTIPLICATION ALGEBRA OF WEIGHTED ALGEBRAS OF DEGREE 4

ABSTRACT In a previous paper,[8] the authors show that there are two main classes of commutative baric -algebras satisfying an equation of the form , where are scalars in the base field . They appear as references (1) and (2) in the body of this paper. Some properties of these classes of algebras are also established in that paper. In the present one, we study their multiplication algebras aiming for a more detailed knowledge of those algebras. Some of the results generalize propositions appearing in[8].