Alicia Labra

On the classification of commutative right-nilalgebras of dimension at most four

Gerstenhaber and Myung (1975) classified all commutative, power-associative nilalgebras of dimension 4. In this article we extend Gerstenhaber and Myungs results by giving a classification of commutative right-nilalgebras of right-nilindex four and dimension at most four, without assuming power-associativity. For quadratically closed fields there is, up to isomorphism, a unique such algebra which is not power-associative in dimension 3, and 7 in dimension 4.

On a class of baric algebras

Abstract It is known that baric algebras satisfying the identity ( x 2 ) 2 = w ( x ) x 3 have idempotent elements and every linear form w : A → K is a multiplicative map. We prove that these algebras are Jordan-Bernstein of order 2 and special train algebras. Moreover, as a corollary we obtain that the train equation of these algebras is x 4 − w ( x ) x 3 = 0, and we give examples of baric algebras satisfying x 4 − w ( x ) x 3 = 0 but not satisfying ( x 2 ) 2 = w ( x ) x 3 .

Evolution algebras and graphs

A digraph is attached to any evolution algebra. This graph leads to some new purely algebraic results on this class of algebras and allows for some new natural proofs of known results. Nilpotency of an evolution algebra will be proved to be equivalent to the nonexistence of oriented cycles in the graph. Besides, the automorphism group of any evolution algebra

ON THE NILPOTENCE OF THE MULTIPLICATION OPERATOR IN COMMUTATIVE RIGHT NIL ALGEBRAS

E

On Nilpotency of Generalized Almost-Jordan Right-Nilalgebras

with

On Plenary Train Algebras of Rank 4

E=E^2

ON LEFT NILALGEBRAS OF LEFT NILINDEX FOUR SATISFYING AN IDENTITY OF DEGREE FOUR

will be shown to be always finite.

A class of Locally Nilpotent Commutative Algebras

ABSTRACT We study conditions under which the identity in a commutative nonassociative algebra A implies is nilpotent where is the multiplication operator for all in A. The separate conditions that we found to be sufficient are (1) dimension four or less, (2) any additional non-trivial identity of degree four, or (3) We assume characteristic

ON SOME JORDAN BARIC ALGEBRAS

We study the variety of algebras A over a field of characteristic ≠ 2, 3, 5 satisfying the identities xy=yx and β {((xx)y)x-((yx)x)x} + γ {((xx)x)y-((yx)x)x}=0, where β, γ are scalars. We do not assume power-associativity. We prove that if A admits a non-degenerate trace form, then A is a Jordan algebra. We also prove that if A is finite-dimensional and solvable, then it is nilpotent. We find three conditions, any of which implies that a finite-dimensional right-nilalgebra A is nilpotent.

SOLVABILITY OF COMMUTATIVE RIGHT-NILALGEBRAS SATISFYING (b(aa))a = b((aa)a)

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.