Antonio Behn

A class of Locally Nilpotent Commutative Algebras

This paper deals with the variety of commutative non associative algebras satisfying the identity

Nilpotency of Commutative Finitely Generated Algebras Satisfying

L_x^3+ \gamma L_{x^3} = 0

Irreducible Representations of Power-associative Train Algebras

, γ ∈ K. In [3] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ ≠ -1, then any such algebra is locally nilpotent. Our results require characteristic ≠ 2, 3.

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

In this article, we prove the nilpotency of commutative nonassociative finitely generated algebras satisfying an identity of type with α + β ≠ 0. Our result requires characteristic ≠ 2, 3, 5.

On Flexible Algebras Satisfying x(yz) = y(zx)

Train algebras were introduced by Etherington in 1939 as an algebraic framework for treating genetic problems. The aim of this paper is to study the representations and irreducible representations of power-associative train algebras of rank 4. There are three families of such algebras and for two of them we prove that every irreducible representation has dimension one over the ground field. For the third family we give an example of an irreducible representation of dimension three.

Semiprimality and Nilpotency of Nonassociative Rings Satisfying x(yz) = y(zx)

We studied the solvability of the algebra which satisfies the polynomial identity (x 2)2 = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x 2)2 = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim F A ≤ 7.

Examples of Commutative Right-Nilalgebras Over Small Fields

In this paper we study flexible algebras (possibly infinite-dimensional) satisfying the polynomial identity x(yz) = y(zx). We prove that in these algebras, products of five elements are associative and commutative. As a consequence of this, we get that when such an algebra is a nil-algebra of bounded nil-index, it is nilpotent. Furthermore, we obtain optimal bounds for the index of nilpotency. Another consequence that we get is that these algebras are associative when they are semiprime.

The Convergence of Difference Boxes

In this article we study nonassociative rings satisfying the polynomial identity x(yz) = y(zx), which we call “cyclic rings.” We prove that every semiprime cyclic ring is associative and commutative and that every cyclic right-nilring is solvable. Moreover, we find sufficient conditions for the nilpotency of cyclic right-nilrings and apply these results to obtain sufficient conditions for the nilpotency of cyclic right-nilalgebras.

Adapted hyperbolic polygons and symplectic representations for group actions on Riemann surfaces

Correa et al. (2003) proved that any commutative right-nilalgebra of nilindex 4 and dimension 4 is nilpotent in characteristic ≠ 2,3. They did not assume power-associativity. In this article we will further investigate these algebras without the assumption on the dimension and providing examples in those cases that are not covered in the classification concentrating mostly on algebras generated by one element.