10 research outputs found
On commutative nilalgebras of low dimension
We prove that every commutative non-associative nilalgebra of dimension ≤7, over a field of characteristic zero or sufficiently large is solvable
Graph immersions with parallel cubic form
We consider non-degenerate graph immersions into affine space whose cubic form is parallel with respect to the Levi-Civita
connection of the affine metric. There exists a correspondence between such
graph immersions and pairs , where is an -dimensional real
Jordan algebra and is a non-degenerate trace form on . Every graph
immersion with parallel cubic form can be extended to an affine complete
symmetric space covering the maximal connected component of zero in the set of
quasi-regular elements in the algebra . It is an improper affine hypersphere
if and only if the corresponding Jordan algebra is nilpotent. In this case it
is an affine complete, Euclidean complete graph immersion, with a polynomial as
globally defining function. We classify all such hyperspheres up to dimension
5. As a special case we describe a connection between Cayley hypersurfaces and
polynomial quotient algebras. Our algebraic approach can be used to study also
other classes of hypersurfaces with parallel cubic form.Comment: some proofs have been simplified with respect to the first versio
On left nilalgebras of left nilindex four satisfying an identity of degree four
We extend the concept of commutative nilalgebras to commutative algebras which are not power associative. We shall study commutative algebras A over fields of characteristic ≠ 2, 3 which satisfy the identities x(x(xx)) = 0 and β{x(y(xx)) - x(x(xy))} + γ{y(x(xx)) - x(x(xy))} = 0. In these algebras the multiplication operator was shown to be nilpotent by Correa, Hentzel and Labra [2]. In this paper we prove that for every x ∈ A we have A(A((xx)(xx))) = 0. We prove that there is an ideal I of A satisfying AI = IA = 0 and A/I is power associative. © World Scientific Publishing Company