Bernstein Algebras Research Articles

On Nuclear Jordan-Bernstein Algebras

The aim of this paper is the construction of free nuclear Jordan-Bernstein algebras, that is, the largest nuclear Jordan-Bernstein algebra generated by a set of generators. In this way, it is proved that every nuclear Jordan-Bernstein algebra is a quotient of an adequate free nuclear Jordan-Bernstein algebra. The same construction may be made without the nuclear assumption and free Jordan-Bernstein algebras may be constructed and related to free nuclear Jordan-Bernstein algebras. The nilpotency index of these algebras is studied, which lets us get a good bound for the nilpotency index in every Jordan-Bernstein algebra and in every nuclear Bernstein algebra. Finally, we can apply the obtained results to get some information about inner derivations of Jordan-Bernstein algebras.

Structure des algèbres de Bernstein

The theory of Bernstein's algebras is revisited. Many results concerning the structure of Bernstein algebras (ideals, nilpotency and so on) are stated. For Bernstein-Jordan algebras, we give a definitive formulation of the theorem which characterizes them. Finally, we emphasize the importance of the invariant L (this ideal doesn't depend on the Peirce decomposition of the algebra) in the study of the structure of Bernstein's algebras.

Low Dimensional Bernstein-Jordan Algebras

Bernstein algebras were introduced by Holgate [5] as an algebraic formulation of the problem of classifying the stationary evolution operators in genetics (see [7]). Since then, many authors have contributed to the study of these algebras, and there is a fairly extensive bibliography on the subject. Known results include classification theorems for some types of Bernstein algebras (for instance, Bernstein algebras of dimension less than or equal to four, see [10, 8, 2]) as well as some other structural results (see for instance [1, 5, 9, 4]). In [3], the authors suggested an approach to the study of the structure of Bernstein algebras through two main ideas: direct products and the related concepts of decomposability and indecomposability, and the reduction of the general problem to the study of Bernstein-Jordan algebras. The procedure depended on getting what was called a reduced Bernstein algebra (which is, in particular, a Jordan algebra), and on the description of the indecomposable factors of the algebra so obtained. In this paper we use these ideas to classify Bernstein-Jordan algebras of low dimension. Thus, in Section 3 we describe reduced Bernstein algebras of dimension less than or equal to 5 through their indecomposable factors. We use this information in Section 4 to recover all Bernstein-Jordan algebras of dimension less than or equal to 5 by gluing together reduced algebras and the trivial ideal that, when factored out, creates reduced algebras. After a Section 1 of preliminaries, where we collect several known facts about Bernstein algebras which will be used later, we devote Section 2 to proving some technical lemmas which will ease the computations of Section 3.

On the Structure of Bernstein Algebras

Journal of the London Mathematical SocietyVolume 51, Issue 1 p. 41-52 Notes and Papers On the Structure of Bernstein Algebras Teresa Cortés, Teresa Cortés Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, SpainSearch for more papers by this authorFernando Montaner, Fernando Montaner Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, SpainSearch for more papers by this author Teresa Cortés, Teresa Cortés Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, SpainSearch for more papers by this authorFernando Montaner, Fernando Montaner Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, SpainSearch for more papers by this author First published: February 1995 https://doi.org/10.1112/jlms/51.1.41Citations: 10 Research partially supported by the DGICYT (PS 90-0129). AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume51, Issue1February 1995Pages 41-52 RelatedInformation

Classification of bernstein algebras of type (3,n- 3)

A classification of Bernstein algebras in dimensions n ⩽ 4 has been made by Holgate in [2], however that article contains no classification up to isomorphism, the problem is solved by Lyubich in [4] when K = R or C, and by Cortes [1] in the general case. Also Lyubich has given in [5] a classification of the regular nonexceptional Bernstein algebra of type (3,n−3) and a classification but not up to isomorphism of nonregular nonexceptional Bernstein algebras of type (3,n − 3) when K = C. The aim of this paper it to characterize, up to isomorphism, Bernstein algebras of type(2, n − 2) and nonexceptional of type(3, n −3) over a infinite commutative field K whose characteristic is different from 2.

Corrigendum et addendum: the Frattini subalgebra of a Bernstein algebra

In a previous paper it is supposed that if A is a Bernstein algebra, every maximal subalgebra, M, verifies that dim M = dim A − 1. This is not true in general. Therefore Proposition 2 in this paper is not correct. However other results there, where this assertion was used, are correct but their proofs need some modifications now.

Special Classes of Second-Order Bernstein Algebras

Special Classes of Second-Order Bernstein Algebras

Bernstein algebras with a lattice isomorphism that does not preserve the nucleus

In this paper Bernstein algebras and isomorphisms between the lattices of subalgebras are studied. Since it is not true, in general, that lattice isomorphisms of Bernstein algebras keep the nucleus of the weight homomorphism, the present aim is to study Bernstein algebras having a lattice isomorphism that does not keep the nucleus

Solvability of the ideal of all weight zero elements in bernstein algebras

We use a computer to verify that the ideal N of all weight zero elements of any (not necessarily finite dimensional) Bernstein algebra is solvable of index ≤4. We also use a computer to verify that N 2 is nilpotent of index ≤9. We give three examples of Bernstein algebras which show that various hypotheses like finite dimensionality, finitely generatedA 2 = A, are separately not enough to force N to be nilpotent.

Minimal polynomial identities of Bernstein algebras. A computer approach

Minimal polynomial identities of Bernstein algebras. A computer approach

Orthogonality in WBJ algebras

The characterization of automorphisms of Bernstein algebras is an open problem. We only know some particular results. Previously we have characterized the automorphisms of quasiorthogonal, orthogonal, and strongly orthogonal weak Bernstein-Jordan algebras. In this paper we work on the minimal dimension with respect to quasiorthogonality, orthogonality, and strong orthogonality. We establish some classification theorems.

Modular Bernstein Algebras

In this paper we give a characterization of Bernstein algebras whose lattices of subalgebras are modular. When the ground field is algebraically closed we prove that such algebras must he genetic and give a complete classification up to isomorphism.

Isomorphisms of Bernstein Algebras

Isomorphisms of Bernstein Algebras

Bernstein algebras with zero derivation algebra

In a paper by A. Wörz the dimension of the derivation algebra of a finite dimensional Bernstein algebra is given and Bernstein algebras having a derivation algebra of maximal dimension are characterized as those with product zero in the kernel (trivial Bernstein algebras). This paper considers the opposite case, that is, Bernstein algebras having a zero derivation algebra. Some conditions needed for a Bernstein algebra to have zero derivation algebra are obtained, and some examples are given.

Les pol( n,m)-algèbres: identités polynômiales symétriques dans des algèbres

Symmetric polynomial identities arise in several algebras. In this work we study nonassociative, noncommutative algebras verifying these kinds of identities. The equivalence between these algebras and algebras satisfying particular monomial relations (in which a linear form appears) is proved. We thus present simplified definitions of some classical baric algebras, for instance, Bernstein's algebras.

Associative bilinear forms in some baric algebras

Let ( A, ω) be a K-algebra, e ϵ A a nontrivial idempotent, K an infinite field whose characteristic is different from 2, and A = K e ⊕ U ⊕ V, where U = { x ϵ ker ω| e x = 1 2 x}. If B : A × A → K is an associative bilinear form, we prove the two following main facts: (1) If A is a kth order Bernstein algebra and B is a nondegenerate form, then A is a kth order quasiconstant algebra and the idempotent is unique. (2) If x 3 - (1 + γ)ω( x) x 2 + γω( x) 2 x = 0 is the rank equation of A and if B is nondegenerate, then A is a Jordan algebra. If 0 ≠ γ ≠ 1, then B , the symmetry of B only depends on the symmetry of B | v , and B ( e 1, e 1) = B ( e, e) for any idempotent elements e, e 1 in A.

A note on the lattice definability of bernstein algebras

We prove that for exclusive and normal Bernstein algebras, the type is determined by the lattice of subalgebras. It is also seen that the lattice of subalgebras does not determine an arbitrary Bernstein algebra up to isomorphism.

Quasiisomorphisms of bernstein algebras∗

In this paper the notion of quasiisomorphism of Bernstein algebras is introduced and studied. The aim of the paper is to get a classification of Bernstein algebras more general than the given by isomorphisms. In the last part of the paper the classification, up to quasiisomorphisms, of Bernstein algebras of dimensions 3 and 4 is given and compared with the corresponding classification up to isomorphism.

On continuous time models in genetic and Bernstein algebras

We discuss the long-time behavior of Andreoli's differential equation for genetic algebras and for Bernstein algebras and show convergence to an equilibrium in both cases. For a class of Bernstein algebras this equilibrium is determined explicitly.

The Frattini subalgebra of a Bernstein algebra

Let A be a finite-dimensional Bernstein algebra over a field K with characteristic not 2. Maximal subalgebras of A are studied, and they are determined if A is a genetic algebra. It is also proved that the intersection of all maximal subalgebras of A (the Frattini subalgebra of A) is always an ideal. Finally the structure of Bernstein algebras with Frattini subalgebra equal to zero is described.