Bernstein Algebras Research Articles

Solution of the Bernstein Problem in the Non-regular Case

This paper describes explicitly all non-regular non-degenerate simplicial stochastic Bernstein algebras. Consequently, the Bernstein problem (S. N. Bernstein, Science Ukraine1 (1992), 14–19) in the non-degenerate case is settled, since the regular and exceptional cases have already been examined by Y. Lyubich in the 1970s. Notice that from this result it is possible to explicitly describe every non-regular simplicial algebra (A,Δ) since the simplicial subalgebra (〈supp(A2)〉, [supp(A2)]) is non-degenerate. Also we prove the relevant Lyubich's conjecture (1992, Yu I. Lyubich, Biomathematics22, 232) in an affirmative way: all normal simplicial stochastic Bernstein algebras are regular.

Nuclear Bernstein Algebras with a Stochastic Basis

The aim of this paper is to prove a conjecture due to Y. Lyubich, according to which a nuclear Bernstein algebra with a stochastic basis is regular.

Invariance of dimension of p-subspaces in bernstein algebras*

The purpose of this paper is to prove that some vector subspaces, called p-subspaces, obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the idempotent used to decompose A. In particular, for Bernstein-Jordan algebras, this fact is true for every such subspace and this implies that all p-subspaces of a Bernstein algebra, contained in V, for A = Ke + U + V, have invariant dimension. Finally we classify all p-subspaces of degree ≥ 3, contained in U, in a Bernstein algebra A, relative to the invariance (or not) of dimension.

Polynomial Identities of Bernstein Algebras of Small Dimension

Let (B,ω) be a Bernstein algebra over a field of zero characteristic andB=eF+̇Ue+̇Zea Peirce decomposition. Regular algebras (that is Bernstein algebras in whichUeZe=Z2e=0) and exclusive algebras (U2e=0) satisfyingUeZe=0 will be considered. Generators of the ideal of polynomial identities for these two classes of Bernstein algebras will be given.

Baric, bernstein and jordan algebras

The aim of this paper is the study of relations between some special classes of baric algebras, as Jordan algebras, Bernstein or second order Bernstein algebras, train algebras and power-associative algebras. It will be proved that the train equation of a Bernstein algebra that is train of rank r is unique, what leads to a characterization of these algebras similar to the well known characterization of a Jordan-Bernstein algebra in terms of a Peirce decomposition.

A radical splitting theorem for bernstein algebras

We introduce the notion of radical in Bernstein algebras and prove a splitting theorem, that is an analog of a well-known statement in classical varieties of algebras. Note that in this situation Bernstein algebras are more similar to solvable Lie and Malcev algebras (see [4], [6]) than to associative, Jordan or Binary Lie ones.Throughout the paper all algebras and vector spaces are finite dimensional over an algebraically closed field k of characteristic 0.

Structure of bernstein populations of type (3, n − 3)

This paper describes explicitly all simplicial stochastic nonregular nonexceptional Bernstein algebras of type (3, n − 3). Consequently, the Bernstein problem of type (3, n − 3) is settled, since the regular and exceptional cases have already been done.

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 .

Higher-order Bernstein algebras given by symmetric bilinear forms

Let ( A, ω) be a kth-order Bernstein algebra and let N be the kernel of ω. This article studies the structure of such algebras in which N 2 has dimension one. The algebras are of two types, I and II, according as N 2 ⊆ U or N 2 ⊈ U. A characterization of the algebras of type I is given. Power associative kth-order Bernstein algebras with dim N 2 = 1 are then considered: they turn out to be Bernstein algebras of at most second order, and multiplication tables for these algebras over the real field are given. Finally, second-order Bernstein algebras of type II are examined and a structure theorem for them is obtained.

Shape identities in bernstein algebras

We study the shape identities arising in the theory of Bernstein algebras. We determine all shape identities of minimal degree for two subclasses of Bernstein algebras, namely, normal Bernstein algebras and exceptional Bernstein algebras.

On nilpotency of the barideal of a bernstein algebra

It is given a new bound for the index of nilpotency of the barideal of a finited generated Bernstein algebra. Also, it is found the index of solvability of the barideal of a Bernstein algebra and a bound of nilpotency for the square of the barideal.

A class of exceptional bernstein algebras associated to graphs1

Given a simple graph G without loops and a field F of characteristic not 2 wr construct an exceptional Bernstein algebra F(G)over F and show that F(G)is indecomposable if and only if G is connected. The duplicate of F(G) is a Bernstein Jordan algebra whose type is deter- mined by the number of points and lines of G. We comment briefly how to extend the theory to the case of multigraphs.

Automorphisms in a class of jordan bernstein algebras of order 2.

Automorphisms in a class of jordan bernstein algebras of order 2.

The Bernstein Problem in Dimension 6

The solution of the Bernstein problem in the regular and exceptional cases, in all dimensionsn, was made by Yu. Lyubich. A. Grishkov proved that there are no nonregular nonexceptional nuclear Bernstein algebras of type (4,2) with stochastic realization and therefore the Bernstein problem of type (4,2) was completely solved by the present author (J. Algebra, to appear). The aim of this paper is to describe explicitly all simplicial stochastic nonexceptional nonregular Bernstein algebras of type (3,3). Since every nonregular nonexceptional Bernstein algebra of dimension 6 is either of type (4,2) or of type (3,3), the Bernstein problem in dimension 6 is completely solved in this paper.

On regular bernstein algebras

In this article we prove a matricial equivalence to the classification of regular Bernstein algebras. On the other hand, we describe all regular nuclear Bernstein algebras with stochastic realization of type ( n + 1, 1).

The Bernstein Problem for Type (n−2, 2)

In a recent paper we solved the Bernstein problem in the nonregular nonexceptional case for the type (3,2). The aim of this paper it is to describe explicitly all simplicial stochastic nonexceptional nonregular nonnuclear Bernstein algebras of type (n−2,2). According to the Lyubich conjecture every nuclear Bernstein algebra with stochastic realization is regular. Consequently, this paper would completely solve the Bernstein problem for the type (n−2,2) if the Lyubich conjecture is proved to be true.

Dupliquée d'une train algèbre et d'une algèbre de Bernstein périodique

We show that if A 2 is a train algebra of rank r, then the duplicate of A is train of rank r + 1. Also, if A 2 is a Bernstein algebra of order n and period p, the commutative duplicate of A is Bernstein of order n + 1 and period p.

Second order Bernstein algebras of dimension 4

The aim of this paper is to characterize, up to isomorphism, 4-dimensional second order Bernstein algebras.

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

The Bernstein Problem in Dimension 5

The aim of this paper is to describe explicitly all simplicial stochastic nonexceptional nonregular Bernstein algebras of dimension 5.