Bernstein Algebras Research Articles

On the structure and classification of Bernstein algebras

Linear algebra tools are used to give a new approach to the open problem of the classification of Bernstein algebras. We prove that any Bernstein algebra [Formula: see text] is isomorphic to a semidirect product [Formula: see text] associated to a commutative algebra [Formula: see text] such that [Formula: see text], for all [Formula: see text] and an idempotent endomorphism [Formula: see text] of [Formula: see text] satisfying two compatibility conditions. The set of types of [Formula: see text]-dimensional Bernstein algebras is parametrized by an explicitly constructed classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups [Formula: see text].

Evolution algebras satisfying train identity of degree 2 and exponent 3

In this paper, we give the necessary and sufficient conditions for an evolution algebra to satisfy a train identity of degree 2 and exponent 3. We show that this class of algebras is a subclass of the Bernstein algebras of order 2. Then, we study the relations existing between these algebras and the Bernstein algebras as well as the power associative algebras. Moreover we give a classification in dimension ≤ 4 of evolution algebras satisfying strictly a train identity of degree 2 and exponent 3. Finally, we describe the derivations and automorphisms of these algebras.

Bernstein algebras that are algebraic and the Kurosh problem

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of Bernstein train algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train).Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the bar-ideal is nil, the Kurosh problem asks whether a finitely generated Bernstein train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case.On the other hand, by results of Yagzhev the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras.

Crossed products of 4-algebras. Applications

A 4-algebra is a commutative algebra A over a field k such that (a2)2=0, for all a∈A. We have proved recently [22] that 4-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let A be a 4-algebra, E a vector space and π:E→A a surjective linear map with V=Ker(π). All 4-algebra structures on E such that π:E→A is an algebra map are described and classified by a global cohomological object GH2(A,V). Any such 4-algebra is isomorphic to a crossed product V#A and GH2(A,V) is a coproduct, over all 4-algebras structures ⋅V on V, of all non-abelian cohomologies Hnab2(A,(V,⋅V)), which are the classifying objects for all extensions of A by V. Several applications and examples are provided: in particular, GH2(A,k) and GH2(k,V) are explicitly computed and the Galois group Gal(V#A/V) of the extension V↪V#A is described.

On Bernstein algebras satisfying chain conditions II

Following a previous work with Boudi, we continue to investigate Bernstein algebras satisfying chain conditions. First, it is shown that a Bernstein algebra A with ascending or descending chain condition on subalgebras is finite-dimensional. We also prove that A is Noetherian (Artinian) if and only if its barideal is. Next, as a generalization of Jordan and nuclear Bernstein algebras, we study whether a Noetherian (Artinian) Bernstein algebra A with a locally nilpotent barideal N is finite-dimensional. The response is affirmative in the Noetherian case, unlike in the Artinian case. This question is closely related to a result by Zhevlakov on general locally nilpotent nonassociative algebras that are Noetherian, for which we give a new proof. In particular, we derive that a commutative nilalgebra of nilindex 3 which is Noetherian or Artinian is finite-dimensional. Finally, we improve and extend some results of Micali and Ouattara to the Noetherian and Artinian cases.

Derivations and dimensionally nilpotent derivations in Lie triple algebras

In this paper, we first study derivations in non nilpotent Lie triple algebras. We determine the structure of derivation algebra according to whether it admits an idempotent or a pseudo-idempotent. We study the multiplicative structure of non nil dimensionally nilpotent Lie triple algebras. We show that when n=2p+1 the adapted basis coincides with the canonical basis of the gametic algebra G(2p+2,2) or this one obviously associated to a pseudo-idempotent and if n=2p then the algebra is either one of the precedent case or a conservative Bernstein algebra.

Dibaric and evolution algebras in biology

We find conditions on ideals of an algebra under which the algebra is dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the real numbers. We introduce a concept of bq-homomorphism (which is given by two linear maps f, g of the algebra to the set of the real numbers) and show that an algebra is dibaric if and only if it admits a non-zero bq-homomorphism. Using the pair (f, g) we define conservative algebras and establish criteria for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein algebra and an algebra induced by a linear operator are introduced and relations between these algebras are studied. For dibaric algebras we describe a dibaric algebra homomorphism and study their properties by bq-homomorphisms of the dibaric algebras. We apply the results to the (dibaric) evolution algebra of a bisexual population. For this dibaric algebra we describe all possible bq-homomorphisms and find conditions under which the algebra of a bisexual population is induced by a linear operator. Moreover, some properties of dibaric algebra homomorphisms of such algebras are studied.

Identités Multilinéaires de Degré 4 pour les Algèbres de Bernstein et de Mutation Non Commutatives

We give the identities generating the spaces of degree 4 multilinear identities satisfied by the non commutative 0th-order Bernstein algebras and the non commutative mutation algebras. For it, we use a method reducing the search for multilinear identities to the resolution of linear systems.

Algèbres de Bernstein régulières

The aim of this paper is to give some further results on regular Bernstein algebras over fields of characteristic other than two. This notion was introduced by Yu. I. Lyubich, who also began the classification of such objects; after Lyubich, many authors worked on this subject. In particular, we study the connection between regular and indecomposable Bernstein algebras.

Derivations in power-associative algebras

In this paper we investigate derivations of a commutative power-associative algebra. Particular cases of stable and partially stable algebras are inspected. Some attention is paid to the Jordan case. Further results are given. Especially, we show that the core of a $n^{th}$-order Bernstein algebra which is power-associative is a Jordan algebra.

Train algebras of degree 2 and exponent 3

In this paper we investigate the structure of weighted algebras satisfying the equation $(x^3)^2 = \omega(x)^3x^3$, a class of algebras properly containing the class of Bernstein algebras. We give the classification of these algebras in dimension three. Some results about the structure of algebras satisfying the more general equation $(x^n)^2 = \omega(x)^nx^n$, for $n\geq 2$, are also obtained.

Power-associative algebras that are train algebras

We investigate the structure of power-associative algebras that are train algebras. We first show the existence of idempotents, which are all principal and absolutely primitive. We then study the train equation involving the Peirce decomposition. When the algebra is finite-dimensional, it turns out that the dimensions of the Peirce components are invariant and that the upper bounds for their nil-indexes are reached for some idempotent. Further, locally train algebras are shown to be train algebras. We then get a complete description of the set of idempotents by giving their explicit formulas, including several illustrative examples. Some attention is paid to the Jordan case, where we discuss conditions forcing power-associative train algebras to be Jordan algebras. It is also shown that finitely generated Jordan train algebras are finite-dimensional. For a n th-order Bernstein algebra of period p , we prove that power-associativity necessitates p = 1 . In this case, there are 2 n − 1 possible train equations, which are explicitly described.

Polynomial Identities for Bernstein Algebras of Simple Mendelian Inheritance

The Bernstein algebra D n G α of simple Mendelian inheritance is the n-fold duplicate of the gametic algebra G α with basis A 1,…, A α and structure constants . We simplify and generalize results of Bernad et al. and Peresi by showing that every identity for G α follows from Costa's shape identity C, and that every identity for the zygotic algebra DG α follows from the recombination identity R. We use computer algebra to determine the identities of degree ≤7 for the copular algebra D 2 G α.

On Bernstein Algebras Satisfying Chain Conditions

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.

On Homogeneous Weighted Algebras

We extend the concept of the “join” to the case of infinitely many weighted algebras. We study the problem of its uniqueness (up to weighted isomorphism) which gives rise to a natural notion of homogeneous weighted algebras. We show that several classes of weighted algebras coming from genetics are homogeneous and that homogeneity is preserved by duplication. Finally, we examine some well-known weighted algebras satisfying identities, as Bernstein, train, and evolution algebras.

On Bernstein Algebras Satisfying Chain Conditions

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.

The bar-radical in power-associative nth-order Bernstein algebras

In this paper we prove that if \( (A, \omega) \) is a power-associative nth-order Bernstein algebra, then the bar-radical of \( (A, \omega) \) is contained in (bar(A))2. When \( (A, \omega) \) is Jordan, then the bar-radical is (bar(A))2.

Some Properties of The Automorphisms of a Bernstein Algebra

In this short note we show that if A is a nuclear Bernstein algebra then the group of automorphisms of M(A), its multiplication algebra, has a proper subgroup isomorphic to Aut A.

TWO NUMERICAL INVARIANTS FOR BERNSTEIN ALGEBRAS

We deal with two numerical invariants for Bernstein algebras, derived from the study of their multiplication algebras: the dimension of these algebras and the maximum rank of their elements.

On idempotents and isomorphisms of multiplication algebras of bernstein algebras

The purpose of this paper is twofold. First of all we characterize Bernstein algebras having a nilpotent kernel using idempotents of their multiplication algebras. Secondly, we describe some properties of Bernstein algebras which are preserved by isomorphisms of their multiplication algebras.