Train algebras of rank 3 with finiteness conditions

Zitan, Fouad

doi:10.1016/j.laa.2009.04.006

Cited by 5 publications

(4 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A special case of locally nilpotent algebras are commutative nilalgebras of nilindex at most 3, whose natural examples are barideals of Bernstein-Jordan algebras and train algebras of rank 3 (see, for instance, [31]). They satisfy the identity x 3 = 0 and so also the Jacobi identity (xy)z + (yz)x + (zx)y = 0.…”

Section: Bernstein Algebras and Locally Nilpotent Nonassociative Alge...mentioning

confidence: 99%

See 1 more Smart Citation

On Bernstein algebras satisfying chain conditions II

Boudi

Zitan

2019

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

Following a previous work with Boudi, we continue to investigate Bernstein algebras satisfying chain conditions. First, it is shown that a Bernstein algebra (A, ω) with ascending or descending chain condition on subalgebras is finitedimensional. We also prove that A is Noetherian (Artinian) if and only if its barideal N = ker(ω) 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.

show abstract

Section: Bernstein Algebras and Locally Nilpotent Nonassociative Alge...mentioning

confidence: 99%

“…Now, if N 2 = 0, then the former case shows that N/N 2 is finite-dimensional. Finally, we apply [31,Lemma 3.2] stating that any commutative algebra N satisfying the identity x 3 = 0 and such that N/N 2 is finitedimensional must be finite-dimensional.…”

Section: Corollary 44mentioning

confidence: 99%

On Bernstein algebras satisfying chain conditions II

Boudi

Zitan

2019

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…Natural examples of Jacobi-Jordan algebras are barideals of Bernstein-Jordan algebras, which are special instances of train algebras of rank 3 (see [17], [18] and the references therein.) However it is not known whether every Jacobi-Jordan algebra arises as a barideal of some Bernstein-Jordan algebra, see [18], section 3. The name of Jordan in the definition of a Jacobi-Jordan algebra is justified by the following observation.…”

Section: Jacobi-jordan Algebrasmentioning

confidence: 99%

Jacobi-Jordan algebras

Burde,

Fialowski

2014

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently, the cohomology and formal 1-parameter deformations of JJ algebras were developed in [4]. The theory of JJ algebras has many applications not only in the study of modern mathematics and theoretical physics [17,19] but also in the study of Bernstein-Jordan algebras and train algebras [18,24,25,29]. For example, JJ algebras are introduced as examples of the more popular and well-referenced Jordan algebras to achieve an axiomatization for the algebra of observables in quantum mechanics [17].…”

Section: Introductionmentioning

confidence: 99%

On cohomology and deformations of Jacobi-Jordan algebras

Yang¹

2022

Preprint

View full text Add to dashboard Cite

In this paper we study cohomology and deformations of Jacobi-Jordan algebras. We develop their formal deformation theory. In particular, we introduce a method to construct a versal deformation for a given Jacobi-Jordan algebra, which can induce all deformations and is unique on the infinitesimal level. We construct formal 1-parameter deformations of Jacobi-Jordan algebras up to dimension 5 and versal deformations for 3-dimensional Jacobi-Jordan algebras.

show abstract

Train algebras of rank 3 with finiteness conditions

Cited by 5 publications

References 22 publications

On Bernstein algebras satisfying chain conditions II

On Bernstein algebras satisfying chain conditions II

Jacobi-Jordan algebras

On cohomology and deformations of Jacobi-Jordan algebras

Contact Info

Product

Resources

About