A historical perspective of Tian’s evolution algebras

We will study evolution algebras A that are free modules of dimension two over domains. We start by making some general considerations about algebras over domains: They are sandwiched between a certain essential D-submodule and its scalar extension over the field of fractions of the domain. We introduce the notion of quasiperfect algebras and we characterize the perfect and quasiperfect evolution algebras in terms of the determinant of its structure matrix. We classify the two-dimensional perfect evolution algebras over domains parametrizing the isomorphism classes by a convenient moduli set.

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“…For instance, consider the group lim →2 D × /(D × ) [3] . Since 2 and 3 are coprime, the transition homomorphisms (•) 2 are isomorphisms.…”

Section: Direct Limitsmentioning

confidence: 99%

“…In [2], the classification of the evolution algebras of dimension two over arbitrary fields is provided. In [3], we can find more information about the evolution of the research in the field of evolution algebras defined in [10]. Our contribution in this work is the study of two-dimensional perfect evolution algebras over domains.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional perfect evolution algebras over domains

2023

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“…Evolution algebras allowed introduce useful algebraic techniques and methods into the investigation of some digraphs because such kind of algebras and weighted digraphs can be canonically identified [13,28] However, a full classification of nilpotent evolution algebras is far from its solution. For review on recent development on evolution algebras, we refer the reader to [11].…”

Section: Introductionmentioning

confidence: 99%

On $\exp(Der(E))$ of nilpotent evolution algebras

Mukhamedov¹,

Khakimov²,

Qaralleh³

2022

Preprint

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“…Later on, evolution algebras are used to model non-Mendelian genetics laws [9][10][11][12][13]. Moreover, these algebras are tightly connected with group theory, the theory of knots, dynamic systems, Markov processes, and graph theory [14][15][16][17][18] and [43,44,46]. Evolution algebras introduced proper algebraic techniques and methods for investigating some digraphs because such algebras and weighted digraphs can be canonically identified [7,19].…”

Section: Introductionmentioning

confidence: 99%

Tensor product of finite dimensional nilpotent evolution algebras

izzat

2022

Preprint

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This paper investigates the tensor product of a finite-dimensional nilpotent evolution algebra. Some properties that translate from tensor products to factors and vice versa have been investigated, including the index of nilpotency and annihilator. The index of nilpotency of the tensor product of two nilpotent evolution algebras with different indexes of nilpotency is determined. Moreover, we investigate the tensorially decomposable of the 4-dimensional nilpotent evolution algebra. In addition, the decomposable nilpotent evolution algebra with the maximal nilindex of nilpotency has been carried out.

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A historical perspective of Tian’s evolution algebras

Cited by 12 publications

References 82 publications

Two-dimensional perfect evolution algebras over domains

Two-dimensional perfect evolution algebras over domains

On $\exp(Der(E))$ of nilpotent evolution algebras

Tensor product of finite dimensional nilpotent evolution algebras

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