SOME PROPERTIES OF EVOLUTION ALGEBRAS

By L. Camacho, J. G'omez, Bakhrom Omirov, R. Turdibaev. 2010

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Abstract. The paper is devoted to the study of finite dimensional com-plex evolution algebras. The class of evolution algebras isomorphic toevolution algebras with Jordan form matrices is described. For finitedimensional complex evolution algebras the criterium of nilpotency is es-tablished in terms of the properties of corresponding matrices. Moreover,it is proved that for nilpotent n-dimensional complex evolution algebrasthe possible maximal nilpotency index is 1 + 2 n−1 . 1. IntroductionIn 20s and 30s of the last century a new object was introduced to math-ematics, which was the product of interactions between Mendelian geneticsand mathematics. Mendel established the basic laws for inheritance, which aresummarized as Mendel’s Law of Segregation and Mendel’s Law of IndependentAssortment. This laws were mathematically formulated by Serebrowsky [10],who was also the first to give an algebraic interpretation of the “ × ” sign,which indicated sexual reproduction. Later Glivenkov [6] used the notion ofMendelian algebras in his work. Also Kostitzin [7] independently introduceda “symbolic multiplication” to express Mendel’s laws. In his several papersEtherington [3]-[5] introduced the formal language of abstract algebra to thestudy of genetics. These algebras, in general, are non-associative.However, in the beginning of the XX century in genetics there were discov-ered several examples of inheritances, where traits do not segregate in accor-dance with Mendel’s laws. In the present day, non-Mendelian genetics is a basiclanguage of molecular genetics. Non-Mendelian inheritance plays an importantrole in several disease processes. Naturally, the question arises: What non-Mendelian genetics offers to mathematics? The evolution algebras, introducedin [12] serve as the answer to this question.The concept of evolution algebras lies between algebras and dynamical sys-tems. Algebraically, evolution algebras are non-associative Banach algebra;dynamically, they represent discrete dynamical systems. Evolution algebras

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Computer Networks and Communications

View Paper: https://www.semanticscholar.org/paper/bfc3e181485b63ccc4e4dbdc7965802971e326e6

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In Algebra and Number Theory

  1. Yolanda Cabrera Casado
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