• Camacho, L.M. (Dpto. Matematica Aplicada I Universidad de Sevilla) ;
  • Gomez, J.R. (Universidad de Sevilla) ;
  • Omirov, B.A. (Institute of Mathematics and Information Technologies) ;
  • Turdibaev, R.M. (Department of Mathematics National University of Uzbekistan)
  • Received : 2011.10.07
  • Published : 2013.09.30


The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criterium of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent $n$-dimensional complex evolution algebras the possible maximal nilpotency index is $1+2^{n-1}$.


  1. J. M. Casas, M. Ladra, B. A. Omirov, and U. A. Rozikov, On evolution algebras, Algebra Colloquium. arXiv:1004.1050v1 (to appear).
  2. J. M. Casas, M. Ladra, and U. A. Rozikov, A chain of evolution, Linear Algebra Appl. 435 (2011), no. 4, 852-870.
  3. I. M. H. Etherington, Genetic algebras, Proc. Roy. Soc. Edinburgh 59 (1939), 242-258.
  4. I. M. H. Etherington, Duplication of linear algebras, Proc. Edinburgh Math. Soc. 2 (1941), no. 6, 222-230.
  5. I. M. H. Etherington, Non-associative algebra and the symbolism of genetics, Proc. Roy. Soc. Edin-burgh. Sect. B. 61 (1941), 24-42.
  6. V. Glivenkov, Algebra Mendelienne comptes rendus, (Doklady) de l'Acad. des Sci. de I'URSS 4 (1936), no. 13, 385-386 (in Russian).
  7. V. A. Kostitzin, Sur les coefficients mendeliens d'heredite, Comptes rendus de l'Acad. des Sci. 206 (1938), 883-885 (in French).
  8. Y. I. Lyubich, Mathematical Structures in Population Genetics, Biomathematics, 22, Springer-Verlag, Berlin, 1992.
  9. E. Mossel, Reconstruction on trees: beating the second eigenvalue, Ann. Appl. Probab. 11 (2001), no. 1, 285-300.
  10. A. Serebrowsky, On the properties of the Mendelian equations, Doklady A.N.SSSR 2 (1934), 33-36 (in Russian).
  11. J. P. Tian, Evolution algebras and their applications, Lecture Notes in Mathematics, 1921, Springer-Verlag, Berlin, 2008.
  12. J. P. Tian and P. Vojtechovsky, Mathematical concepts of evolution algebras in non-Mendelian genetics, Quasigroups Related Systems 14 (2006), no. 1, 111-122.

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