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On nilpotent index and dibaricity of evolution algebras. (English) Zbl 1343.17021
The concept of evolution algebra, popularized under this name by J. P. Tian [Evolution algebras and their applications. Lecture Notes in Mathematics 1921. Berlin: Springer (2008; Zbl 1136.17001)], was first introduced by I. M. H. Etherington [Proc. R. Soc. Edinb., Sect. B, Biol. 61, 24–42 (1941; Zbl 0063.01290), p. 34] to study algebraically the self-fertilization and was used later by P. Holgate [J. Math. Biol. 6, 197–206 (1978; Zbl 0387.92008)] for the partial selfing.
An algebra $$E$$ on a field $$K$$ is an evolution algebra if it admits a basis $$\left(e_{i}\right)$$ such that $$e_{i}e_{j}=\delta_{ij}\sum_{k}a_{ik}e_{k}$$. In a previous work [J. M. Casas et al., Algebra Colloq. 21, No. 2, 331–342 (2014; Zbl 1367.17026)], the authors showed that an $$n$$-dimensional evolution algebra $$E$$ is nil if and only if the canonically associated matrix $$A=\left(a_{ij}\right)$$ of the structural constants of $$E$$ is strictly upper triangular. In the present paper, they prove that an $$n$$-dimensional nilpotent evolution algebra $$E$$ has maximal nilpotent index $$2^{n-1}+1$$ if and only if the matrix of structural constants $$\left(a_{ij}\right)_{1\leq i,j\leq n}$$ is strictly upper triangular and verify the condition $$a_{12}a_{23}\ldots a_{n-1,n}\neq0$$.
From this, they deduce the classification of finite dimensional complex evolution algebras with maximal nilpotent index and for any integer $$0<s<n$$ they give examples of $$n$$-dimensional nilpotent evolution algebras with nilpotent index $$2^{n-s}+1$$ and $$2^{\max\left\{ s-1,n-s\right\} }+1$$. They conclude by showing that nilpotent evolution algebras are not dibaric and they give a necessary and sufficient condition for two-dimensional real evolution algebras to be dibaric.

MSC:
 17D92 Genetic algebras
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References:
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