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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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