The Jacobson radical of an evolution algebra

  • M. Victoria Velasco

    Universidad de Granada, Spain
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In this paper we characterize the maximal modular ideals of an evolution algebra AA in order to describe its Jacobson radical, Rad(A)(A). We characterize semisimple evolution algebras (i.e. those such that Rad(A)={0})(A)=\{0\}) as well as radical ones. We introduce two elemental notions of spectrum of an element aa in an evolution algebra AA, namely the spectrum σA(a)\sigma^{A}(a) and the mm-spectrum σmA(a)\sigma _{m}^{A}(a) (they coincide for associative algebras, but in general σA(a)σmA(a),\sigma ^{A}(a)\subseteq \sigma_{m}^{A}(a), and we show examples where the inclusion is strict). We prove that they are non-empty and describe σA(a)\sigma ^{A}(a) and σmA(a)\sigma _{m}^{A}(a) in terms of the eigenvalues of a suitable matrix related with the structure constants matrix of A.A. We say AA is mm-semisimple (respectively spectrally semisimple) if zero is the unique ideal contained into the set of aa in AA such that σmA(a)={0}\sigma _{m}^{A}(a)=\{0\} (respectively σA(a)={0}\sigma^{A}(a)=\{0\}). In contrast to the associative case (where the notions of semisimplicity, spectrally semisimplicty and mm-semisimplicity are equivalent) we show examples of mm-semisimple evolution algebras AA that, nevertheless, are radical algebras (i.e Rad(A)=A(A)=A). Also some theorems about automatic continuity of homomorphisms will be considered.

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M. Victoria Velasco, The Jacobson radical of an evolution algebra. J. Spectr. Theory 9 (2019), no. 2, pp. 601–634

DOI 10.4171/JST/257