In this paper we characterize the maximal modular ideals of an evolution algebra in order to describe its Jacobson radical, Rad. We characterize semisimple evolution algebras (i.e. those such that Rad as well as radical ones. We introduce two elemental notions of spectrum of an element in an evolution algebra , namely the spectrum and the -spectrum (they coincide for associative algebras, but in general and we show examples where the inclusion is strict). We prove that they are non-empty and describe and in terms of the eigenvalues of a suitable matrix related with the structure constants matrix of We say is -semisimple (respectively spectrally semisimple) if zero is the unique ideal contained into the set of in such that (respectively ). In contrast to the associative case (where the notions of semisimplicity, spectrally semisimplicty and -semisimplicity are equivalent) we show examples of -semisimple evolution algebras that, nevertheless, are radical algebras (i.e Rad). Also some theorems about automatic continuity of homomorphisms will be considered.
Cite this article
M. Victoria Velasco, The Jacobson radical of an evolution algebra. J. Spectr. Theory 9 (2019), no. 2, pp. 601–634DOI 10.4171/JST/257