Theoretical aspects and applications of the evolution operator of evolution algebras

• Autores: Victor Manuel Gómez Sousa
• Directores de la Tesis: Desamparados Fernández Ternero (dir. tes.), Juan Núñez-Valdés (dir. tes.)
• Lectura: En la Universidad de Sevilla ( España ) en 2024
• Idioma: inglés
• Número de páginas: 118
• Enlaces
  • Tesis en acceso abierto en: Idus
• Resumen

  • The present manuscript deals with different aspects of the theory of evolution algebras, all of them related to the evolution operator. The distribution of the manuscript is detailed below.

    In Chapter 2, we have presented those preliminary concepts and results on evolution algebras that are necessary to develop the following chapters.

    In Chapter 3, we deal with associative evolution algebras, providing a classification up to rearrangement of the basis and later up to isomorphism. In addition, we show that for a non-degenerate evolution algebras $E$, $E$ is associative if and only if $E$ es unitary. Finally, we study the space of derivations of these algebras.

    Next, in Chapter 4, we deal with the problem of the non-uniqueness of the evolution operator (as a cause of the non-uniqueness of the natural basis) and study common properties and relations between these operators.

    Finally, in Chapters 5 and 6, we classify evolution algebras whose evolution operator is an homomorphism or a derivation, respectively, up to permutations in the natural basis. For this purpose, we discriminate between the degenerate and the non-degenerate case. Furthermore, in chapter 5 we study the solvability problem in these evolution algebras, proving that it is equivalent to the nilpotency of the structure matrix.