Evolution algebras of arbitrary dimension and their decompositions

Abstract

We study evolution algebras of arbitrary dimension. We analyze in deep the notions of evolution subalgebras, ideals and non-degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras. We also prove the existence and unicity of a direct sum decomposition into irreducible components for every non-degenerate evolution algebra. When the algebra is degenerate, the uniqueness cannot be assured.The graph associated to an evolution algebra (relative to a natural basis) will play a fundamental role to describe the structure of the algebra. Concretely, a non-degenerate evolution algebra is irreducible if and only if the graph is connected. Moreover, when the evolution algebra is finite-dimensional, we give a process (called the fragmentation process) to decompose the algebra into irreducible components.2010 Mathematics Subject Classification. Primary 17A60, 05C25.

Power-Associative Evolution Algebras

“…If α = 0, then N = e 1 , v 2 , v 3 , v 5 ⊕ e 4 , w , direct sum of evolution algebras. We deduce that α = 0 and N is isomorphic to N 6,25 (α) : e 2 1 = e 2 + e 3 , e 2 2 = e 5 e 2 3 = −e 5 , e 2 4 = α(e 2 + e 3 ) + e 6 , e 2 5 = 0, e 2 6 = 0 with α ∈ F * and its type is [2,2,2]. We have ann(N ) = e 5 , e 6 ; ann 2 (N ) = e 2 , e 3 , e 5 , e 6 and ann 3 (N ) = N .…”

“…b) For α 42 = 0 and α 52 = 0, we find again case a) by permuting vectors e 4 and e 5 . 1.1.3) Type [1,2,3] leads to α 42 α 52 = 0. We have ann 2 (N ) = v 2 , v 3 , v 6 and ann 3 (N ) = N .…”

“…We have: ann(N ) = v 6 , v 2 , v 3 ∈ ann 2 (N ) and e 1 ∈ ann 3 (N ). Since e 2 4 , e 2 5 ∈ v 2 , v 3 , v 6 then the possible types of N are: [1,4,1], [1,3,2] • dim (U 3 ⊕ U 1 ) 2 = 1 =⇒ α 46 = 0. N ≃ N 6,20 (α, β) : e 2 1 = e 2 + e 3 , e 2 2 = e 6 , e 2 3 = −e 6 , e 2 4 = α(e 2 + e 3 ), e 2 5 = βe 6 , e 2 6 = 0 with α, β ∈ F * .…”

“…In ( [2], p.127), the authors exhibit a necessary condition for an evolution algebra to be powerassociative. They show in particular that the only power-associative evolution algebras are such that a 2 ii = a ii (i = 1, .…”

Evolution algebras, automorphisms, and graphs

“…Evolution algebras were introduced in 2006 by Tian and Vojtechovsky (see [10]) and present many connections with other fields like graph theory, group theory or Markov chains, to mention a few (see Tian's monograph [9]). They have received considerable attention in the last years (see [1] and the references therein).…”