Evolution algebras with one-dimensional square

Abstract

Evolution algebras with one dimensional square are classified using the theory of inner product spaces. More precisely, for $A$ an evolution algebra with $\dim (A^2)=1$ and $a$ a generator of $A^2$, the product of $A$ is given by $xy=⟨x,y⟩a$. Three broad classes of algebras are obtained: (1) $a\in Ann(A)$; (2) $a\notin \text{Ann}(A)$ and $a$ is isotropic relative to $⟨\cdot ,\cdot ⟩$; (3) $a\notin \text{Ann}(A)$ and $a$ is anisotropic relative to $⟨\cdot ,\cdot ⟩$.