Commutative algebras with one-dimensional square

Abstract

In this paper, we classify and study commutative algebras having a one-dimensional square. In finite dimension (see Theorem 3.9) besides some cases (which are all associative and nilpotent with nilpotency index 3), the algebras with zero annihilator are either of symplectic type (appearing only in characteristic 2), or evolution algebras. In infinite dimension, ruling out the associative case, we prove that our algebras are either of symplectic type or evolution algebras provided some technical conditions are satisfied (see Theorem 3.12). Our main tool is the theory of inner product spaces and quadratic forms. More precisely, if A denotes an evolution algebra with dim (A2)=1 and a a generator of A2, then A admits an inner product {·, · } such that the product of A is given by xy = {x, y}a. There are three classes to consider:

(1) a ∈ Ann(A); (2) a ∉ Ann(A) and a is isotropic relative to {·, ·}; (3) a ∉ Ann(A) and a is anisotropic relative to {·, ·}

The isomorphism problem among these objects is investigated. For some of these algebras, we have also determined the existence of faithful associative representations in certain Clifford algebras.