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 $$\dim (A^2) = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and a a generator of $$A^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , then A admits an inner product $$\langle \cdot , \cdot \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> such that the product of A is given by $$xy = \langle x, y\rangle a$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>⟩</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> . There are three classes to consider: $$a\in \textrm{Ann}(A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mtext>Ann</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ; $$a\notin \textrm{Ann}(A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∉</mml:mo> <mml:mtext>Ann</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and a is isotropic relative to $$\langle \cdot , \cdot \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> ; $$a\notin \textrm{Ann}(A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∉</mml:mo> <mml:mtext>Ann</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and a is anisotropic relative to $$\langle \cdot , \cdot \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> . 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.