Algebras Generated by Two Quadratic Elements

Abstract

Let K be a field of any characteristic and let R be an algebra generated by two elements satisfying quadratic equations. Then R is a homomorphic image of F = K ⟨x, y | x 2 + ax + b = 0, y 2 + cy + d = 0⟩ for suitable a, b, c, d ∈ K. We establish that F can be embedded into the 2 × 2 matrix algebra with entries from the polynomial algebra over the algebraic closure of K and that F and satisfy the same polynomial identities as K-algebras. When the quadratic equations have double zeros, our result is a partial case of more general results by Ufnarovskij, Borisenko, and Belov from the 1980s. When each of the equations has different zeros, we improve a result of Weiss, also from the 1980s.