Algebras generated by two quadratic elements

TL;DR

This paper studies algebras generated by two elements satisfying quadratic equations over any field, showing they can be embedded into 2x2 matrix algebras over polynomial rings and satisfy the same identities as matrix algebras.

Contribution

It proves that such algebras are homomorphic images of a specific algebra that embeds into matrix algebras, extending previous results and providing new insights into their polynomial identities.

Findings

F can be embedded into M_2(E[t])

F and M_2(E) satisfy the same polynomial identities

Results extend and improve upon 1980s work by Weiss and others

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 in K. We establish that F can be embedded into the 2x2 matrix algebra M_2(E[t]) with entries from the polynomial algebra E[t] over the algebraic closure E of K and that F and M_2(E) 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 1980's. When each of the equations has different zeros, we improve a result of Weiss, also from the 1980's.