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.

Key Words:

• Algebras with polynomial identity
• Embedding in matrices
• Free products of algebras
• Idempotents
• Quadratic elements

2000 Mathematics Subject Classification:

• 15A30
• 16R20
• 16S15
• 16S50

ACKNOWLEDGMENTS

The first named author is grateful to the Department of Mathematics of the University of Miskolc for the warm hospitality during his visit when a part of this project was carried out.

The second named author was supported by OTKA of Hungary No. K61007.

The third named author was supported by the National Research Foundation of South Africa under Grant No. UID 61857. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and, therefore, the National Research Foundation does not accept any liability in regard thereto.

Notes

Communicated by A. Smoktunowicz