Complex Structures in Real Algebras. I. Two-Dimensional Commutative Case

Abstract

We study complex structures in real two-dimensional commutative algebras and show their connection with homotopy properties of the multiplications. An application to the Riccati equation in rank three algebras is also discussed.

Key Words:

• Non-associative algebras
• Polynomial equations in algebra
• Complex structures
• Quadratic maps

1991 Mathematics Subject Classification:

• Primary 58E05, 58E09
• Secondary 35J20

Acknowledgments

We are thankful to Bernstein and Walcher who improved our understanding of the subject. We are also grateful to G. Belitskii, B. Kunyavskii and M. Muzychuk for useful discussions. Finally, we are grateful to an anonymous referee for important remarks and suggestions.

The first author is grateful to the Alexander von Humboldt Foundation for their support.