Abstract
We study complex involutive algebras generated by a single nonselfadjoint idempotent and use them to construct a family of algebras, which we call planar Lyapunov algebras. As our main result, we prove that every 2-dimensional commutative real algebra whose homogeneous Riccati differential equation is stable at the origin must be isomorphic either to an algebra with zero multiplication or to some planar Lyapunov algebra.
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