On an extremal property of Jordan algebras of Clifford type

TL;DR

This paper characterizes Jordan algebras of Clifford type as the unique class of finite-dimensional unital commutative algebras with an associative positive definite form that contain a nontrivial idempotent of minimal length.

Contribution

It establishes a geometric extremal property that uniquely identifies Jordan algebras of Clifford type among such algebras.

Findings

Existence of a nonzero idempotent with minimal length in the algebra.

Equality in the length inequality characterizes Jordan algebras of Clifford type.

Provides a geometric criterion for identifying Clifford type Jordan algebras.

Abstract

If $V$ is a finite-dimensional unital commutative (maybe nonassociative) algebra carrying an associative positive definite bilinear form then there exist a nonzero idempotent $c\ne e$ ($e$ being the algebra unit) of the shortest possible length $|c|^2$. In particular, $|c|^2\le \frac12|e|^2$. We prove that the equality holds exactly when $V$ is a Jordan algebra of Clifford type.