Nonassociative rings and algebras Definition

Formally real Jordan algebra

Definition

Let $J$ be a Jordan algebra over $\R$ , with product $(x, y) \mapsto xy$ , and write $x^2 = xx$ . $J$ is formally real if for every $n \ge 1$ and all $x_1, \ldots, x_n \in J$ , the equation $x_1^2 + \cdots + x_n^2 = 0$ implies $x_1 = \cdots = x_n = 0$ .

Remarks

Example: the real symmetric $n \times n$ matrices with $X \circ Y = \frac{1}{2}(XY + YX)$ , so that $X \circ X = X^2$ . For symmetric $X$ , $\tr(X^2) = \sum_{i, j} X_{ij}^2$ . If $X_1^2 + \cdots + X_n^2 = 0$ , taking traces gives $\sum_k \sum_{i, j} (X_k)_{ij}^2 = 0$ , so every $X_k$ is $0$ .

Non-example: the commutative associative algebra $\R[\varepsilon]/(\varepsilon^2)$ is a Jordan algebra, but $\varepsilon^2 = 0$ with $\varepsilon \ne 0$ .

Every formally real Jordan algebra is power-associative and has no nonzero nilpotent elements.

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Jordan algebra concept
Formally real Jordan algebra definition

Tag	`JN3K5K2Z7Z`
URI	`m2://JN3K5K2Z7Z`
Kind	definition
Writing	W0 stub
MSC	17C20, 17C90
Defines	Formally real Jordan algebra
Uses	Jordan algebra
Changed	3 November 2026

Formally real Jordan algebra

Definition

Remarks

Reading path

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