Peirce decomposition of a Jordan algebra

Core idea

Let $e$ be an idempotent in a Jordan algebra $J$ over a field of characteristic different from $2$ . Its Peirce spaces are

J_\lambda(e)=\{x\in J:e\circ x=\lambda x\}, \qquad \lambda\in\left\{0,\tfrac12,1\right\}.

The Peirce decomposition theorem says that multiplication by $e$ is diagonalizable with these possible eigenvalues and gives the direct sum

J=J_0(e)\oplus J_{1/2}(e)\oplus J_1(e).

Peirce multiplication rules

The Jordan product respects the decomposition through

\begin{aligned} J_1\circ J_1&\subseteq J_1,& J_0\circ J_0&\subseteq J_0,& J_1\circ J_0&=0,\\ (J_0+J_1)\circ J_{1/2}&\subseteq J_{1/2},&& J_{1/2}\circ J_{1/2}&\subseteq J_0+J_1. \end{aligned}

Thus $J_1(e)$ and $J_0(e)$ are Jordan subalgebras, while the middle space mediates between them.

For a Euclidean Jordan algebra the three summands are mutually orthogonal for the trace inner product.

Several frame idempotents

A Jordan frame refines the single-idempotent splitting into diagonal lines and pair spaces. This produces the six-summand description of $H_3(\mathbb K)$ in the frame decomposition.

References

Nathan Jacobson, Structure and Representations of Jordan Algebras, American Mathematical Society, 1968, Chapter III, §1. Publisher record.
Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994, Chapter IV, §1. Publisher record.