Trace form of a Euclidean Jordan algebra

Definition

Let $J$ be a Euclidean Jordan algebra. Its Jordan trace form is

\tau(x,y)=\operatorname{tr}_J(x\circ y),

where $\operatorname{tr}_J(x)$ is the sum of the Jordan eigenvalues of $x$ .

Fundamental properties

The form $\tau$ is symmetric, positive definite, and associative:

\tau(x\circ y,z)=\tau(x,y\circ z).

Thus it is a canonical compatible Euclidean inner product. A Euclidean Jordan algebra may be presented with another compatible inner product; on each simple ideal, every such form is a positive scalar multiple of the canonical trace form. Different simple summands may carry different positive scalings.

Matrix and spin-factor cases

On $\mathfrak h_n(\mathbb R)$ , $\mathfrak h_n(\mathbb C)$ , and $\mathfrak h_n(\mathbb H)$ , the Jordan trace is the ordinary real matrix trace, and

\tau(X,Y)=\operatorname{Re}\operatorname{Tr}(XY).

For $J(V)=\mathbb R\oplus V$ ,

\operatorname{tr}_J(\lambda,u)=2\lambda,\qquad \tau((\lambda,u),(\mu,v)) =2(\lambda\mu+\langle u,v\rangle).

Operator-trace convention

Another bilinear form, also sometimes called “the trace form,” is

\beta(x,y)=\operatorname{Tr}(L_{x\circ y}), \qquad L_x(z)=x\circ z.

Here $\operatorname{Tr}$ is the trace of a vector-space endomorphism. On a simple Euclidean Jordan algebra of dimension $N$ and rank $r$ ,

\beta(x,y)=\frac Nr\,\tau(x,y).

The conventions differ only by a positive factor on each simple ideal, but a formula should identify its normalization.

Every Jordan automorphism preserves spectral eigenvalues and hence $\operatorname{tr}_J$ and $\tau$ . This realizes $\operatorname{Aut}(J)$ as a closed subgroup of the orthogonal group of the trace form.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. Publisher record.