Derivation of a Jordan algebra

Definition

Let $J$ be a Jordan algebra. A derivation of $J$ is a linear map $D:J\to J$ satisfying

D(x\circ y)=D(x)\circ y+x\circ D(y) \qquad(x,y\in J).

The vector space of all derivations is denoted $\operatorname{Der}(J)$ . With the commutator bracket $[D,E]=D\circ E-E\circ D$ , it is a Lie algebra.

Infinitesimal automorphisms

For a finite-dimensional real Jordan algebra, the automorphism group is a Lie group and

\operatorname{Lie}(\operatorname{Aut}(J))=\operatorname{Der}(J).

Indeed, differentiating a curve of product-preserving automorphisms gives the Leibniz rule. Conversely, the exponential $\exp(tD)$ of a derivation is a one-parameter group of Jordan automorphisms. If $J$ is unital, every derivation fixes the unit infinitesimally: $D(1)=0$ .

Inner derivations

Write $L_x(y)=x\circ y$ . In a Jordan algebra, each commutator

[L_x,L_y]=L_xL_y-L_yL_x

is a derivation. Derivations in the linear span of these commutators are called inner derivations. This terminology is Jordan-theoretic: it should not be confused with the associative-algebra formula $a\mapsto xa-ax$ , since the Jordan product itself is commutative.

For finite-dimensional semisimple Jordan algebras over a field of characteristic zero, every derivation is inner. In a Euclidean Jordan algebra, derivations are skew-adjoint for the canonical trace inner product, reflecting compactness of the automorphism group.

References

Nathan Jacobson, Structure and Representations of Jordan Algebras, American Mathematical Society, 1968, Chapter VIII. Publisher record.
Richard D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, 1966, Chapter IV. Publisher record.