Derivation Algebras and Multiplication Algebras of Semi-Simple Jordan Algebras

Abstract

In this note we investigate the Lie algebra of derivations and the Lie algebra $\mathfrak{L}$ generated by the multiplications in any semi-simple Jordan algebra (with a finite basis) over a field of characteristic 0. We show that the derivation algebra $\mathfrak{D}$ possesses a certain ideal $\mathfrak{F}$ consisting of derivations that we call inner and that $\mathfrak{F}$ is also a subalgebra of the Lie multiplication algebra $\mathfrak{L}$. For semi-simple algebras we prove that $\mathfrak{F} = \mathfrak{D}$ This result is a consequence of a general theorem (Theorem 1) on derivations of semi-simple non-associative algebras of characteristic 0. It can be seen that another easy consequence of our general theorem is the known result that the derivations of semi-simple associative algebras are all inner.¹ Our method can be applied in other cases, too (for example, alternative algebras), but we shall not discuss these here.