Derivation Algebras and Multiplication Algebras of Semi-Simple Jordan Algebras

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.

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