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.
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© 1989 Birkhäuser Boston
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Jacobson, N. (1989). Derivation Algebras and Multiplication Algebras of Semi-Simple Jordan Algebras. In: Nathan Jacobson Collected Mathematical Papers. Contemporary Mathematicians. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3694-8_4
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DOI: https://doi.org/10.1007/978-1-4612-3694-8_4
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8215-0
Online ISBN: 978-1-4612-3694-8
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