Mutations of Alternative Algebras pp 136-181 | Cite as

# Automorphisms and Derivations

## Abstract

A central topic of interest in the theory of nonassociative algebras is the study of automorphisms and derivations of algebras. As is well known, many linear algebraic and Lie groups and their Lie algebras are obtained from the automorphism groups and derivation algebras of certain nonassociative algebras. If *B* is a finitedimensional algebra over a field *F*, then there is a natural relationship between the automorphism group Aut *B* and the derivation algebra Der *B* of *B*. The Lie algebra of Aut *B*, viewed as a linear algebraic group over *F*, is a subalgebra of Der *B*. If *F* is the field of real numbers or complex numbers, then Aut *B* is a Lie subgroup of the linear group *GL*(*B*) and Der *B* is the Lie algebra of Aut *B* [**SW1**].

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