Orthogonal conjugacies in associative and Lie algebras

Earl J. Taft
1964 Transactions of the American Mathematical Society  
Introduction. If A is a (finite-dimensional) associative algebra over a field F with radical R such that A/R is separable (semi simple, and remains semisimple under any extension of F), then the well-known Wedderburn principal theorem states that A is a semi-direct sum T+ R, where Tis a subalgebra of A isomorphic to A/R. Tis a maximal separable subalgebra of A and will be also called a Wedderburn factor of A. The Malcev theorem (see [6] ) states that any two maximal separable subalgebras of A
more » ... subalgebras of A are conjugate in A via an inner automorphism given by conjugation by an element 1 -z, where zeR (A need not contain an identity). Let G be a semisimple group of automorphisms and antiautomorphisms of A (see §2 for definitions and terminology). In certain circumstances, G will leave invariant a Wedderburn factor of A (see [7; 8] and Theorem 1 of §3 for an exact description). In [9], we proved a uniqueness theorem for G-invariant Wedderburn factors for F of characteristic 0 and G finite. In [11], this was generalized to characteristic F not two and G finite of order not divisible by the characteristic of F. In §3 we generalize this result to arbitrary semisimple G and characteristic F not two (Theorem 2 and Corollary 1). It is shown that any two G-invariant Wedderburn factors of A are G-orthogonally conjugate in A. Let Lbe a Lie algebra over a field F of characteristic 0. Then, if R denotes the radical (maximal solvable ideal) of L, then the well-known Levi theorem (see [5] ) says that L is a semi-direct sum T+ R where Tis a subalgebra of L isomorphic to L/R. Tis a maximal semisimple subalgebra of Land will also be called a Levi factor of L. The Malcev-Harish-Chandra theorem (see [4; 6]) states that any two Levi factors of L are conjugate by an automorphism exp(Ad z) of L, where z is in the nil radical (maximal nilpotent ideal) of L. If G is a semisimple group of automorphisms of L, then L will contain G-invariant Levi factors (see [7] ). In [10], we proved a uniqueness theorem for G-invariant Levi
doi:10.1090/s0002-9947-1964-0163930-7 fatcat:4judrdidujdmlefjmsdklfw57i