Some Results on the Classification of Triple Systems |
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a e k algebra system Aut L yes Aut S,o Aut*L automorphism groups bijective correspondence bilinear form central subgroup Chapter characteristic zero classes of k-forms classes of polarized Corollary 4.12 correspondence between elements dim ker direct sum dual Dynkin diagram elements of H¹(G exact sequences Faulkner diagram finite dimensional follows from Lemma G-group Galois descent Galois group H¹ G Hence homomorphism inner derivations irreducible components irreducible L-modules irreducible submodules Jordan pair Jordan triple system k-isomorphism k-scalar isomorphism classes Lie algebra Lie module triple Lie triple systems M₁ and M₂ M₁ M₂ M₁t M₂ with M₁ module triple system nonassociative algebra nondegenerate nonzero normal subgroup polarized k-forms Proof R(M₁ R(M₁,M₂ R(M₂ scalar Schur's lemma semisimple simple Lie triple standard embedding subalgebra subgroup of Aut submodules Suppose triple product twisted Lie module x,y e x₁ yes Aut απ ε ε