On the Cartan–Jacobson Theorem

Abstract

The well-known Cartan-Jacobson theorem claims that the Lie algebra of derivations of a Cayley algebra is central simple if the characteristic is not 2 or 3. In this paper we have studied these two cases, with the following results: if the characteristic is 2, the theorem is also true, but, if the characteristic is 3, the derivation algebra is not simple. We have also proved that in this last case, there is a unique nonzero proper seven-dimensional ideal, which is a central simple Lie algebra of type A 2 , and the quotient of the derivation algebra modulo this ideal turns out to be isomorphic, as a Lie algebra, to the ideal itself. The original motivation of this work was a series of computer-aided calculations which proved the simplicity of derivation algebras of Cayley algebras in the case of characteristic not 3. These computations also proved the existence of a unique nonzero proper ideal

The Magic Square and Symmetric Compositions

“…The argument above shows then that ker The Z 2 × Z 2 -grading of the Lie algebra g = g(S, S ) constructed above induces a Z 2 gradation given by g0 = g (0,0) ⊕ g (1,0) and g1 = g (0,1) ⊕ g (1,1) . The structure of g0 is given by: Corollary 3.5 Let (S, * , q) and (S , * , q ) be two symmetric composition algebras and let g = g(S, S ) and g0 as above.…”

“…The roots in g (0,0) are ±ε i ± ε j , 1 ≤ i < j ≤ 4. Since g (1,0) = S ⊗ S with the natural action of o(4, 4) ∼ = tri(S, * , q) on S; the roots in g (1,0) are ±ε i ± (δ 1 + δ 2 ), 1 ≤ i ≤ 4, and the roots in g (0,1) and in g (1,1) are obtained by applying to the roots in g (1,0) the triality automorphisms on the Dynkin diagram of D 4 that fixes ε 2 − ε 3 and permutes cyclically ε 1 − ε 2 , ε 3 − ε 4 and ε 3 + ε 4 , while substituting δ 1 + δ 2 by δ 1 and δ 2 . As a consequence, the roots in g (0,1) (respectively g (1,1) ) are 1 2 (±ε 1 ± ε 2 ± ε 3 ± ε 4 ) ± δ 1 (resp.…”

“…The right most column represents the corresponding extended Dynkin diagram X (1) N with marked nodes, where X N is the entry in the third column. Recall that over an algebraically closed field of characteristic 0, any order three automorphism of a simple Lie algebra of type X N is determined, up to conjugation, by some marked nodes in X (1) N such that the sum of the labels of the nodes is 3 [12,Theorem 8.6]. A Z in the third column means a one-dimensional central ideal.…”

New simple Lie superalgebras in characteristic 3

“…is the trivial product by Eqs. (2.27a) and (2.27b), or J = k with n(α) = α 3 for any α ∈ k, because then t (α) = 3α = 0 for any α ∈ k, so that the trace form is trivial (this has to do with the fact that the exceptional Lie algebra of type G 2 is no longer simple in characteristic 3-see, for instance, [3]). Theorem 2.16 and Brown's classification of the Freudenthal triple systems in characteristic 3 [6] yield: ; or (iii) dim T = 2, and hence T is described in item (ii) of Proposition 2.7; or (iv) up to isomorphism, T is the symplectic triple system described in Proposition 2.14.…”

Simple $(-1,-1)$ Balanced Freudenthal Kantor Triple Systems

“…Hence there is a bilinear nondegenerate form ( | ) : 1 for any a ∈ W and u ∈W . This bilinear form determines h and allows us to identifyW with the dual W * .…”