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
A Jordan algebra J over a field k of characteristic 2 becomes a 2-Lie algebra L(J ) with Lie product [x, y] = x • y and squaring x [2] = x 2 . We determine the precise ideal structure of L(J ) in case J is simple finite-dimensional and k is algebraically closed. We also decide which of these algebras have smooth automorphism groups. Finally, we study the derivation algebra of a reduced Albert algebra J = H 3 (O, k) and show that Der J has a unique proper nonzero ideal V J , isomorphic to L(J )/k · 1 J , with quotient Der J/V J independent of O. On the group level, this gives rise to a special isogeny between the automorphism group of J and that of the split Albert algebra, whose kernel is the infinitesimal group determined by V J .
The aim of this work is the description of the isomorphism classes of all Leavitt path algebras coming from graphs satisfying Condition (Sing) with up to three vertices. In particular, this classification recovers the one achieved by Abrams et al. [1] in the case of graphs whose Leavitt path algebras are purely infinite simple. The description of the isomorphism classes is given in terms of a series of invariants including the K 0 group, the socle, the number of loops with no exits and the number of hereditary and saturated subsets of the graph.2000 Mathematics Subject Classification. Primary 16D70.
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