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

We find all the fine group gradings on the real forms of the Albert algebra and of the exceptional Lie algebras g 2 and f 4 . ͑pO s ͒, J ͑0,0,e͒ = R 3 , J ͑0,0,g͒ = 0, ͑g e͒ J ͑0,1,g͒ = 0 ͑͑pO s ͒ g ͒, 053516-19 Gradings on g 2 and f 4 J. Math. Phys. 51, 053516 ͑2010͒ This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.12.232.124 On: Wed, 26 Nov 2014 20:30:56͑2͒ If J is compact, ͑i͒ the Z 2 5 -grading on JªA ͑1,1,1͒ ͑pO͒, J ͑0,0,e͒ = R 3 , J ͑0,0,g͒ = 0, ͑g e͒ J ͑0,1,g͒ = 0 ͑͑pO͒ g ͒, J ͑1,0,g͒ = 1 ͑͑pO͒ g ͒, J ͑1,1,g͒ = 2 ͑͑pO͒ g ͒. ͑3͒ If J is nonsplit and noncompact, ͑i͒ the Z 2 5 -grading on JªA ͑−1,1,−1͒ ͑pO͒, J ͑0,0,e͒ = R 3 , J ͑0,0,g͒ = 0, ͑g e͒ J ͑0,1,g͒ = 0 ͑͑pO͒ g ͒, J ͑1,0,g͒ = 1 ͑͑pO͒ g ͒, J ͑1,1,g͒ = 2 ͑͑pO͒ g ͒; ͑ii͒ the Z 2 3 ϫ Z-grading on JªA ͑−1,1,−1͒ ͑pO͒, J ͑−2,e͒ = ͗2͑e 0 − e 2 ͒ − 1 ͑1͒͘, J ͑−2,g͒ = 0, ͑g e͒ J ͑−1,g͒ = ͗ 0 ͑x͒ − 2 ͑x͒: x ͑pO͒ g ͘, J ͑0,e͒ = ͗e 1 ,e 0 + e 2 ͘, J ͑0,g͒ = ͗ 1 ͑x͒: x ͑pO͒ g ͘, ͑g e͒ J ͑1,g͒ = ͗ 0 ͑x͒ + 2 ͑x͒: x ͑pO͒ g ͘, J ͑2,e͒ = ͗2͑e 0 − e 2 ͒ + 1 ͑1͒͘, J ͑2,g͒ = 0, ͑g e͒.Also, in the same way, we can have a detailed description of the fine gradings on the real forms of f 4 .

The fine abelian group gradings on the simple exceptional classical Lie superalgebras over algebraically closed fields of characteristic 0 are determined up to equivalence.

The structure theory of separable complex L*-algebras was given by Schue in [10] and [11]. In [3] Balachandran makes a study of infinite-dimensional complex topologically simple L*-algebras of classical type and poses the question whether these algebras exhaust the class of all infinite-dimensional complex topologically simple L*-algebras. In this paper we give an affirmative answer by determining all the complex topologically simple infinite-dimensional L*-algebras. The case of the real L*-algebras was studied previously in [4], [9] and [12] also under the separability condition. Applying the result of Balachandran, our result yields the structure theory for real L*-algebras. The main tool used here is the ‘approximation’ of the L*-algebra by topologically simple separable L*-algebras via an ultraproduct construction.

We associate a square to any two-dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behavior of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic group, getting in this form a new algebraic invariant. The study of associative representations of evolution algebras is also started and we get faithful representations for most two-dimensional evolution algebras. In some cases, we prove that faithful commutative and associative representations do not exist, giving rise to the class of what could be termed as “exceptional” evolution algebras (in the sense of not admitting a monomorphism to an associative algebra with deformed product).