We describe all the fine group gradings, up to equivalence, on the Lie algebra d 4 . This problem is equivalent to finding the maximal abelian diagonalizable subgroups of the automorphism group of d 4 . We prove that there are fourteen by using two different viewpoints. The first approach is computational: we get a full description of the gradings by using a particular implementation of the automorphism group of the Dynkin diagram of d 4 and some algebraic groups stuff. The second approach, more qualitative, emphasizes some algebraic aspects, as triality, and it is mostly devoted to gradings involving the outer automorphisms of order three.

In this paper we provide some new tools for the study of finite-dimensional absolute-valued algebras. We introduce homotopy notions in this field and develop some of their applications. Next, we parametrize these algebras by spin groups and study their isomorphisms. Finally, we introduce a duplication process for the construction of absolute-valued algebras.

No abstract

A case of oat-cell carcinoma arising in the pericardium of a 51-year-old woman is described. The patient had multiple nodes; the largest was 2 x 1 cm. Two years later the patient presented with a tumor on her lower gum; this measured 2 mm and had similar characteristics to the previous one. The immunohistochemical study showed strong positivity for neuron-specific enolase. From review of the literature, it may be concluded that this is the first report of oat-cell carcinoma occurring in the pericardium.

No abstract