Also, we have preferred to keep this work elementary in the sense of not going into the theory of affine group schemes, which would probably have smoothened the characteristic zero hypothesis. Special lecture on the recent spectacular developments concerning the Poincar Conjecture. We have determined the automorphism group of □ as a tool for our study. Logo of Instituto de Ciencias Matemticas (ICMAT). For V = K4, there are only three such decompositions, providing three equivalence classes of fine gradings on □. It turns out that under suitable hypothesis on K, the fine gradings on □ are related to certain decompositions of V as orthogonal direct sums of non-isotropic lines and hyperbolic planes. When K = ℝ and (V, Q) = (ℝ4, Q) is the Minkowsky space, □ is the Poincaré algebra. The Lie algebra of this algebraic group is denoted by □. If we fix a field K, and (V, Q) is a K-vector space with a non-degenerate quadratic form Q, denoting by O(V, Q) the corresponding orthogonal group, we can consider the matrix group ( 1 V 0 O ( V, Q ) ). We study fine gradings on a class of Lie algebras containing the generalised Poincaré algebras.
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