Fine gradings on the Poincaré algebra

We study fine gradings on a class of Lie algebras containing the generalised Poincaré algebras. 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 ) ) ....

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Bibliographic Details
Published in:Mathematical proceedings of the Royal Irish Academy 2017-01, Vol.117A (2), p.39-45
Main Authors: Pablo Alberca Bjerregaard, Dolores Martín Barquero, Cándido Martín González, Daouda Ndoye
Format: Article
Language:English
Subjects:
Abstract algebra
Algebra
Automorphisms
Equivalence relation
Geometric planes
Homomorphisms
Linear algebra
Mathematical theorems
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Summary:We study fine gradings on a class of Lie algebras containing the generalised Poincaré algebras. 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 ) ) . The Lie algebra of this algebraic group is denoted by p. When K = R and (V, Q) = (R4, Q) is the Minkowsky space, p is the Poincaré algebra. It turns out that under suitable hypothesis on K, the fine gradings on p are related to certain decompositions of V as orthogonal direct sums of non-isotropic lines and hyperbolic planes. For V = K 4, there are only three such decompositions, providing three equivalence classes of fine gradings on p. We have determined the automorphism group of p as a tool for our study. 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.
ISSN:1393-7197
2009-0021
DOI:10.3318/pria.2017.117.05