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The holonomy Lie algebras of neutral metrics in dimension four. (English) Zbl 1016.53039
Summary: The Lie algebra isomorphism between \(su(1,1) \times su(1,2)\) and \(o(2,2)\) is used to obtain a list of subalgebras of the latter. The resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the Lie algebra of the holonomy group of a neutral metric in four dimensions. The conclusions, taken in conjunction with previously known results, furnish a classification of such Lie subalgebras of \(o(2,2)\), with only one case remaining unresolved.

MSC:
53C29 Issues of holonomy in differential geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
17B05 Structure theory for Lie algebras and superalgebras
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