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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
##### Keywords:
subalgebras; Lie algebra; holonomy group; neutral metric
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