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The Killing superalgebra of 10-dimensional supergravity backgrounds

We construct the Killing superalgebra of supersymmetric backgrounds of 10-dimensional heterotic and type II supergravities and prove that it is a Lie superalgebra. We also show that if the fraction of supersymmetry preserved by the background is greater than 1/2, in the heterotic case, or greater than 3/4 in the type II case, then the background is locally homogeneous.