Spencer Cohomology and 11-Dimensional Supergravity

We recover the classification of the maximally supersymmetric bosonic backgrounds of 11-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the Z-graded subalgebrash= h- 2⊕ h- 1⊕ h0 of the Poincaré superalgebra g= g- 2⊕ g- 1⊕ g0= V⊕ S⊕ so(V) which differ only in zero degree, that is h0⊂ g0 and hj= gj for j< 0. Aside from the Poincaré superalgebra itself and its Z-graded subalgebras, there are only three other Lie superalgebras, which are the symmetry superalgebras of the non-flat maximally supersymmetric backgrounds. In passing we identify the gravitino variation with (a component of) a Spencer cocycle.