Cohomology of graded lie algebras of maximal class with coefficients in the adjoint representation

We compute explicitly the adjoint cohomology of two ℕ-graded Lie algebras of maximal class (infinite-dimensional filiform Lie algebras) m0 and m2. It is known that up to an isomorphism there are only three ℕ-graded Lie algebras of maximal class. The third algebra from this list is the “positive” part L 1 of the Witt (or Virasoro) algebra, and its adjoint cohomology was computed earlier by Feigin and Fuchs.

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