Cohomology of graded Lie algebras of maximal class

It was shown by A. Fialowski that an arbitrary infinite-dimensional N-graded “filiform type” Lie algebra with one-dimensional homogeneous components gi such that [g1,gi]=gi+1,∀i⩾2 over a field of zero characteristic is isomorphic to one (and only one) Lie algebra from three given ones: m0,m2,L1, where the Lie algebras m0 and m2 are defined by their structure relations: m0: [e1,ei]=ei+1, ∀i⩾2 and m2: [e1,ei]=ei+1, ∀i⩾2, [e2,ej]=ej+2, ∀j⩾3 and L1 is the “positive” part of the Witt algebra.In the present article we compute the cohomology H∗(m0) and H∗(m2) with trivial coefficients, give explicit formulas for their representative cocycles and describe the multiplicative structure in the cohomology. Also we discuss the relations with combinatorics and representation theory. The cohomology H∗(L1) was calculated by L. Goncharova in 1973.