Cohomology of graded Lie algebras of maximal class

It was shown by A. Fialowski that an arbitrary infinite-dimensional -graded “filiform type” Lie algebra with one-dimensional homogeneous components such that over a field of zero characteristic is isomorphic to one (and only one) Lie algebra from three given ones: , where the Lie algebras and are defined by their structure relations: : [e1,ei]=ei+1, i 2 and : [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 and 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.