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 g= ⊕ ∞ i=1 gi 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. Introduction N-graded Lie algebras are closely related to nilpotent Lie algebras, for instance, a finite-dimensional N-graded Lie algebra g must to be nilpotent. Infinite-dimensional ones are also called residual nilpotent Lie algebras. M. Vergne studied in [12] nilpotent Lie algebras with the maximal possible nilindex s(g) = dim g−1 (by nilindex s(g) we mean the length of the descending central series {Cg} of g). M. Vergne called them filiform Lie algebras. In her study the N-graded filiform Lie algebra m0(n) played a special role. This Lie algebra is defined by its basis e1, . . . , en and non-trivial commutator relations: [e1, ei] = ei+1, i = 2, . . . , n−1. Evidently, m0(n) is generated by e1 and e2. Another example of N-graded twogenerated filiform Lie algebra is m2(n): [e1, ei] = ei+1, i = 2, . . . , n−1, [e2, ei] = ei+2, i = 3, . . . , n−2. A. Fialowski classified in [4] all infinite-dimensional N-graded two-generated Lie algebras g = ⊕igi with one-dimensional homogeneous components gi. In particular, there are only three algebras in her list satisfying the ”filiform property”: [g1, gi] = gi+1, ∀i. They are m0,m2, L1, where m0,m2 denote infinite-dimensional analogues of m0(n),m2(n), respectively and L1 is the ”positive” part of the Witt or Virasoro algebra. The classification of finite-dimensional N-graded filiform Lie algebras over a field of zero characteristic was done in [9]. A. Shalev and E. Zelmanov defined in [11] the coclass (which might be infinity) of a finitely generated and residually nilpotent Lie algebra g, in analogy with the case 1991 Mathematics Subject Classification. 17B56, 17B70, 17B10, 17B65, 05A17.