Homology and Cohomology of the Lamplighter Lie Algebra

It is shown that the lamplighter Lie algebra iota over the field of rational numbers, introduced by S. Ivanov, R. Mikhailov, and A. Zaikovskii, is isomorphic to the infinite-dimensional naturally graded Lie algebra of maximal class m(0). Y. Felix and A. Murillo proved that its q-dimensional homology H-q(iota, Q) is infinite-dimensional. However, they failed to completely calculate the spaces H-q(iota, Q), q >= 3. In this paper, an infinite basis of the bigraded homology H-*,H-*(iota, Q) is explicitly constructed using the results of D. Millionshchikov and A. Fialowski on the cohomology H*(iota,Q).