Homology and Cohomology of the Lamplighter Lie Algebra

It is shown that the lamplighter Lie algebra $\mathfrak l$ 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 $\mathfrak m_0$. Y. Félix and A. Murillo proved that its $q$-dimensional homology $H_q(\mathfrak l,\mathbb Q)$ is infinite-dimensional. However, they failed to completely calculate the spaces $H_q(\mathfrak l,\mathbb Q)$, $q\ge 3$. In this paper, an infinite basis of the bigraded homology $H_{*,*}(\mathfrak l,\mathbb Q)$ is explicitly constructed using the results of D. Millionshchikov and A. Fialowski on the cohomology $H^*(\mathfrak l,\mathbb Q)$.