Homology and Cohomology of the Lamplighter Lie Algebra

D. V. Millionshchikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 166-176

Résumé

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 $H^*(\mathfrak l,\mathbb Q)$.

Détail

Keywords: homology, cohomology, lamplighter group, pronilpotent completion, $\mathfrak {sl}_2$-module.
Mots-clés : Lie algebra of maximal class