2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103742A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras $H(2\colon\n;ω_2)$ (of dimension one less than a power of $p$) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of $H(2\colon\n;ω_2)$ with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).36 pages, to be published in J. Austral. Math. Soc. Ser. ARings and Algebras17B50; 17B70, 17B56, 17B65text