Automorphism groups and derivation algebras of Hamiltonian Lie algebras

In this paper, we compute the automorphism group and derivation algebra of the Hamiltonian Lie algebra ${\mathcal{H}}_N$ and its derived subalgebra ${\mathcal{H}}_N^{\mathrm{\prime }}$, where $N$ is an even positive integer. The automorphism groups are shown to be ${\mathbf{G}\mathbf{S}\mathbf{p}}_N(\mathbb{Z})⋉({\mathbb{K}}^{\times }{)}^N$ for both Lie algebras and we prove that all derivations are inner for the Hamiltonian Lie algebra, also we compute the full derivation space for the derived subalgebra of Hamiltonian Lie algebra. Finally we compute the second cohomology group of Hamiltonian Lie algebra.