Automorphism groups and derivation algebras of Hamiltonian Lie algebras

TL;DR

This paper determines the automorphism groups, derivation algebras, and second cohomology of Hamiltonian Lie algebras, revealing their structure and inner derivations.

Contribution

It explicitly computes the automorphism groups and derivation algebras of Hamiltonian Lie algebras and their derived subalgebras, including the second cohomology.

Findings

Automorphism groups are isomorphic to GSp_N(Z)⋉(K^×)^N.

All derivations of the Hamiltonian Lie algebra are inner.

The second cohomology group of the Hamiltonian Lie algebra is computed.

Abstract

In this paper, we compute the automorphism group and derivation algebra of the Hamiltonian Lie algebra $\mathcal{H}_{N}$ and its derived subalgebra $\mathcal{H}_{N}'$, where $N$ is an even positive integer. The automorphism groups are shown to be $\mathbf{GSp}_{N}(\mathbb{Z})\ltimes (\mathbb{\mathbb{K}}^{\times})^{N}$ 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.