On derivations of the Heisenberg algebra

Derivations of the Heisenberg algebra $\mathcal H$ and some related questions are studied. The ideas and the language of formal differential geometry are used. It is proved that all derivations of this algebra are inner. The main subalgebras of the Lie algebra $\mathfrak D(\mathcal H)$ of all derivations of $\mathcal H$ are distinguished, and their properties are studied. It is shown that the algebra $\mathcal H$ interpreted as a Lie algebra (with the commutator as the Lie bracket) forms a one-dimensional central extension of $\mathfrak D(\mathcal H)$.