On the Cohomologies of the Heisenberg Algebras

This paper studies the cohomologies of the Heisenberg algebras as one-dimensional central extensions of commutative Lie algebras generated by nondegenerate 2-cocycles. Section 1 contains a general description of the cohomologies of one-dimensional central extensions of Lie algebras and typical examples. In Section 2, the general description is applied to explicitly evaluate the cohomologies of the Heisenberg algebra in important cases. Appendix describes the main properties of linear operations related to the generating 2-cocycle that are useful in particular calculations.