Let \(\mathfrak {g}\) be the vector space with basis \((j_k)_{k \in \mathbb {Z}}\) over \(\mathbb {K}\), considered as an abelian Lie algebra. The bilinear map \({\gamma }_{\mathfrak {hei}} \colon \mathfrak {g}\times \mathfrak {g}\to \mathbb {K}\) given on basis elements by

is a Lie algebra 2-cocycle, \({\gamma }_{\mathfrak {hei}} \in C^2(\mathfrak {g},\mathbb {K})\). We call \({\gamma }_{\mathfrak {hei}}\) the Heisenberg cocycle.

Let \(\mathbb {K}\) be a field of characteristic zero. The Heisenberg algebra \(\mathfrak {hei}\) is the Lie algebra over \(\mathbb {K}\) obtained as the central extension of the abelian Lie algebra \(\mathfrak {g}\) with basis \((j_k)_{k \in \mathbb {Z}}\), corresponding to the Heisenberg cocycle \({\gamma }_{\mathfrak {hei}} \in C^2(\mathfrak {g},\mathbb {K})\).

The Heisenberg algebra \(\mathfrak {hei}\) has a basis consisting of \((J_k)_{k \in \mathbb {Z}}\) and \(K\), with Lie brackets determined by the following

for \(k,l \in \mathbb {Z}\).