A Heisenberg group (or Weyl-Heisenberg group) is a Lie group integrating a Heisenberg Lie algebra.
There are several such, and so the conventions in the literature vary slightly as to which one to pick by default.
The Heisenberg group historically originates in and still has its strongest ties to quantum physics: there it is a group of unitary operators acting on the space of states induced from those observables on a linear phase space – a symplectic vector space – which are given by linear or by constant functions. So any Heisenberg group is a subgroup of a group of observables in certain simple examples of quantum mechanical systems.
A Heisenberg group is a Lie group whose Lie algebra is a Heisenberg Lie algebra.
We spell out some special cases in detail.
The simplest non-trivial example of a Heisenberg group is the unique simply connected Lie integration of the Heisenberg Lie algebra $Heis(\mathbb{R}^2, \omega = d p \wedge dq)$ on the the canonical 2-dimensional symplectic vector space $\mathbb{R}^2$ with canonical coordinates $(p,q)$.
This Heisenberg Lie algebra is generated from 3 elements, here to be denoted $\mathbf{p}, \mathbf{q},\mathbf{e}$, subject to the single non-trivial Lie bracket
The corresponding Heisenberg group is usually denoted $H_3$:
The underlying smooth manifold of this Lie group is the Cartesian space $\mathbb{R}^3$. A general element may be written as
with $a, b , c\in \mathbb{R}$. In terms of this notation the product in the group is given (by the Baker-Campbell-Hausdorff formula) by
A discrete quotient group of this, which still has the same Lie algebra, has as underlying manifold $\mathbb{R}^2 \times U(1)$ (the second factor being the circle group), with the projection
being quotienting by $\mathbb{Z}$: $U(1) \simeq \mathbb{R} / \mathbb{Z}$.
If the circle group is instead thought of as the unit circle in the complex plane, then this quotient is thought of as the exponential map $\exp(2 \pi i (-)) : \mathbb{R} \to U(1)$. In terms of this the group elements in the quotient read
and their product is
While the Lie algebra is still the same real Heisenberg Lie algebra as before, it is now suggestive to write the Lie bracket as
This is the way the relation appears in texts on quantum physics.
Generally, there is a Heisenberg group $H(V, \omega)$ associated to any symplectic vector space $(V, \omega)$.
Regard $V$ with its abelian group structure underlying its vector space structure.
The Heisenberg group $H(V,\omega)$ is the space $V \times U(1)$ (for $U(1)$ the circle group) equipped with the group product
A symplectic vector space $(V, \omega)$ is in particular a symplectic manifold. Accordingly its algebra of smooth functions $C^\infty(V)$ is a Poisson algebra. The Lie algebra underlying the Poisson algebra contains the Heisenberg Lie algebra as the subspace with is the direct sum of the linear functions $V^* \hookrightarrow C^\infty(V)$ and the constant functions $\marthbb{R} \hookrightarrow C^\infty(V)$.
For more details in this at Heisenberg Lie algebra the section Relation to Poisson algebra.
By the above, the Heisenberg group is a subgroup of the group that integrates the Poisson bracket. The latter is a central extension of the group of Hamiltonian symplectomorphisms.
(Of course, on a contractible symplectic manifold such as a symplectic vector space, every symplectomorphism is automatically a Hamiltonian symplectomorphism.)
The additive group on the Cartesian space $\mathbb{R}^2$ with group operation
carries a degree-2 group cocycle $\omega$ with values in $\mathbb{R}$ given by
The cocycle condition for this is the identity
The Heisenberg group $H_3$ is the group extension of $\mathbb{R}^2$ by this cocycle.
See Stone-von Neumann theorem.
The automorphism group of the Heisenberg group is the symplectic group.
An original account is in
A textbook account is in section II.3 of
Discussion in the context of geometric quantization is in
Discussion of the representation theory of the infinite-dimensional Heisenberg group is in
The Heisenberg group for the phase space $U(1)$-Chern-Simons theory on an arbitrary Riemann surface (and its relation to skein relations and theta functions) is discussed in
Razvan Gelca, Alejandro Uribe, From classical theta functions to topological quantum field theory (arXiv:1006.3252, slides pdf)
Razvan Gelca, Alastair Hamilton, Classical theta functions from a quantum group perspective (arXiv:1209.1135)