Suppose we are given a commutative unital ring $k$ and a module $V$ over $k$ equipped with a skew-symmetric bilinear form
(Typically, one requires $\omega$ to be non-degenerate, see below, but this is not needed for the following definition).
The Heisenberg Lie algebra $Heis(V, \omega)$ corresponding to $(V,\omega)$ is the Lie algebra given by the $k$-module $V\oplus k$ together with the unit $k \hookrightarrow V\oplus k$, $s\mapsto (0,s) =: s 1$ and Lie $k$-algebra bracket
The notion of Heisenberg algebra arose in the study of quantization by tools of symplectic geometry:
A special case of the above definition is that where $(V,\omega)$ a symplectic vector space (hence $k$ a field and $\omega$ non-degenerate).
In this case the Heisenberg algebra is a sub-Lie algebra of the Lie algebra underlying the Poisson algebra of $(V,\omega)$. For more on this see below.
We discuss a generalization of the notion of Heisenberg Lie algebra from ordinary symplectic geometry to a notion of Heisenberg Lie n-algebra in higher geometric quantization of n-plectic geometry. See at Heisenberg Lie n-algebra for more.
The following definition is naturally motivated from the fact that:
The ordinary Heisenberg Lie algebra is the sub-Lie algebra of the Poisson bracket Lie algebra, the one underlying the corresponding Poisson algebra (see below) on the constant and linear functions.
The generalization of Poisson brackets to Poisson Lie n-algebras in n-plectic geometry for all $n$ is established (see there).
In view of this, the following definition takes the Heisenberg Lie $n$-algebra to be the sub-Lie $n$-algebra of the Poisson Lie n-algebra on the linear and constant differential forms.
First we need the following definition, which is elementary, but nevertheless worth making explicit once.
Let $n \in \mathbb{N}$, let $(V, \omega)$ be an n-plectic vector space.
The corresponding $n$-plectic manifold is the n-plectic manifold $(V, \mathbf{\omega})$, with $V$ now the canonical smooth manifold structure on the given vector space, and with
the differential form obtained by left (right) translating $\omega$ along $V$.
Explicitly, for all vector fields $\{v_i \in \Gamma(T V)\}_{i = 1}^n$ and all points $x \in V$ we set
Here on the right – and in all of the following – we are using that every tangent space $T_x V$ of $V$ is naturally identified with $V$ itself
Let $n \in \mathbb{N}$, let $(V, \omega)$ be an n-plectic vector space and let $(V, \mathbf{\omega})$ be the corresponding n-plectic manifold.
The Heisenberg Lie $n$-algebra $Heis(V,\omega)$ is the sub-Lie n-algebra of the Poisson Lie n-algebra $\mathcal{P}(V, \omega)$ on those differential forms which are either linear or constant (with respect to left/right translation on $V$).
All one has to observe is:
This is indeed a sub-Lie $n$-algebra.
We need to check that the linear and constant forms are closed under the L-infinity algebra brackets of $\mathcal{P}(V, \omega)$.
The only non-trivial such brackets are the unary one, and the ones on elements all of degree 0.
The unary bracket is given by the de Rham differential. Since this sends a linear form to a constant form and a constant form to 0, our sub-complex is closed under this.
Similarly, the brackets on elements all in degree 0 is given by contraction of $\mathbf{\omega}$ with the Hamiltonian vector fields of linear or constant forms. Since $\mathbf{\omega}$ is a constant form, and since the de Rham differential of a linear or constant form is constant (or even 0), these Hamiltonian vector fields are necessarily constant. Hence their contraction with $\mathbf{\omega}$ gives a constant form.
We discuss how the notion of Heisenberg Lie algebra relates to that of Poisson algebra.
For $(X, \omega)$ a symplectic vector space, there is a natural Lie algebra homomorphism
exhibiting the Heisenberg Lie algebra as a sub-Lie algebra of that underlying the Poisson algebra $\mathcal{P}(V,\omega)$ of $V$.
Namely, it is the sub-Lie algebra on the linear functions and the constant functions.
Let $(V, \omega)$ be a symplectic vector space over the real numbers. Using the canonical isomorphism $\phi : T V \simeq V \times V$ of the tangent bundle of $V$ with the projection $p_1 : V \times V \to V$, we obtain from the bilinear form $\omega$ a differential 2-form $\mathbf{\omega} \in \Omega^2(V)$ by the assignment
for all $\mathbf{v}, \mathbf{w} \in \Gamma(T V)$.
This way $(V, \mathbf{\omega})$ is a symplectic manifold and thus comes with a Poisson algebra. Write $\mathcal{P}(V,\mathbf{\omega})$ for the Lie algebra underlying the Poisson algebra of $(V, \mathbf{\omega})$.
Its underlying vector space is the space $C^\infty(V)$ of smooth functions $V \to \mathbb{R}$. To every element $f \in C^\infty(V)$ is associated its Hamiltonian vector field $\mathbf{v}_f \in \Gamma(T X)$, defined (uniquely, due to the non-degeneracy of $\omega$) by the equation
In terms of this, the Lie bracket of the Poisson algebra is defined to be
Inside all smooth functions sit the linear functions $V \to \mathbb{R}$, which form the dual vector space to $V$:
By the non-degeneracy of $\omega$, for every $f \in V^*$ there is an element $v_f \in V$ such that
Moreover, the canonical extension $\mathbf{v}_f$ of $v_f$ to a vector field on $V$ is a Hamiltonian vector field for $f$
It follows that the Lie bracket of two linear functions $f,g$ in the Poisson algebra is
Notice that on the right we have a constant function on $V$.
Write $\rho_2 : \mathbb{R} \hookrightarrow C^\infty(V)$ for the inclusion of the constant functions, and write
Then, by the above, the inclusion
induces a Lie algebra homomorphism
which exhibits the Heisenberg Lie algebra as a sub-Lie algebra of that underlying the Poisson algebra.
As for any Lie algebra one has Lie integration of the Heisenberg Lie algebra to a Lie group. This is called the Heisenberg group (of the given symplectic vector space).
In the case of standard symplectic form on the Cartesian space $\mathbf{R}^{2n}$, the universal enveloping algebra of the Heisenberg Lie algebra is an associative algebra $\mathcal{U}(Heis(\mathbb{R}^{2n}))$. The quotient of this that identifies the central elements of the Heisenberg Lie algebra with multiples of the identity element is the Weyl algebra on $n$ generators.
Given any Hopf algebra, one can define its Heisenberg double, which generalized the Heisenberg-Weyl algebra, which corresponds to the case when the Hopf algebra is the polynomial algebra.
Lecture notes on standard material include
(section 4 in) Gordon, Infinite-dimensional Lie algebras, Lecture notes, Edinburgh (2008) (pdf)
Teruji Thomas, Geometric quantization II: Prequantization and the Heisenberg group (pdf), section 4 (relating to geometric quantization)
A categorification of the Heisenberg algebra is considered in
An $n$-fold categorification of the Lie algebra underlying the Poisson algebra (and hence including the Weil algebra) for all $n$ to a Lie n-algebra is considered in n-plectic geometry,