∞-Lie theory (higher geometry)
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:
Poisson bracket Lie algebra, the one underlying the corresponding Poisson algebra (see below) on the constant and linear functions.
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
Mikhail Khovanov, Heisenberg algebra and a graphical calculus (arXiv:1009.3295)
Owen Gwilliam, Rune Haugseng, Linear Batalin-Vilkovisky quantization as a functor of ∞-categories (arXiv:1608.01290)
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,