Heisenberg Group - On Symplectic Vector Spaces

On Symplectic Vector Spaces

The general abstraction of a Heisenberg group is constructed from any symplectic vector space. For example, let (V,ω) be a finite dimensional real symplectic vector space (so ω is a nondegenerate skew symmetric bilinear form on V). The Heisenberg group H(V) on (V,ω) (or simply V for brevity) is the set V×R endowed with the group law

The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence

Any symplectic vector space admits a Darboux basis {e_j,fk}_{1 ≤ j,k ≤ n} satisfying ω(e_j,fk) = δ_jk and where 2n is the dimension of V (the dimension of V is necessarily even). In terms of this basis, every vector decomposes as

The qa and p_a are canonically conjugate coordinates.

If {e_j, fk}_{1 ≤ j,k ≤ n} is a Darboux basis for V, then let {E} be a basis for R, and {e_j, fk, E}_{1 ≤ j,k ≤ n} is the corresponding basis for V×R. A vector in H(V) is then given by

and the group law becomes

Because the underlying manifold of the Heisenberg group is a linear space, vectors in the Lie algebra can be canonically identified with vectors in the group. The Lie algebra of the Heisenberg group is given by the commutation relation

or written in terms of the Darboux basis

and all other commutators vanish.

It is also possible to define the group law in a different way but which yields a group isomorphic to the group we have just defined. To avoid confusion, we will use u instead of t, so a vector is given by

and the group law is

An element of the group : can then be expressed as a matrix

which gives a faithful matrix representation of H(V). The u in this formulation is related to t in our previous formulation by, so that the t value for the product comes to

as before.

The isomorphism to the group using upper triangular matrices relies on the decomposition of V into a Darboux basis, which amounts to a choice of isomorphism V ≅ U ⊕ U*. Although the new group law yields a group isomorphic to the one given higher up, the group with this law is sometimes referred to as the polarized Heisenberg group as a reminder that this group law relies on a choice of basis (a choice of a Lagrangian subspace of V is a polarization).

To any Lie algebra, there is a unique connected, simply connected Lie group G. All other connected Lie groups with the same Lie algebra as G are of the form G/N where N is a central discrete group in G. In this case, the center of H(V) is R and the only discrete subgroups are isomorphic to Z. Thus H(V)/Z is another Lie group which shares this Lie algebra. Of note about this Lie group is that it admits no faithful finite dimensional representations; it is not isomorphic to any matrix group. It does however have a well-known family of infinite-dimensional unitary representations.