Editing Symplectic vector space (section)

==Heisenberg group==
{{main|Heisenberg group}}
A [[Heisenberg group]] can be defined for any symplectic vector space, and this is the typical way that [[Heisenberg group]]s arise.

A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative [[Lie algebra]], meaning with trivial Lie bracket. The Heisenberg group is a [[central extension (mathematics)|central extension]] of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the [[canonical commutation relation]]s (CCR), and a Darboux basis corresponds to [[canonical coordinate]]s&nbsp;– in physics terms, to [[momentum operator]]s and [[position operator]]s.

Indeed, by the [[Stone–von Neumann theorem]], every representation satisfying the CCR (every representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one.

Further, the [[group ring|group algebra]] of (the dual to) a vector space is the [[symmetric algebra]], and the group algebra of the Heisenberg group (of the dual) is the [[Weyl algebra]]: one can think of the central extension as corresponding to quantization or [[Deformation quantization|deformation]].

Formally, the symmetric algebra of a vector space ''V'' over a field ''F'' is the group algebra of the dual, {{nowrap|1=Sym(''V'') := ''F''[''V''<sup>∗</sup>]}}, and the Weyl algebra is the group algebra of the (dual) Heisenberg group {{nowrap|1=''W''(''V'') = ''F''[''H''(''V''<sup>∗</sup>)]}}. Since passing to group algebras is a [[contravariant functor]], the  central extension map {{nowrap|''H''(''V'') → ''V''}} becomes an inclusion {{nowrap|Sym(''V'') → ''W''(''V'')}}.