Heisenberg group Symplectic vector space

a heisenberg group can defined symplectic vector space, , typical way heisenberg groups arise.

a vector space can thought of commutative lie group (under addition), or equivalently commutative lie algebra, meaning trivial lie bracket. heisenberg group central extension of such commutative lie group/algebra: symplectic form defines commutation, analogously canonical commutation relations (ccr), , darboux basis corresponds canonical coordinates – in physics terms, momentum operators , position operators.

indeed, stone–von neumann theorem, every representation satisfying ccr (every representation of heisenberg group) of form, or more unitarily conjugate standard one.

further, group algebra of (the dual to) vector space symmetric algebra, , group algebra of heisenberg group (of dual) weyl algebra: 1 can think of central extension corresponding quantization or deformation.

formally, symmetric algebra of v group algebra of dual, sym(v) := k[v], , weyl algebra group algebra of (dual) heisenberg group w(v) = k[h(v)]. since passing group algebras contravariant functor, central extension map h(v) → v becomes inclusion sym(v) → w(v).