Lie algebras MOC

Heisenberg algebra

In the general formulation used in conformal field theory, a Heisenberg algebra over is a nilpotent Lie algebra whose 1-dimensional centre is its commutator ideal #m/def/lie

Assuming is countable, one may impose a -grading with for and given above, giving abelian subalgebras

so that are maximal abelian subalgebras of . An alternating bilinear form on is given by

which is nondegenerate on and for all , so one may form bases of and of satisfying the Heisenberg commutation relations

and

for .

Properties

1. , since otherwise the centre would be trivial (not 1-dimensional)
2. If is finite, then it is odd

Examples

• Standard Heisenberg algebra for QM
• Natural Heisenberg algebras

See also

• Modules of a Heisenberg algebra
  • Heisenberg module
• Heisenberg group

#state/tidy | #lang/en | #SemBr