Heisenberg Lie algebra
E503521
Lie algebra
central extension
nilpotent Lie algebra
non-abelian Lie algebra
two-step nilpotent Lie algebra
The Heisenberg Lie algebra is a fundamental nilpotent Lie algebra generated by position and momentum operators with a central element, encoding the canonical commutation relations that underlie quantum mechanics and harmonic analysis.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Heisenberg group | 2 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Lie algebra
ⓘ
central extension ⓘ nilpotent Lie algebra ⓘ non-abelian Lie algebra ⓘ two-step nilpotent Lie algebra ⓘ |
| associatedWith |
harmonic analysis
ⓘ
quantum mechanics ⓘ representation theory ⓘ symplectic geometry ⓘ |
| encodes | canonical commutation relations ⓘ |
| hasAbelianQuotientByCenter | true ⓘ |
| hasAutomorphismGroupContaining | symplectic group Sp(2n,R) NERFINISHED ⓘ |
| hasBasis | {X_1,…,X_n,Y_1,…,Y_n,Z} ⓘ |
| hasCanonicalForm | [q_i,p_j] = iħ δ_{ij} 1 (in physics notation) ⓘ |
| hasCenter | span{Z} ⓘ |
| hasCentralElement | Z ⓘ |
| hasCommutationRelation |
[X_i,X_j] = 0
ⓘ
[X_i,Y_j] = δ_{ij} Z ⓘ [X_i,Z] = 0 ⓘ [Y_i,Y_j] = 0 ⓘ [Y_i,Z] = 0 ⓘ |
| hasDerivedAlgebra | span{Z} ⓘ |
| hasDimension | 2n+1 ⓘ |
| hasLowerCentralSeriesLength | 2 ⓘ |
| hasNaturalGrading | deg(X_i)=deg(Y_i)=1, deg(Z)=2 ⓘ |
| hasOneDimensionalCenter | true ⓘ |
| hasUniqueIrreducibleUnitaryRepresentationUpToEquivalence | true (for fixed central character) ⓘ |
| isCarnotAlgebra | true ⓘ |
| isCentralExtensionOf | abelian Lie algebra R^{2n} ⓘ |
| isGeneratedBy |
momentum operators
ⓘ
position operators ⓘ |
| isGraded | true ⓘ |
| isModelFor | canonical quantization ⓘ |
| isNilpotent | true ⓘ |
| isPerfect | false ⓘ |
| isPrototypeOf | nilpotent Lie algebra used in analysis ⓘ |
| isSemisimple | false ⓘ |
| isSimple | false ⓘ |
| isSolvable | true ⓘ |
| isStepTwoNilpotent | true ⓘ |
| isStratified | true ⓘ |
| isUnimodular | true ⓘ |
| namedAfter | Werner Heisenberg NERFINISHED ⓘ |
| overField |
complex numbers
ⓘ
real numbers ⓘ |
| playsRoleIn | Stone–von Neumann theorem NERFINISHED ⓘ |
| relatedTo | Heisenberg group NERFINISHED ⓘ |
| usedIn |
Fourier analysis on non-commutative groups
ⓘ
sub-Riemannian geometry ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Heisenberg group
this entity surface form:
Heisenberg group