hasLieAlgebra
P28830
predicate
Indicates that one mathematical structure is associated with, or gives rise to, a specific Lie algebra capturing its infinitesimal or tangent-level structure.
All labels observed (11)
| Label | Occurrences |
|---|---|
| hasLieAlgebra canonical | 19 |
| LieAlgebra | 5 |
| LieAlgebraDescription | 3 |
| LieAlgebraDefinedAs | 2 |
| LieAlgebraIsomorphicTo | 1 |
| associatedLieGroup | 1 |
| canInduceLieAlgebraOverField | 1 |
| definesLieAlgebraOn | 1 |
| hasLieAlgebraCondition | 1 |
| hasLieAlgebraStructure | 1 |
| hasTangentSpaceAtIdentity | 1 |
Description generation (PDg)
The one-sentence description above was generated by prompting gpt-5.1 with the predicate name and this instruction.
Instruction
Given a predicate that represents a relationship or action between entities, generate a one-sentence description explaining its meaning. # Instructions Focus on describing the relationship, not the entities themselves. # Response Format Begin the description with \' Indicates...\'
Input
Predicate: hasLieAlgebra
Generated description
Indicates that one mathematical structure is associated with, or gives rise to, a specific Lie algebra capturing its infinitesimal or tangent-level structure.
Sample triples (36)
| Subject | Object |
|---|---|
| Poincaré group |
Poincaré algebra
ⓘ
linked to:
Poincaré group
|
| Lorentz group | so(1,3) ⓘ |
| E(n) | e(n) ⓘ |
| Lie ring | true via predicate surface "canInduceLieAlgebraOverField" ⓘ |
| Lie subgroup | Lie subalgebra of the ambient Lie algebra via predicate surface "hasTangentSpaceAtIdentity" ⓘ |
| AdS isometry group SO(2,d) | so(2,d) ⓘ |
| SU(3) | su(3) ⓘ |
|
SO(3)
linked to:
rotation group SO(3)
|
so(3) via predicate surface "LieAlgebra" ⓘ |
|
SO(3)
linked to:
rotation group SO(3)
|
ℝ³ with cross product via predicate surface "LieAlgebraIsomorphicTo" ⓘ |
| SL(2,C) | sl(2,C) ⓘ |
|
SU(2)
linked to:
rotation group SU(2)
|
su(2) ⓘ |
| orthogonal group O(n) | skew-symmetric n×n real matrices ⓘ |
| affine group of R^n | affine Lie algebra of R^n ⓘ |
| affine group of R^n | R^n ⋊ gl(n,R) via predicate surface "hasLieAlgebraStructure" ⓘ |
|
SO(n)
linked to:
special orthogonal group SO(n)
|
so(n) via predicate surface "LieAlgebra" ⓘ |
|
SO(n)
linked to:
special orthogonal group SO(n)
|
skew-symmetric n×n real matrices via predicate surface "LieAlgebraDescription" ⓘ |
| U(1) | iR ⓘ |
| Poisson bracket | space of smooth functions on a Poisson manifold via predicate surface "definesLieAlgebraOn" ⓘ |
| orthogonal group O(n+1,2) | 𝔰𝔬(n+1,2) ⓘ |
|
SU(n)
linked to:
special unitary group SU(n)
|
su(n) via predicate surface "LieAlgebra" ⓘ |
|
SU(n)
linked to:
special unitary group SU(n)
|
traceless skew-Hermitian n×n complex matrices via predicate surface "LieAlgebraDescription" ⓘ |
|
GL(n,ℝ)
linked to:
general linear group GL(n,R)
|
gl(n,ℝ) NERFINISHED ⓘ |
|
GL(n,ℝ)
linked to:
general linear group GL(n,R)
|
all n×n real matrices via predicate surface "LieAlgebraDescription" ⓘ |
|
SL(n,ℝ)
linked to:
special linear group SL(n,R)
|
sl(n,ℝ) ⓘ |
| SO(2,d-1) | so(2,d-1) ⓘ |
| Spin(2,d) | \mathfrak{so}(2,d) ⓘ |
|
GL(n,ℂ)
linked to:
general linear group GL(n,C)
|
𝔤𝔩(n,ℂ) via predicate surface "LieAlgebra" ⓘ |
|
GL(n,ℂ)
linked to:
general linear group GL(n,C)
|
all n×n complex matrices with usual commutator bracket via predicate surface "LieAlgebraDefinedAs" ⓘ |
|
SL(n,ℂ)
linked to:
special linear group SL(n,C)
|
sl(n,ℂ) ⓘ |
|
SL(n,ℂ)
linked to:
special linear group SL(n,C)
|
trace zero matrices via predicate surface "hasLieAlgebraCondition" ⓘ |
| PSL(2,ℝ) | sl(2,ℝ) NERFINISHED ⓘ |
| sl(2,C) | SL(2,C) via predicate surface "associatedLieGroup" NERFINISHED ⓘ |
| SL(2,R) | sl(2,R) via predicate surface "LieAlgebra" ⓘ |
| SL(2,R) | 2×2 real matrices with trace 0 via predicate surface "LieAlgebraDefinedAs" ⓘ |
| metaplectic group | symplectic Lie algebra ⓘ |
|
PSL(2,ℂ)
linked to:
PSL(2,\mathbb{C})
|
sl(2,ℂ) ⓘ |