non-compact Lie group
C7234
concept
A non-compact Lie group is a Lie group whose underlying topological space is not compact, meaning it is a smooth group manifold that is unbounded or not closed in the sense of compactness.
All labels observed (7)
| Label | Occurrences |
|---|---|
| matrix group | 22 |
| isometry group | 3 |
| non-compact Lie group canonical | 3 |
| real Lie group | 3 |
| conformal group | 1 |
| non-compact group | 1 |
| unitary group | 1 |
Description generation (CDg)
The one-sentence description above was generated by prompting gpt-5.1 with the class name and this instruction.
Instruction
generate a one-sentence description for a given conceptual class. # Response Format Return only the sentence: "Description: [one-sentence description of the conceptional class]"
Input
Class: non-compact Lie group
Generated description
A non-compact Lie group is a Lie group whose underlying topological space is not compact, meaning it is a smooth group manifold that is unbounded or not closed in the sense of compactness.
Instances (29)
| Instance | Via concept surface |
|---|---|
| Euclidean group | isometry group |
| E(n) | isometry group |
| AdS isometry group SO(2,d) | isometry group |
|
modular group PSL(2,Z)
surface form:
PSL(2,ℤ)
|
matrix group |
|
rotation group SO(3)
surface form:
SO(3)
|
matrix group |
| SL(2,C) | matrix group |
| Poincaré group | — |
| Lorentz group | matrix group |
|
rotation group SU(2)
surface form:
SU(2)
|
matrix group |
| ISO(n) | matrix group |
| orthogonal group O(n) | matrix group |
| affine group of R^n | matrix group |
|
special orthogonal group SO(n)
surface form:
SO(n)
|
matrix group |
| U(1) | unitary group |
| orthogonal group O(n+1,2) | matrix group |
| SO(2,d-1) | matrix group |
|
special unitary group SU(n)
surface form:
SU(n)
|
matrix group |
|
general linear group GL(n,R)
surface form:
GL(n,ℝ)
|
matrix group |
|
special linear group SL(n,R)
surface form:
SL(n,ℝ)
|
matrix group |
|
general linear group GL(n,C)
surface form:
GL(n,ℂ)
|
matrix group |
|
special linear group SL(n,C)
surface form:
SL(n,ℂ)
|
matrix group |
| PSL(2,ℤ/Nℤ) | matrix group |
| SL(2,ℤ) | matrix group |
| PSL(2,ℝ) | real Lie group |
| Pauli group | matrix group |
| SL(2,R) | matrix group |
|
PSL(2,\mathbb{C})
surface form:
PSL(2,ℂ)
|
— |
| SL(2,7) | matrix group |
| PGL(2,7) | matrix group |