linear algebraic group
C44535
concept
A linear algebraic group is a group that is also an affine algebraic variety, whose group operations (multiplication and inversion) are given by regular polynomial maps when the group is realized as a closed subgroup of some general linear group GLₙ.
All labels observed (5)
| Label | Occurrences |
|---|---|
| linear algebraic group canonical | 6 |
| algebraic group | 1 |
| algebraic group constructions | 1 |
| linear algebraic groups | 1 |
| orthogonal 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: linear algebraic group
Generated description
A linear algebraic group is a group that is also an affine algebraic variety, whose group operations (multiplication and inversion) are given by regular polynomial maps when the group is realized as a closed subgroup of some general linear group GLₙ.
Instances (9)
| Instance | Via concept surface |
|---|---|
| orthogonal group O(n) | — |
| Chevalley groups | linear algebraic groups |
| orthogonal group O(n+1,2) | orthogonal group |
|
special linear group SL(n,R)
surface form:
SL(n,ℝ)
|
— |
|
general linear group GL(n,C)
surface form:
GL(n,ℂ)
|
— |
|
special linear group SL(n,C)
surface form:
SL(n,ℂ)
|
— |
| SL(2,R) | — |
| Langlands dual group | algebraic group |
|
PSL(2,\mathbb{C})
surface form:
PSL(2,ℂ)
|
— |