special linear group SL(n,R)
E593510
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
Observed surface forms (3)
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Lie algebra
ⓘ
Lie group ⓘ linear algebraic group ⓘ matrix group ⓘ |
| actsOn |
upper half-plane
ⓘ
ℝ^n ⓘ |
| appearsIn |
algebraic groups
ⓘ
differential geometry ⓘ representation theory ⓘ theory of Lie groups ⓘ |
| consistsOf | n×n real matrices with determinant 1 ⓘ |
| definedAs |
{A ∈ M_n(ℝ) | det(A) = 1}
ⓘ
{X ∈ M_n(ℝ) | tr(X) = 0} ⓘ |
| hasCenter |
{I} for n odd
ⓘ
{±I} for n even ⓘ |
| hasDimension |
3
ⓘ
n^2 − 1 ⓘ n^2 − 1 ⓘ |
| hasFundamentalGroup | ℤ for n = 2 ⓘ |
| hasIdentityElement | I_n ⓘ |
| hasLieAlgebra | sl(n,ℝ) ⓘ |
| hasMaximalCompactSubgroup |
SO(2)
NERFINISHED
ⓘ
SO(n) NERFINISHED ⓘ |
| hasOperation | matrix multiplication ⓘ |
| hasRealRank | n − 1 ⓘ |
| hasRealStructureConstants | true ⓘ |
| isClosedUnder |
matrix multiplication
ⓘ
taking inverses ⓘ |
| isConnected | true for n ≥ 2 ⓘ |
| isDefinedByEquation | det(A) = 1 in M_n(ℝ) ⓘ |
| isDeterminantOnePartOf | GL(n,ℝ) NERFINISHED ⓘ |
| isLinear | true ⓘ |
| isNonCompact | true for n ≥ 2 ⓘ |
| isNormalSubgroupOf | GL(n,ℝ) NERFINISHED ⓘ |
| isPathConnected | true for n ≥ 2 ⓘ |
| isPerfectGroup | true for n ≥ 2 ⓘ |
| isRealPointsOf | algebraic group SL_n over ℝ ⓘ |
| isSemisimple | true ⓘ |
| isSimpleAsLieGroup | true for n ≥ 2 except low-dimensional isomorphisms ⓘ |
| isSimplyConnected | false for n ≥ 2 ⓘ |
| isSpecialCaseOf | SL(n,ℝ) NERFINISHED ⓘ |
| isSubgroupOf | GL(n,ℝ) NERFINISHED ⓘ |
| isSubsetOf | GL(n,ℝ) NERFINISHED ⓘ |
| isUnimodular | true ⓘ |
| kernelOf | det : GL(n,ℝ) → ℝ^× ⓘ |
| preserves |
Lebesgue measure on ℝ^n up to normalization
ⓘ
volume on ℝ^n ⓘ |
| quotientBy | {±I} is PSL(2,ℝ) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.