Statements (50)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:group_of_people
gptkb:Lie_group |
| gptkbp:centralTo |
set of scalar matrices with n-th roots of unity
|
| gptkbp:connects |
yes, if field is R or C
|
| gptkbp:definedIn |
the group of n x n matrices with determinant 1
|
| gptkbp:dimensions |
n^2 - 1
|
| gptkbp:field |
complex numbers
real numbers finite field arbitrary field |
| gptkbp:generation |
elementary matrices with determinant 1
|
| gptkbp:hasSubgroup |
general linear group GL(n)
|
| gptkbp:heldBy |
gptkb:Weyl_group
gptkb:algebraic_geometry gptkb:Lie_group orthogonal group classical group reductive group semisimple group connected group non-compact (over R or C) root system of type A_{n-1} split simple algebraic group trivial center for n=2 over most fields |
| gptkbp:homomorphism |
determinant map to multiplicative group of field
|
| https://www.w3.org/2000/01/rdf-schema#label |
special linear group SL(n)
|
| gptkbp:importantFor |
gptkb:algebraic_geometry
number theory physics representation theory |
| gptkbp:isNonAbelian |
for n > 2
|
| gptkbp:isSimple |
n > 2 and field not of characteristic 2
|
| gptkbp:kernelOf |
determinant map from GL(n)
|
| gptkbp:notation |
gptkb:SL(n)
SL_n |
| gptkbp:order |
depends on field and n
|
| gptkbp:relatedTo |
gptkb:orthogonal_group_O(n)
gptkb:unitary_group_U(n) general linear group GL(n) projective special linear group PSL(n) |
| gptkbp:usedIn |
gptkb:gauge_theory
gptkb:topology differential geometry modular forms quantum mechanics automorphic forms study of symmetry |
| gptkbp:bfsParent |
gptkb:SO(n)
gptkb:special_orthogonal_group_SO(n) |
| gptkbp:bfsLayer |
6
|