Classification of simple Lie groups Simple Lie group

1 classification of simple lie groups

1.1 full classification
1.2 compact lie groups

1.2.1 series
1.2.2 b series

1.2.2.1 c series


1.2.3 d series
1.2.4 exceptional cases

classification of simple lie groups

full classification

simple lie groups classified. classification stated in several steps, namely:

classification of simple complex lie algebras classification of simple lie algebras on complex numbers dynkin diagrams.
classification of simple real lie algebras each simple complex lie algebra has several real forms, classified additional decorations of dynkin diagram called satake diagrams, after ichirô satake.
classification of centerless simple lie groups every (real or complex) simple lie algebra





g




{\displaystyle {\mathfrak {g}}}

, there unique centerless simple lie group



g


{\displaystyle g}

lie algebra





g




{\displaystyle {\mathfrak {g}}}

, has trivial center.
classification of simple lie groups

one can show fundamental group of lie group discrete commutative group. given (nontrivial) subgroup



k
⊂

π

1


(
g
)


{\displaystyle k\subset \pi _{1}(g)}

of fundamental group of lie group



g


{\displaystyle g}

, 1 can use theory of covering spaces construct new group







g
~




k




{\displaystyle {\tilde {g}}^{k}}





k


{\displaystyle k}

in center. (real or complex) lie group can obtained applying construction centerless lie groups. note real lie groups obtained way might not real forms of complex group. important example of such real group metaplectic group, appears in infinite-dimensional representation theory , physics. when 1 takes



k
⊂

π

1


(
g
)


{\displaystyle k\subset \pi _{1}(g)}

full fundamental group, resulting lie group







g
~




k
=

π

1


(
g
)




{\displaystyle {\tilde {g}}^{k=\pi _{1}(g)}}

universal cover of centerless lie group



g


{\displaystyle g}

, , connected. in particular, every (real or complex) lie algebra corresponds unique connected , connected lie group






g
~





{\displaystyle {\tilde {g}}}

lie algebra, called connected lie group associated





g


.


{\displaystyle {\mathfrak {g}}.}

compact lie groups

every simple lie algebra has unique real form corresponding centerless lie group compact. turns out connected lie group in these cases compact. compact lie groups have particularly tractable representation theory because of peter-weyl theorem. simple complex lie algebras, centerless compact lie groups classified dynkin diagrams (first classified wilhelm killing , Élie cartan).

for infinite (a, b, c, d) series of dynkin diagrams, connected compact lie group associated each dynkin diagram can explicitly described matrix group, corresponding centerless compact lie group described quotient subgroup of scalar matrices.

a series

a1, a2, ...

ar has associated connected compact group special unitary group, su(r + 1) , associated centerless compact group projective unitary group pu(r + 1).

b series

b2, b3, ...

br has associated centerless compact groups odd special orthogonal groups, so(2r + 1). group not connected however: universal (double) cover spin group.

c series

c3, c4, ...

cr has associated connected group group of unitary symplectic matrices, sp(r) , associated centerless group lie group psp(r) = sp(r)/{i, -i} of projective unitary symplectic matrices.

d series

d4, d5, ...

dr has associated compact group special orthogonal groups, so(2r) , associated centerless compact group projective special orthogonal group pso(2r) = so(2r)/{i, -i}. b series, so(2r) not connected; universal cover again spin group, latter again has center (cf. article).

the diagram d2 2 isolated nodes, same a1 ∪ a1, , coincidence corresponds covering map homomorphism su(2) × su(2) so(4) given quaternion multiplication; see quaternions , spatial rotation. so(4) not simple group. also, diagram d3 same a3, corresponding covering map homomorphism su(4) so(6).

exceptional cases

in addition 4 families above, there 5 so-called exceptional dynkin diagrams g2, f4, e6, e7, , e8. of these have associated connected , centerless compact groups, although these not easy describe in terms of matrix groups infinite series ai, bi, ci , di above.

see e7½.