Editing Simple Lie group (section)

==Overview of the classification==
A<sub>''r''</sub> has as its associated simply connected compact group the [[special unitary group]], [[Special unitary group|SU(''r'' + 1)]] and as its associated centerless compact group the projective unitary group [[Projective unitary group|PU(''r'' + 1)]].

B<sub>''r''</sub> has as its associated centerless compact groups the odd [[special orthogonal group]]s, [[Special orthogonal group|SO(2''r'' + 1)]]. This group is not simply connected however: its universal (double) cover is the [[spin group]].

C<sub>''r''</sub> has as its associated simply connected group the group of [[symplectic group|unitary symplectic matrices]], [[Symplectic group|Sp(''r'')]] and as its associated centerless group the Lie group {{math|1=PSp(''r'') = Sp(''r'')/{{mset|I, −I}}}} of projective unitary symplectic matrices. The symplectic groups have a double-cover by the [[metaplectic group]].

D<sub>''r''</sub> has as its associated compact group the even [[special orthogonal group]]s, [[Special orthogonal group|SO(2''r'')]] and as its associated centerless compact group the projective special orthogonal group {{math|1=PSO(2''r'') = SO(2''r'')/{{mset|I, −I}}}}. As with the B series, SO(2''r'') is not simply connected; its universal cover is again the [[spin group]], but the latter again has a center (cf. its article).

The diagram D<sub>2</sub> is two isolated nodes, the same as A<sub>1</sub> &cup; A<sub>1</sub>, and this coincidence corresponds to the covering map homomorphism from SU(2) × SU(2) to SO(4) given by [[quaternion]] multiplication; see [[Quaternions and spatial rotation#Pairs of unit quaternions as rotations in 4D space|quaternions and spatial rotation]]. Thus SO(4) is not a simple group. Also, the diagram D<sub>3</sub> is the same as A<sub>3</sub>, corresponding to a covering map homomorphism from SU(4) to SO(6).

In addition to the four families ''A''<sub>''i''</sub>, ''B''<sub>''i''</sub>, ''C''<sub>''i''</sub>, and ''D''<sub>''i''</sub> above, there are five so-called exceptional Dynkin diagrams [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and [[E8 (mathematics)|E<sub>8</sub>]]; these exceptional Dynkin diagrams also have associated simply connected and centerless compact groups. However, the groups associated to the exceptional families are more difficult to describe than those associated to the infinite families, largely because their descriptions make use of [[exceptional object]]s. For example, the group associated to G<sub>2</sub> is the automorphism group of the [[octonion]]s, and the group associated to F<sub>4</sub> is the automorphism group of a certain [[Albert algebra]].

See also [[E7 1/2|{{tmath|\color{Blue} E_{7\frac 1 2} }}]].