Lie Class | Group | # of generators | isomorphisms | extended Dynkin Diagram | discrete center |
---|---|---|---|---|---|

A_{n} | SU(n + 1) | (n + 1)^{2} - 1 | A_{1} ~ B_{1} |
Z_{n + 1} | |

B_{n} | SO(2n + 1) | n (2n + 1) | B_{1} ~ C_{1} |
||

C_{n} | Sp(2n) | n (2n + 1) | C_{2} ~ B_{2} |
Z_{2} | |

D_{n} | SO(2n) | n (2n - 1) | D_{2} ~ A_{1} + A_{1} |
Z_{2} | |

D_{3} ~ A_{3} | |||||

E_{6} | E_{6} | 78 | |||

E_{7} | E_{7} | 133 | Z_{2} | ||

E_{8} | E_{8} | 248 | Z_{3} | ||

F_{4} | F_{4} | 52 | |||

G_{2} | G_{2} | 14 |

Notes:

- The rank n is the dimension of the Cartan subalgebra (the maximal Abelian subalgebra), whose eigenvalues in the adjoint representation are the roots.

- SU(n) leaves invariant the quadratic form z
_{i}zbar^{i}; SO(n) leaves invariant the quadratic form x_{i}x^{i}; Sp(n) leaves invariant the quadratic form in 2n dimensions x_{i}x'^{n + 1 - i}- x'_{i}x^{n + 1 - i}.

- In the (extended) Dynkin Diagrams, removing a root yields a (maximal) subgroup.

©2008, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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