Compact Lie Groups
Lie groups form a very nice class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
- the circle group T and the torus groups Tn,
- the orthogonal groups O(n), the special orthogonal group SO(n) and its covering spin group Spin(n),
- the unitary group U(n) and the special unitary group SU(n),
- the symplectic group Sp(n),
- the compact forms of the exceptional Lie groups: G2, F4, E6, E7, and E8,
The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies).
Read more about this topic: Compact Group
Famous quotes containing the words compact, lie and/or groups:
“Take pains ... to write a neat round, plain hand, and you will find it a great convenience through life to write a small and compact hand as well as a fair and legible one.”
—Thomas Jefferson (1743–1826)
“Nature herself has not provided the most graceful end for her creatures. What becomes of all these birds that people the air and forest for our solacement? The sparrow seems always chipper, never infirm. We do not see their bodies lie about. Yet there is a tragedy at the end of each one of their lives. They must perish miserably; not one of them is translated. True, “not a sparrow falleth to the ground without our Heavenly Father’s knowledge,” but they do fall, nevertheless.”
—Henry David Thoreau (1817–1862)
“Some of the greatest and most lasting effects of genuine oratory have gone forth from secluded lecture desks into the hearts of quiet groups of students.”
—Woodrow Wilson (1856–1924)