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:
“The powers of the federal government ... result from the compact to which the states are parties, [and are] limited by the plain sense and intention of the instrument constituting that compact.”
—James Madison (1751–1836)
“The interest in life does not lie in what people do, nor even in their relations to each other, but largely in the power to communicate with a third party, antagonistic, enigmatic, yet perhaps persuadable, which one may call life in general.”
—Virginia Woolf (1882–1941)
“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists.”
—Thomas S. Kuhn (b. 1922)