Relationship of Simple Lie Algebras To Groups
Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology, by computing the fundamental group of G (an abelian group: a Lie group is an H-space). This was done by Élie Cartan.
For an example, take the special orthogonal groups in even dimension. With the non-identity matrix −I in the center, these aren't actually simple groups; and having a twofold spin cover, they aren't simply-connected either. They lie 'between' G* and G, in the notation above.
Read more about this topic: Simple Lie Group
Famous quotes containing the words relationship of, relationship, simple, lie and/or groups:
“Only men of moral and mental force, of a patriotic regard for the relationship of the two races, can be of real service as ministers in the South. Less theology and more of human brotherhood, less declamation and more common sense and love for truth, must be the qualifications of the new ministry that shall yet save the race from the evils of false teaching.”
—Fannie Barrier Williams (1855–1944)
“Strange and predatory and truly dangerous, car thieves and muggers—they seem to jeopardize all our cherished concepts, even our self-esteem, our property rights, our powers of love, our laws and pleasures. The only relationship we seem to have with them is scorn or bewilderment, but they belong somewhere on the dark prairies of a country that is in the throes of self-discovery.”
—John Cheever (1912–1982)
“It is not the simple statement of facts that ushers in freedom; it is the constant repetition of them that has this liberating effect. Tolerance is the result not of enlightenment, but of boredom.”
—Quentin Crisp (b. 1908)
“The rebel, unlike the revolutionary, does not attempt to undermine the social order as a whole. The rebel attacks the tyrant; the revolutionary attacks tyranny. I grant that there are rebels who regard all governments as tyrannical; nonetheless, it is abuses that they condemn, not power itself. Revolutionaries, on the other hand, are convinced that the evil does not lie in the excesses of the constituted order but in order itself. The difference, it seems to me, is considerable.”
—Octavio Paz (b. 1914)
“Women over fifty already form one of the largest groups in the population structure of the western world. As long as they like themselves, they will not be an oppressed minority. In order to like themselves they must reject trivialization by others of who and what they are. A grown woman should not have to masquerade as a girl in order to remain in the land of the living.”
—Germaine Greer (b. 1939)