Definition
- G is a connected semisimple real Lie group.
- is the Lie algebra of G
- is the complexification of .
- θ is a Cartan involution of
- is the corresponding Cartan decomposition
- is a maximal abelian subalgebra of
- Σ is the set of restricted roots of, corresponding to eigenvalues of acting on .
- Σ+ is a choice of positive roots of Σ
- is a nilpotent Lie algebra given as the sum of the root spaces of Σ+
- K, A, N, are the Lie subgroups of G generated by and .
Then the Iwasawa decomposition of is
and the Iwasawa decomposition of G is
The dimension of A (or equivalently of ) is called the real rank of G.
Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a (disconnected) maximal compact subgroup provided the center of G is finite.
The restricted root space decomposition is
where is the centralizer of in and is the root space. The number is called the multiplicity of .
Read more about this topic: Iwasawa Decomposition
Famous quotes containing the word definition:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)