Advanced Structure Theory

  • Anthony W. Knapp
Part of the Progress in Mathematics book series (PM, volume 140)


The first main results are that simply connected compact semisimple Lie groups are in one-one correspondence with abstract Cartan matrices and their associated Dynkin diagrams and that the outer automorphisms of such a group correspond exactly to automorphisms of the Dynkin diagram. The remainder of the first section prepares for the definition of a reductive Lie group: A compact connected Lie group has a complexification that is unique up to holomorphic isomorphism. A semisimple Lie group of matrices is topologically closed and has finite center.

Reductive Lie groups G are defined as 4-tuples (G, K,θ, B) satisfying certain compatibility conditions. Here G is a Lie group, K is a compact subgroup, B is an involution of the Lie algebra go of G, and B is a bilinear form on go. Examples include semisimple Lie groups with finite center, any connected closed linear group closed under conjugate transpose, and the centralizer in a reductive group of a θ stable abelian subalgebra of the Lie algebra. The involution θ, which is called the “Cartan involution” of the Lie algebra, is the differential of a global Cartan involution Θ of G. In terms of Θ, G has a global Cartan decomposition that generalizes the polar decomposition of matrices.

A number of properties of semisimple Lie groups with finite center generalize to reductive Lie groups. Among these are the conjugacy of the maximal abelian subspaces of the —1 eigenspace p0 of θ, the theory of restricted roots, the Iwasawa decomposition, and properties of Cartan subalgebras. The chapter addresses also some properties not discussed in Chapter VI, such as the K Ap K decomposition and the Bruhat decomposition. Here A p is the analytic subgroup corresponding to a maximal abelian subspace of p0.

The degree of disconnectedness of the subgroup M p = Z K (A p) controls the disconnectedness of many other subgroups of G. The most complete description of M p is in the case that G has a complexification, and then serious results from Chapter V about representation theory play a decisive role.

Parabolic subgroups are closed subgroups containing a conjugate of M p A p N p. They are parametrized up to conjugacy by subsets of simple restricted roots. A Cartan subgroup is defined to be the centralizer of a Cartan subalgebra. It has only finitely many components, and each regular element of G lies in one and only one Cartan subgroup of G. When G has a complexification, the component structure of Cartan subgroups can be identified in terms of the elements that generate M p.

A reductive Lie group G that is semisimple has the property that G/K admits a complex structure with G acting holomorphically if and only if the centralizer in go of the center of the Lie algebra to of K is just to. In this case, G/K may be realized as a bounded domain in some ℂn by means of the Harish-Chandra decomposition. The proof of the Harish-Chandra decomposition uses facts about parabolic subgroups. The spaces G/K of this kind may be classified easily by inspection of the classification of simple real Lie algebras in Chapter VI.


Parabolic Subgroup Cartan Subalgebra Cartan Subgroup Iwasawa Decomposition Parabolic Subalgebra 
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© Anthony W. Knapp 1996

Authors and Affiliations

  • Anthony W. Knapp
    • 1
  1. 1.Department of MathematicsState University of New York at Stony BrookStony BrookUSA

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