Lie Groups Beyond an Introduction pp 291-371 | Cite as

# Structure Theory of Semisimple Groups

## Abstract

Every complex semisimple Lie algebra has a compact real form, as a consequence of a particular normalization of root vectors whose construction uses the Isomorphism Theorem of Chapter II. If go is a real semisimple Lie algebra, then the use of a compact real form of (g_{0})^{ℂ} leads to the construction of a “Cartan involution” *θ* of go. This involution has the property that if go = *t* _{0} ⊕ p_{0} is the corresponding eigenspace decomposition or “Cartan decomposition”, then to ⊕ *i*po is a compact real form of (g_{0})^{ℂ}. Any two Cartan involutions of go are conjugate by an inner automorphism. The Cartan decomposition generalizes the decomposition of a classical matrix Lie algebra into its skew-Hermitian and Hermitian parts.

If *G* is a semisimple Lie group, then a Cartan decomposition g_{0} = t_{0} ⊕ p_{0} of its Lie algebra leads to a global decomposition *G = K* exp p_{0}, where *K* is the analytic subgroup of *G* with Lie algebra go. This global decomposition generalizes the polar decomposition of matrices. The group *K* contains the center of *G* and, if the center of *G* is finite, is a maximal compact subgroup of *G*.

The Iwasawa decomposition *G = K AN* exhibits closed subgroups *A* and *N* of *G* such that *A* is simply connected abelian, *N* is simply connected nilpotent, *A* normalizes *N*, and multiplication from *K* × *A* × *N* to *G* is a diffeomorphism onto. This decomposition generalizes the Gram-Schmidt orthogonalization process. Any two Iwasawa decompositions of *G* are conjugate. The Lie algebra a_{0} of *A* may be taken to be any maximal abelian subspace of p_{0}, and the Lie algebra of *N* is defined from a kind of root-space decomposition of g_{0} with respect to a_{0}. The simultaneous eigenspaces are called “restricted roots”, and the restricted roots form an abstract root system. The Weyl group of this system coincides with the quotient of normalizer by centralizer of a_{0} in *K*.

A Cartan subalgebra of g_{0} is a subalgebra whose complexification is a Cartan sub-algebra of (g_{0})^{ℂ}. One Cartan subalgebra of g_{0} is obtained by adjoining to the above ao a maximal abelian subspace of the centralizer of a_{0} in t_{0}. This Cartan subalgebra is *θ* stable. Any Cartan subalgebra of g_{0} is conjugate by an inner automorphism to a *θ* stable one, and the subalgebra built from a_{0} as above is maximally noncompact among all *θ* stable Cartan subalgebras. Any two maximally noncompact Cartan subalgebras are conjugate, and so are any two maximally compact ones. Cayley transforms allow one to pass between any two *θ* stable Cartan subalgebras, up to conjugacy.

A Vogan diagram of g_{0} superimposes certain information about the real form g_{0} on the Dynkin diagram of (go)^{ℂ}. The extra information involves a maximally compact *θ* stable Cartan subalgebra and an allowable choice of a positive system off roots. The effect of *θ* on simple roots is labeled, and imaginary simple roots are painted if they are “noncompact,” left unpainted if they are “compact”, Such a diagram is not unique for g_{0}, but it determines go up to isomorphism. Every diagram that looks formally like a Vogan diagram arises from some g_{0}.

Vogan diagrams lead quicakly to a classification of all simple real Lie algebras, the only diffcuylty being eliminating the redundancy in the chosice of postitigve system of roots. This difficylty is resolved by the Borel and de Siebenthal Theotem. Using a succession of Catkey transforms to pass form a maximally compact Cartan subalgebre to a maximally non compact Cartan subalgebre, on e readily identifies the restricted roots for each simple real Lie alebra.

## Keywords

Simple Root Dynkin Diagram Cartan Subalgebra Semisimple Group Iwasawa Decomposition## Preview

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