Lie group, semi-simple

A connected Lie group that does not contain non-trivial connected solvable (or, equivalently, connected Abelian) normal subgroups. A connected Lie group is semi-simple if and only if its Lie algebra is semi-simple (cf. Lie algebra, semi-simple). A connected Lie group $ G $ is said to be simple if its Lie algebra is simple, that is, if $ G $ does not contain non-trivial connected normal subgroups other than $ G $ . A connected Lie group is semi-simple if and only if it splits into a locally direct product of simple non-Abelian normal subgroups.

The classification of semi-simple Lie groups reduces to the local classification, that is, to the classification of semi-simple Lie algebras (cf. Lie algebra, semi-simple), and also to the global classification of the Lie groups $ G $ that correspond to a given semi-simple Lie algebra $ \mathfrak g $ .

In the case of Lie groups over the field $ \mathbf C $ of complex numbers the main result of the local classification is that every simply-connected simple non-Abelian complex Lie group is isomorphic to one of the groups $ \mathop{\rm SL}\nolimits _{n+1} ( \mathbf C ) $ , $ n \geq 1 $ , $ \mathop{\rm Spin}\nolimits _{n} ( \mathbf C ) $ , $ n \geq 5 $ (the universal covering of the group $ \mathop{\rm SO}\nolimits _{n} ( \mathbf C ) $ ), $ \mathop{\rm Sp}\nolimits _{n} ( \mathbf C ) $ , $ n \geq 3 $ (see Classical group), or one of the exceptional complex Lie groups (see Lie algebra, exceptional). The global classification of the Lie groups corresponding to a semi-simple Lie algebra $ \mathfrak g $ over $ \mathbf C $ goes as follows. Let $ \mathfrak h $ be a Cartan subalgebra of $ \mathfrak g $ and let $ \Sigma $ be the root system of $ \mathfrak g $ with respect to $ \mathfrak h $ . To every semi-simple Lie group $ G $ with Lie algebra $ \mathfrak g $ corresponds a lattice $ \Gamma (G) \subset \mathfrak h $ that is the kernel of the exponential mapping $ \mathop{\rm exp}\nolimits : \ \mathfrak h \rightarrow G $ . In particular, if $ G $ is simply connected, then $ \Gamma (G) $ coincides with the lattice $ \Gamma _{0} = \Gamma _{0} ( \mathfrak g ) $ generated by the elements $ 2 \pi i H _ \alpha $ , $ \alpha \in \Sigma $ (see Lie algebra, semi-simple), and if $ G $ is a group without centre (an adjoint group), then $ \Gamma (G) $ is the lattice $$ \Gamma _{1} = \Gamma _{1} ( \mathfrak g ) = \{ {X \in \mathfrak h} : {\alpha (X) \in 2 \pi i \mathbf Z \textrm{ for all } \alpha \in \Sigma} \} . $$ In the general case $ \Gamma _{0} \subset \Gamma (G) \subset \Gamma _{1} $ . For any additive subgroup $ M \subset \mathfrak h $ satisfying the condition $ \Gamma _{0} \subset M \subset \Gamma _{1} $ there is a unique (up to isomorphism) connected Lie group $ G $ with Lie algebra $ \mathfrak g $ such that $ \Gamma (G) = M $ . The centre of $ G $ is isomorphic to $ \Gamma _{1} / \Gamma (G) $ , and for the fundamental group one has: $$ \pi _{1} (G) \cong \Gamma (G) / \Gamma _{0} . $$ The quotient group $ Z _ {\mathfrak g} = \Gamma _{1} / \Gamma _{0} $ (the centre of the simply-connected Lie group with Lie algebra $ \mathfrak g $ ) is finite and for the different types of simple Lie algebras $ \mathfrak g $ it has the following form:

<tbody> </tbody> $ \mathfrak g $ $ A _{n} $ $ B _{n} $ $ C _{n} $ $ D _{2n} $ $ D _{2n+1} $ $ E _{6} $ $ E _{7} $ $ E _{8} $ , $ F _{4} $ , $ G _{2} $ $ Z _{\mathfrak g} $ $ \mathbf Z _{n+1} $ $ \mathbf Z _{2} $ $ \mathbf Z _{2} $ $ \mathbf Z _{2} \oplus \mathbf Z _{2} $ $ \mathbf Z _{4} $ $ \mathbf Z _{3} $ $ \mathbf Z _{2} $ $ 0 $

The order of the group $ \Gamma _{1} / \Gamma _{0} $ is the same as the number of vertices with coefficient 1 in the extended Dynkin diagram of $ \mathfrak g $ ; discarding one of the vertices gives the Dynkin diagram. A similar classification holds for compact real semi-simple Lie groups, each of which is imbedded in a unique complex semi-simple Lie group as a maximal compact subgroup (see Lie group, compact).

The global classification of non-compact real semi-simple Lie groups can be carried out in a similar but more complicated way. In particular, the centre $ Z _ {\mathfrak g} $ of the simply-connected Lie group corresponding to a semi-simple Lie algebra $ \mathfrak g $ over $ \mathbf R $ can be calculated as follows. Let $ \mathfrak g = \mathfrak k + \mathfrak p $ be the Cartan decomposition, where $ \mathfrak k $ is a maximal compact subalgebra of $ \mathfrak g $ and $ \mathfrak p $ is its orthogonal complement with respect to the Killing form, let $ \theta $ be the corresponding involutive automorphism, extended to $ \mathfrak g ^ {\mathbf C} $ , $ \mathfrak h $ the Cartan subalgebra of $ \mathfrak g ^ {\mathbf C} $ containing a Cartan subalgebra $ \mathfrak h ^ \prime \subset \mathfrak k $ , $ \theta _{0} $ an automorphism of $ \mathfrak g ^ {\mathbf C} $ that coincides with $ \theta $ on the roots with respect to $ \mathfrak h $ and extended to the root vectors in an appropriate way, and $ \mathfrak g _{0} = \mathfrak k _{0} + \mathfrak p _{0} $ the Cartan decomposition of the real form $ \mathfrak g \subset \mathfrak g ^ {\mathbf C} $ corresponding to $ \theta _{0} $ . Then $ Z _{\mathfrak g} \cong \Gamma _{1} ( \mathfrak k _{0} ) / \Gamma _{0} [ \mathfrak k ,\ \mathfrak k ] $ (see [3], where this group is calculated for all types of simple algebras $ \mathfrak g $ over $ \mathbf R $ ).

Every complex semi-simple Lie group $ G $ has the unique structure of an affine algebraic group compatible with the analytic structure specified on it, and any analytic homomorphism of $ G $ to an algebraic group is rational. The corresponding algebra of regular functions on $ G $ coincides with the algebra of holomorphic representation functions. On the other hand, a non-compact real semi-simple Lie group does not always admit a faithful linear representation — the simplest example is the simply-connected Lie group corresponding to the Lie algebra $ \mathop{\rm sl}\nolimits ( 2 ,\ \mathbf R ) $ . If $ \mathfrak g $ is a semi-simple Lie algebra over $ \mathbf R $ , then in the centre $ Z _{\mathfrak g} $ of the simply-connected group $ G _{0} $ corresponding to $ \mathfrak g $ there is a smallest subgroup $ {\mathbf L} ( \mathfrak g ) $ , called the linearizer, such that $ G _{0} / {\mathbf L} ( \mathfrak g ) $ is isomorphic to a linear semi-simple Lie group. If $ \mathfrak u = \mathfrak k + i \mathfrak u $ is the compact real form of $ \mathfrak g ^{\mathbf C} $ , then $$ {\mathbf L} ( \mathfrak g ) \cong \Gamma _{0} ( \mathfrak u ) \cap \mathfrak h ^ \prime / \Gamma _{0} ( [ \mathfrak k ,\ \mathfrak k ] ) $$ (see [3], where this group is calculated for all types of simple Lie algebras $ \mathfrak g $ ).

References

[1]	J.F. Adams, "Lectures on Lie groups" , Benjamin (1969) MR0252560 Zbl 0206.31604
[2]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[3]	A.I. Sirota, A.S. Solodovnikov, "Noncompact semisimple Lie groups" Russian Math. Surveys , 18 : 3 (1963) pp. 85–140 Uspekhi Mat. Nauk , 18 : 3 (1963) pp. 87–144 MR155929 Zbl 0132.02101

Comments

The (infinite-dimensional) representation theory of semi-simple Lie groups over $ \mathbf R $ has been created for a large part by Harish-Chandra. See also the excellent survey of Harish-Chandra's work by V.S. Varadarajan in the collected works [a1].

The $ \mathop{\rm mod}\nolimits \ p $ reductions of semi-simple Lie groups over $ \mathbf Z $ are called the Chevalley groups (cf. Chevalley group), and from them most of the finite simple groups can be obtained (with the exception of the alternating group and the 26 sporadic groups, cf. Sporadic simple group). For a survey of the structure and representation theory of the Chevalley groups see [a2].

References

[a1]	Harish-Chandra, "Collected works" , 1–4 , Springer (1984) Zbl 0699.62084 Zbl 0653.01018
[a2]	R.W. Carter, "Finite groups of Lie type: Conjugacy classes and complex characters" , Wiley (Interscience) (1985)
[a3]	J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1974) MR0343214 Zbl 0281.53034
[a4]	G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) MR0207883 Zbl 0131.02702