Lie group, compact

A compact group that is a finite-dimensional real Lie group. Compact Lie groups can be characterized as finite-dimensional locally connected compact topological groups.

If $ G ^{0} $ is the connected component of the identity of a compact Lie group $ G $ , then the group of connected components $ G / G ^{0} $ is finite. The study of the structure of connected compact Lie groups is a basic topic in the theory of Lie groups.

The following examples of connected compact Lie groups play an important role in the general structure theory of compact Lie groups.

1) The multiplicative group $ T ^{1} $ of all complex numbers of modulus 1.

2) The group $ \mathop{\rm SU}\nolimits (n) $ of all complex unitary matrices of order $ n $ with determinant 1.

3) The group $ \mathop{\rm SO}\nolimits (n) $ of all real orthogonal matrices of order $ n $ with determinant 1.

4) The group $ \mathop{\rm Sp}\nolimits (n) $ of all matrices $ X \in \mathop{\rm SU}\nolimits (2n) $ for which $ X J X ^{t} = J $ , where $$ J = \left \| \begin{array}{cc} 0 &{E _{n}} \\ {- E _{n}} & 0 \\ \end{array} \ \right \| , $$ $ {}^{t} $ is the transposition sign and $ E _{n} $ is the unit matrix of order $ n $ .

A complete classification of connected compact Lie groups was obtained in the works of E. Cartan [1] and H. Weyl [2]. It is as follows.

There are two basic types of connected compact Lie groups.

1) Connected commutative compact Lie groups. These are precisely the tori, that is, groups of the form $ T ^{n} = T ^{1} \times \dots \times T ^{1} $ ( $ n $ factors) (cf. also Torus).

2) Connected semi-simple compact Lie groups (see Lie group, semi-simple). If $ G $ is a connected semi-simple compact Lie group, then the universal covering group $ G tilde $ of $ G $ is also a compact Lie group (Weyl's theorem). The centre $ Z $ of $ G tilde $ is finite, and all connected Lie groups locally isomorphic to $ G $ are compact and are, up to isomorphism, the groups of the form $ G tilde / D $ , where $ D \subset Z $ . The Lie algebras of semi-simple compact Lie groups can be intrinsically characterized among all finite-dimensional real Lie algebras as algebras with negative-definite Killing form.

The two basic types of connected Lie groups indicated above determine the structure of arbitrary compact Lie groups. Namely, the latter are, up to isomorphism, all groups of the form $ ( G \times T ) / D $ , where $ G $ is a connected simply-connected compact Lie group with centre $ Z $ , $ T $ is a torus and $ D $ is a finite subgroup of the group $ Z \times T $ which intersects $ T $ only in the identity. The Lie algebras of arbitrary compact Lie groups can also be characterized intrinsically among all finite-dimensional real Lie algebras: they are precisely the Lie algebras $ \mathfrak g $ that have a positive-definite scalar product $ ( , ) $ such that $ ( [ x ,\ y ] ,\ z ) + ( y ,\ [ x ,\ z ] ) = 0 $ for any $ x ,\ y ,\ z \in \mathfrak g $ . They are called compact Lie algebras.

Thus, the classification of connected compact Lie groups reduces to the classification of connected simply-connected semi-simple compact Lie groups (or, equivalently, to the classification of semi-simple compact Lie algebras) and a description of their centres. It turns out that semi-simple compact Lie algebras are in one-to-one correspondence with semi-simple complex Lie algebras (and therefore with reduced root systems, cf. Root system). Namely, if $ \mathfrak g $ is a semi-simple compact Lie algebra, then its complexification $ \mathfrak g _ {\mathbf C} = \mathfrak g \otimes _ {\mathbf R} \mathbf C $ is semi-simple. Conversely, in any semi-simple Lie algebra over $ \mathbf C $ there is a unique (up to conjugacy) compact real form (cf. Form of an (algebraic) structure). In particular, the final result on the classification of simple compact Lie algebras and the connected simply-connected compact Lie groups corresponding to them is the following. There are four infinite series of so-called classical simple compact Lie algebras that correspond to the following series of irreducible reduced root systems: $ A _{n} $ , $ n \geq 1 $ , $ B _{n} $ , $ n \geq 2 $ , $ C _{n} $ , $ n \geq 3 $ , and $ D _{n} $ , $ n \geq 4 $ . They are, respectively, the Lie algebras of the groups $ \mathop{\rm SU}\nolimits ( n + 1 ) $ , $ \mathop{\rm SO}\nolimits ( 2 n + 1) $ , $ \mathop{\rm Sp}\nolimits (n) $ , and $ \mathop{\rm SO}\nolimits (2n) $ . Apart from these there are the five so-called exceptional simple compact Lie algebras corresponding to the root systems of the types $ G _{2} $ , $ F _{4} $ , $ E _{6} $ , $ E _{7} $ , and $ E _{8} $ . Any compact simple Lie algebra is isomorphic to one of these Lie algebras, and they themselves are not pairwise isomorphic to one another. The compact Lie groups $ \mathop{\rm SU}\nolimits (n) $ and $ \mathop{\rm Sp}\nolimits (n) $ , $ n \geq1 $ , are connected and simply connected. The group $ \mathop{\rm SO}\nolimits (n) $ , $ n \geq 3 $ , is connected but not simply connected. Its universal covering is called the spinor compact Lie group and is denoted by $ \mathop{\rm Spin}\nolimits (n) $ . The centre of a connected simply-connected semi-simple compact Lie group coincides with the centre of the corresponding simply-connected complex Lie group (see Lie group, semi-simple).

Any compact Lie group has a faithful linear representation; the image of such a representation is a real algebraic group. Any compact Lie group $ G $ has a complexification $ G _ {\mathbf C} $ (see Complexification of a Lie group). Moreover, $ G _ {\mathbf C} $ is a complex reductive algebraic group (cf. Reductive group) whose affine algebra $ A _{G} $ can be described as the algebra of all representation functions on $ G $ , that is, continuous complex-valued functions $ f $ such that the linear envelope of the translates of $ f $ by elements of $ G $ is finite-dimensional. The algebra $ A _{G} $ has a natural real structure and so it determines an algebraic group over $ \mathbf R $ . The real points of this group form $ G $ , and the complex points form $ G _ {\mathbf C} $ . The group $ G $ is a maximal compact subgroup in $ G _ {\mathbf C} $ . As a result one obtains a one-to-one correspondence between the classes of isomorphic compact Lie groups and reductive algebraic groups over $ \mathbf C $ .

Any compact Lie group is a real-analytic group. Compact complex-analytic groups are also called complex compact Lie groups. Any connected complex compact Lie group (as a complex Lie group) is isomorphic to a complex torus $ \mathbf C ^{n} / \Gamma $ , where $ \Gamma $ is a discrete subgroup of rank $ 2n $ in $ \mathbf C ^{n} $ , and (as a real Lie group) it is isomorphic to $ T ^{2n} $ . Two complex tori $ \mathbf C ^{n} / \Gamma _{1} $ and $ \mathbf C ^{n} / \Gamma $ are isomorphic (as complex Lie groups) if and only if $ \Gamma = g ( \Gamma _{1} ) $ for some $ g \in \mathop{\rm GL}\nolimits _{n} ( \mathbf C ) $ .

References

[1]	E. Cartan, "La topologie des groupes de Lie" , Hermann (1936) Zbl 0016.10401 Zbl 0015.20401 Zbl 62.0441.03 Zbl 62.0441.02
[2]	H. Weyl, Uspekhi Mat. Nauk , 4 (1938) pp. 201–246
[3]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[4]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1955) Zbl 0068.02102
[5]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
[6]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[7]	A.L. Onishchik, E.B. Vinberg, "Seminar on algebraic groups and Lie groups, 1967/68" , Moscow (1969) (In Russian)
[8]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[9]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[10]	J.F. Adams, "Lectures on Lie groups" , Benjamin (1969) MR0252560 Zbl 0206.31604

References

[a1]	Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) MR0781344 Zbl 0581.22009