Lie group

A group $ G $ having the structure of an analytic manifold such that the mapping $ \mu : \ ( x ,\ y ) \rightarrow x y ^{-1} $ of the direct product $ G \times G $ into $ G $ is analytic. In other words, a Lie group is a set endowed with compatible structures of a group and an analytic manifold. A Lie group is said to be real, complex or $ p $ -adic, depending on the field over which its analytic manifold is considered. Henceforth, as a rule, real Lie groups are considered (every complex Lie group is naturally endowed with the structure of a real Lie group by means of restriction of the ground field; for Lie groups over $ p $ -adic number fields see Lie group, $ p $ -adic; Analytic group).

Examples of Lie groups. The general linear group $ \mathop{\rm GL}\nolimits ( n ,\ \mathbf R ) $ over the field $ \mathbf R $ of real numbers (see also Linear group) and its subgroups, closed in the natural Euclidean topology (J. von Neumann, 1927).

The main concepts of the theory of Lie groups were introduced into mathematics in the 1870-s by S. Lie. Lie groups arose in connection with the problem of the solvability of differential equations by quadratures and research into continuous transformation groups. The successful application of group theory to the solution of algebraic equations of higher degrees, which manifested itself in the creation of Galois theory, suggested an attempt to construct an analogue of Galois theory for differential equations. Although groups took up a position in the theory of differential equations somewhat different from that in the theory of algebraic equations, this led to the creation of the theory of Lie groups, and also to the theory of algebraic groups, which has deep connections with many branches of mathematics. Lie groups were originally defined as local transformation groups of the $ n $ -dimensional space $ \mathbf R ^{n} $ (or $ \mathbf C ^{n} $ ) that depend analytically on a finite system of parameters, and it was required that the parameters of a product of transformations be expressible in terms of the parameters of the factors by means of analytic functions. Later on, mathematicians turned to the abstract consideration of Lie groups, but also from the local point of view (see Lie group, local). Systematic research into the global structure of Lie groups was first begun by E. Cartan and H. Weyl. The first modern account of the theory of Lie groups was given in 1938 by L.S. Pontryagin (see [1]).

Does the replacement of analyticity of the manifold $ G $ and the mapping $ \mu $ by differentiability lead to an extension of the class of Lie groups? This question was solved by Lie: If $ \mu $ is twice continuously differentiable, then $ G $ is a Lie group. Hilbert's fifth problem turned out to be considerably more complicated: If $ G $ is an $ n $ -dimensional topological manifold and the mapping $ \mu : \ ( x ,\ y ) \rightarrow x y ^{-1} $ is continuous, is $ G $ a Lie group? For compact groups this problem was affirmatively solved by von Neumann in 1933, and for locally compact Abelian groups by Pontryagin in 1934. In the general case an affirmative answer was obtained in 1952 by A.M. Gleason, D. Montgomery and L. Zippin (see [4], and also [18]). Thus one can define a Lie group as a topological group whose topological space is a finite-dimensional (or locally Euclidean) manifold. This is very important for the general theory of topological groups.

A subset $ H $ of a Lie group $ G $ is called a subgroup (more precisely, a Lie subgroup) if $ H $ is a subgroup of the abstract group $ G $ and a submanifold of the analytic manifold $ G $ . A morphism between Lie groups $ G _{1} $ and $ G _{2} $ is an analytic mapping $ f : \ G _{1} \rightarrow G _{2} $ that is a homomorphism of abstract groups; if $ f $ is also bijective and $ f ^ {\ -1} $ is analytic, then $ f $ is called an isomorphism of Lie groups; in the case when $ f $ is locally bijective (around the identity $ e $ ) one says that the Lie groups $ G _{1} $ and $ G _{2} $ are locally isomorphic. The dimension of a Lie group $ G $ is the dimension of $ G $ as an analytic manifold. Henceforth only finite-dimensional Lie groups are considered, although many results can be generalized to the case of Banach Lie groups (cf. Lie group, Banach). Let $ H $ be a closed normal subgroup of a finite-dimensional Lie group $ G $ . Then the quotient group $ G / H $ can be endowed with the structure of an analytic manifold such that $ G / H $ becomes a Lie group and the canonical mapping $ G \rightarrow G / H $ is a morphism.

The correspondence between Lie groups and Lie algebras.

The main method of research in the theory of Lie groups is the infinitesimal method created by Lie. This method makes it possible to reduce the study of such a complicated object as a Lie group largely to the study of a purely algebraic object, a Lie algebra. To every Lie group $ G $ corresponds a Lie algebra $ L (G) $ , constructed as follows (see also Lie algebra of an analytic group). A left-invariant vector field on $ G $ is a vector field that is invariant with respect to the differentials of left translations, that is, $ X $ is a left-invariant vector field if $ ( d L _{g} ) X (h) = X (g h) $ for any $ g ,\ h \in G $ , where $ L _{g} (h) = g h $ . The left-invariant vector fields on $ G $ form a vector space that can be identified with the tangent space $ T _{e} (G) $ at the identity $ e $ of the group $ G $ , by associating with the field $ X $ its value at $ e $ . If $ X ,\ Y \in T _{e} (G) $ , then the Lie bracket $ X \cso Y - Y \cso X $ is also a left-invariant field, and this defines in $ T _{e} (G) $ a bilinear operation with respect to which $ T _{e} (G) $ becomes the Lie algebra $ L (G) $ (here $ \cso $ denotes composition of vector fields, regarded as derivations of the algebra of infinitely-differentiable real-valued functions on the manifold $ G $ ). A more explicit construction of the commutation operation $ [ X ,\ Y ] $ in $ L (G) $ is as follows: Let $ x (t) ,\ y (t) $ be the integral curves of the fields $ X ,\ Y $ in $ G $ passing through the identity $ e $ of the group. Then $ [ X ,\ Y ] $ is the tangent vector at $ e $ to the curve $$ z (s) = x ( \sqrt s ) y ( \sqrt s ) x ( \sqrt s ) ^{-1} y ( \sqrt s ) ^{-1} . $$ Recovering a Lie group $ G $ from its Lie algebra $ L (G) $ is possible by the exponential mapping $ \mathop{\rm exp}\nolimits \ : \ L (G) \rightarrow G $ , which associates with a field $ X \in L (G) $ the element $ x (1) $ of its integral curve $ x (t) $ . If $ G $ is a linear Lie group, that is, a subgroup of the general linear group $ \mathop{\rm GL}\nolimits ( n ,\ \mathbf R ) $ , then $ L (G) $ can be identified with a subalgebra of the general matrix Lie algebra $ L ( n ,\ \mathbf R ) $ and the exponential mapping takes the form $$ \mathop{\rm exp}\nolimits \ X = \sum _{m=0} ^ \infty \frac{1}{m!} X ^{m} . $$ The mapping $ \mathop{\rm exp}\nolimits \ : \ L (G) \rightarrow G $ is analytic and a local isomorphism, and so in some neighbourhood of the identity of the group $ G $ it defines a local chart (canonical coordinates). According to the Campbell–Hausdorff formula the multiplication in $ G $ in canonical coordinates, that is, the locally defined mapping $$ ( X ,\ Y ) \rightarrow \mathop{\rm exp}\nolimits ^{-1} ( \mathop{\rm exp}\nolimits \ X \ \mathop{\rm exp}\nolimits \ Y ) , X ,\ Y \in L (G) , $$ can be given in terms of operations in the Lie algebra $ L (G) $ . Thus, locally a Lie group is completely determined by its Lie algebra.

The correspondence between Lie groups and Lie algebras has deep functorial properties. A Lie group is determined by its Lie algebra up to a local isomorphism; in particular, if two Lie groups $ G _{1} $ and $ G _{2} $ are connected and simply connected, then the isomorphy of their Lie algebras implies $ G _{1} \cong G _{2} $ . Arcwise-connected subgroups of a Lie group $ G $ correspond one-to-one to subalgebras of the Lie algebra $ L (G) $ . Let $ f : \ G _{1} \rightarrow G _{2} $ be a morphism of Lie groups. Then the differential of this morphism at the identity is a homomorphism of Lie algebras: $$ d f _{e} : \ L ( G _{1} ) \rightarrow L ( G _{2} ) . $$ In general, not every homomorphism $ L ( G _{1} ) \rightarrow L ( G _{2} ) $ has the form $ d f _{e} $ , but if $ G _{1} $ is simply connected this is the case. An arcwise-connected subgroup $ H $ of a connected Lie group $ G $ is normal if and only if $ L (H) $ is an ideal of the Lie algebra $ L (G) $ ; if in addition $ H $ is closed in $ G $ , then $$ L ( G / H ) \cong L (G) / L (H) . $$ By construction, the Lie algebra $ L (G) $ of a given Lie group $ G $ is an analytic invariant. In reality $ L (G) $ is a topological invariant; this follows immediately from the following theorem of Cartan: A continuous homomorphic mapping of a (real) Lie group $ G $ into a Lie group $ H $ is a morphism. For complex Lie groups this assertion is not always true, although it holds for $ p $ -adic Lie groups (see [3]). The automorphism group $ \mathop{\rm Aut}\nolimits (G) $ of a connected Lie group $ G $ is a Lie group which can be identified with a Lie subgroup of $ \mathop{\rm Aut}\nolimits ( L (G) ) $ . In particular, if the Lie group $ G $ is simply-connected, then $$ \mathop{\rm Aut}\nolimits (G) \cong \mathop{\rm Aut}\nolimits ( L (G) ) \textrm{ and } L ( \mathop{\rm Aut}\nolimits (G) ) \cong D ( L (G) ) , $$ where $ D ( L (G) ) $ denotes the Lie algebra of derivations of the algebra $ L (G) $ . The correspondence $$ g \rightarrow A d (g) = d _{e} ( \mathop{\rm Int}\nolimits (g) ) , $$ where $ \mathop{\rm Int}\nolimits (g) $ is the inner automorphism implemented by the element $ g \in G $ , is called the adjoint representation of the Lie group $ G $ ; its differential is the adjoint representation $ x \rightarrow \mathop{\rm ad}\nolimits \ x $ of the Lie algebra $ L (G) $ . Cf. also Adjoint representation of a Lie group.

The global structure of Lie groups.

The existence of a global Lie group with a given real Lie algebra was proved in 1930 by Cartan. He also showed that a closed subgroup of a real Lie group is a Lie subgroup. Two types of Lie groups play a special role, namely: semi-simple and solvable ones (see Lie group, semi-simple; Lie group, solvable). A connected Lie group $ G $ is said to be semi-simple if it does not contain connected solvable normal subgroups other than the identity; if $ G $ does not contain non-trivial connected normal subgroups, it is said to be simple. The Lie algebra $ L (G) $ of a semi-simple, simple or solvable Lie group $ G $ is, respectively, a semi-simple, simple or solvable Lie algebra. The investigation of arbitrary Lie groups essentially reduces to the study of semi-simple and solvable Lie groups. Any Lie group $ G $ has a largest connected solvable normal subgroup, called the solvable radical and denoted by $ R (G) $ . In $ G $ there are maximal semi-simple subgroups. If $ S $ is one of them, then $ G = S \cdot R (G) $ , and all maximal semi-simple subgroups are conjugate; if $ G $ is simply connected, then $ S \cap R (G) = (e) $ and the product is semi-direct (the Levi–Mal'tsev theorem). The existence of this decomposition was first proved by E. Levi in 1905 for complex Lie algebras, and the conjugacy of the semi-simple components was established by A.I. Mal'tsev in 1942 (see [16], [3], and also Levi–Mal'tsev decomposition).

The most general fact about solvable Lie groups was obtained by Lie: Any connected solvable linear group over the field $ \mathbf C $ can be transformed to triangular form; that is, the description of connected solvable Lie groups reduces to the description of subgroups of the general triangular group $ T (n) \subset \mathop{\rm GL}\nolimits ( n ,\ \mathbf C ) $ . A detailed investigation of solvable subgroups was undertaken by Mal'tsev in [16].

In the study of the structure of semi-simple Lie groups an important role is played by their maximal compact subgroups, studied by Cartan in close connection with the theory of symmetric spaces (see [10]). According to Cartan's classical theorem, maximal compact subgroups of a semi-simple Lie group $ G $ are conjugate; if $ B $ is a maximal compact subgroup of $ G $ , there is a submanifold $ E \subset G $ , analytically isomorphic to a Euclidean space, such that $ G = B E $ and the mapping $ B \times E \rightarrow B E $ , $ ( b ,\ e ) \rightarrow b e $ , is an isomorphism of analytic manifolds. Thus, the topological structure of $ G $ is determined by the topological structure of $ B $ . Mal'tsev [16] extended Cartan's theorem to arbitrary connected Lie groups. Another decomposition of a connected Lie group into a product of a maximal compact subgroup and a Euclidean space was found by K. Iwasawa (see Iwasawa decomposition).

Linear representability.

From the very beginning of the development of the theory of Lie groups it was clear that arbitrary Lie groups are close to linear Lie groups. Lie proved that in many cases Lie groups are locally isomorphic to linear Lie groups. The general theorem was obtained by I.D. Ado in 1935: Any Lie group is locally isomorphic to a linear Lie group (see [15]). At the same time it is not difficult to give examples of Lie groups that are not linear, e.g. the simply-connected covering group of $ \mathop{\rm SL}\nolimits ( 2 ,\ \mathbf R ) $ , or (in the case of the field $ \mathbf C $ ) a complex compact torus. If $ G $ is a simply-connected solvable Lie group, then any Lie subgroup of it is simply connected and isomorphic to a linear Lie group. In the general case the following criterion has been found for linear representability [16]: A connected Lie group $ G $ is linear if and only if its radical $ R (G) $ and semi-simple quotient group $ G / R (G) $ are linear; in turn, for $ R (G) $ to be linearly representable it is necessary and sufficient that its commutator subgroup should be simply connected, and the linearity of the semi-simple Lie group $ G / R (G) $ depends on the structure of its centre. Compact, and also complex semi-simple, Lie groups are not only linear, but also linear algebraic groups (cf. Linear algebraic group) .

Classification.

One of the main problems in the theory of Lie groups is that of classifying arbitrary connected Lie groups up to isomorphism. In the class of all locally isomorphic connected Lie groups that have the same Lie algebra there is a unique simply-connected Lie group $ G _{0} $ , and any Lie group $ G $ of this class is isomorphic to $ G _{0} / N $ where $ N $ is a discrete central normal subgroup. Therefore, the classification of Lie groups reduces to the classification of finite-dimensional Lie algebras and the calculation of the centres of simply-connected Lie groups. On the other hand, it reduces to the classification of two fundamentally different types of groups: semi-simple and solvable (see Lie group, semi-simple, Lie group, solvable). At first glance solvable Lie groups have a simpler structure and their classification, it would seem, should not be difficult. However, this impression is deceptive and up to now (1988) there is no hope of obtaining a classification of solvable Lie groups. By contrast, semi-simple Lie groups have been completely classified. A complete classification of complex semi-simple Lie algebras was obtained by W. Killing in 1888–1890 (see [1], [3]). Since a complex semi-simple Lie algebra is a direct sum of simple subalgebras, it is sufficient to classify simple Lie algebras. It turns out that there are only nine types of complex simple Lie algebras, namely the four infinite series $$ A _{n} , n\geq 1, B _{n} , n \geq 2 , C _{n} , n \geq 3 , D _{n} , n \geq 4 , $$ and the five exceptional algebras $$ G _{2} , F _{4} , E _{6} , E _{7} , E _{8} $$ (see also Lie algebra, semi-simple). To the infinite series of complex simple Lie algebras correspond the classical linear Lie groups. The corresponding simply-connected groups have the form: type $ A _{n} $ — $ \mathop{\rm SL}\nolimits ( n+1 ,\ \mathbf C ) $ ; type $ B _{n} $ — $ \mathop{\rm Spin}\nolimits (f _{2n+1} ) $ , the spinor group corresponding to a non-singular quadratic form $ f _{2n+1} $ of dimension $ 2n + 1 $ ; type $ C _{n} $ — the symplectic group of degree $ 2n $ ; type $ D _{n} $ — $ \mathop{\rm Spin}\nolimits ( f _{2n} ) $ . It is not difficult to compute the centres of these groups. For example, the centre of $ \mathop{\rm SL}\nolimits ( n+1 ,\ \mathbf C ) $ is the cyclic group of order $ n+1 $ , and the centres of $ \mathop{\rm Spin}\nolimits ( f _{2n+1} ) $ and the symplectic group are the cyclic group of order 2. In this way one obtains a classification of complex semi-simple Lie groups. The classification of real semi-simple Lie groups turns out to be much more complicated and depends on the classification of their real forms (cf. also Form of an algebraic group). The most important point here is the existence for any complex semi-simple group $ G $ of a unique compact real form $ B $ ; this implies that the Lie algebra $ L (G) $ is isomorphic to $ L (B) \otimes _ {\mathbf R} \mathbf C $ , that is, it is obtained by complexifying the Lie algebra $ L (B) $ . Relying on this, Cartan in 1914 obtained a complete classification of the real forms of complex semi-simple Lie groups. In terms of Galois cohomology this is equivalent to a description of the set $ H ^{1} ( \mathbf R ,\ \mathop{\rm Aut}\nolimits (G) ) $ (see also Linear algebraic group).

Later Killing's method was perfected by Cartan and Weyl, which gave the possibility of solving a number of other classification problems, and also to develop the important theory of representations of Lie groups. A classification of semi-simple subgroups of the classical complex simple Lie groups has been obtained (see [17]).

Modern development and applications.

In the 1950-s a new step in the development of the theory of Lie groups was begun, which manifested itself, in particular, in the creation of the theory of algebraic groups (see Linear algebraic group). Earlier C. Chevalley (see ) had explained in detail the algebraic nature of the fundamental results of Lie group theory. The application of methods of algebraic geometry made it possible to illuminate these classical results in a new way and revealed new deep connections with the theory of functions, number theory, etc. The theory of $ p $ -adic Lie groups (cf. Lie group, $ p $ -adic) had a significant development (see [3], [6]). Lie groups are connected in practice with all main branches of mathematics: with geometry and topology through the theory of Lie transformation groups (cf. Lie transformation group), with analysis through the theory of linear representations, etc. The various applications of Lie groups to physics and mechanics are also extremely important.

References

[1]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[2]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[4]	H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502
[5]	E. Wigner, "Group theory and its applications to the quantum mechanics of atomic spectra" , Acad. Press (1959) (Translated from German) MR0106711
[6]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[7]	R.G. Steinberg, "Lectures on Chevalley groups" , Yale Univ. Press (1967) MR0476871 MR0466335 Zbl 0307.22001 Zbl 1196.22001
[8]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1962) Zbl 0106.02701
[9]	M. Hamermesh, "Group theory and its application to physical problems" , Addison-Wesley (1962) MR0136667 Zbl 0100.36704
[10]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038
[11]	N.G. Chebotarev, "The theory of Lie groups" , Moscow-Leningrad (1940) (In Russian)
[12a]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[12b]	C. Chevalley, "Théorie des groupes de Lie" , 2–3 , Hermann (1951–1955) MR0068552 MR0051242 MR0019623 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[13]	G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) MR0207883 Zbl 0131.02702
[14]	D. Montgomery, L. Zippin, "Topological transformation groups" , Interscience (1964) MR0379739 MR1529355 MR0084705 MR0073104 MR0002529 Zbl 0323.57023 Zbl 0073.25802 Zbl 0068.01904 Zbl 0025.23701 Zbl 66.0959.03
[15]	I.D. Ado, "Note on the representation of finite continuous groups by means of linear substitutions" Izv. Fiz.-Mat. Obsch. (Kazan') , 7 (1935) pp. 1–43 (In Russian)
[16]	A.I. Mal'tsev, "On the theory of Lie groups in the large" Mat. Sb. , 16 (58) (1945) pp. 163–190 (In Russian) Zbl 0061.04602
[17]	A.I. Mal'tsev, "On semisimple subgroups of Lie groups" Transl. Amer. Math. Soc. (1) , 9 (1950) pp. 172–213 Izv. Akad. Nauk SSSR Ser. Mat. , 8 : 4 (1944) pp. 143–174
[18]	"Hilbert problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479
[19]	H. Freudenthal, H. de Vries, "Linear Lie groups" , Acad. Press (1969) MR0260926 Zbl 0377.22001

Comments

For an early account of Lie groups see also [a1]. Concerning Ado's theorem one may also consult [a2]. See [10] for Cartan's classification of real forms of complex semi-simple Lie groups.

The theorem that a $ C ^{2} $ group is analytic is due to F. Schur (1893), working out less detailed statements of Lie on this matter. The Campbell–Hausdorff formula also goes back to Schur (1891).

The main new development since the 1950-s is the creation of the representation theory of non-compact semi-simple Lie groups, for a large part by Harish-Chandra [a3].

The definition of local isomorphy of two Lie groups given in the main article above is not quite the usual one. Usually, two Lie groups $ G _{1} $ , $ G _{2} $ are called locally isomorphic if there are neighbourhoods of the identity $ U _{1} $ of $ G _{1} $ and $ U _{2} $ of $ G _{2} $ such that there is an isomorphism $ f : \ U _{1} \rightarrow U _{2} $ of analytic manifolds for which $ x ,\ y ,\ x y \in U _{1} $ implies $ f (x) f (y) = f ( x y ) $ and $ u ,\ v ,\ u v \in U _{2} $ implies $ f ^ {\ -1} (u) f ^ {\ -1} (v) = f ^ {\ -1} ( u v ) $ .

References

[a1]	W. Mayer, T.Y. Thomas, "Foundations of the theory of Lie groups" Ann. of Math. , 36 (1935) pp. 770–822 MR1503252 Zbl 0012.05502 Zbl 61.0474.01
[a2]	I.D. Ado, "The representation of Lie algebras by matrices" Transl. Amer. Math. Soc. (1) , 9 (1962) pp. 308–327 Uspekhi Mat. Nauk. , 2 (1947) pp. 159–173 MR0030946 MR0027753
[a3]	Harish-Chandra, "Collected works" , 1–4 , Springer (1984) Zbl 0699.62084 Zbl 0653.01018