Жатық топтар және дифференциалдық манифольдтар

Group that is also a differentiable manifold with group operations that are smooth

In mathematics, a Lie group (pronounced /l//iː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For any rotation of the circle, there exists the same symmetry, and concatenation of such rotations makes them into the circle group, an archetypal example of a Lie group. Lie groups are widely used in many parts of modern mathematics and physics. Lie groups were first found by studying matrix subgroups contained in or , the groups of invertible matrices over or These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.

Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra. Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance preserving transformations of the Euclidean space , conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G structure, where G is a Lie group of "local" symmetries of a manifold. Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the representations of the Lie group (or of its Lie algebra) are especially important. Representation theory is used extensively in particle physics. Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory. In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; p adic Lie groups play an important role, via their connections with Galois representations in number theory.

A real Lie group is a group that is also a finite dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication

means that μ is a smooth mapping of the product manifold G × G into G. The two requirements can be combined to the single requirement that the mapping

be a smooth mapping of the product manifold into G.

We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by

with a fixed irrational number, is a subgroup of the torus that is not a Lie group when given the subspace topology. If we take any small neighborhood of a point in , for example, the portion of in is disconnected. The group winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of
The group can, however, be given a different topology, in which the distance between two points is defined as the length of the shortest path in the group joining to In this topology, is identified homeomorphically with the real line by identifying each element with the number in the definition of With this topology, is just the group of real numbers under addition and is therefore a Lie group. The group is an example of a "Lie subgroup" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.

A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: ), and holomorphic maps. Similarly, using an alternate metric completion of , one can define a p adic Lie group over the p adic numbers, a topological group which is also an analytic p adic manifold, such that the group operations are analytic. In particular, each point has a p adic neighborhood. Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, Montgomery and Zippin showed that if G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group (see also Hilbert–Smith conjecture). If the underlying manifold is allowed to be infinite dimensional (for example, a Hilbert manifold), then one arrives at the notion of an infinite dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups. The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups. This categorical point of view leads also to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement.

A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup G of the general linear group such that
for some neighborhood V of the identity element e in G, the topology on V is the subspace topology of and V is closed in G has at most countably many connected components. (For example, a closed subgroup of ; that is, a matrix Lie group satisfies the above conditions.) Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:
Given a Lie group G in the usual manifold sense, the Lie group–Lie algebra correspondence (or a version of Lie's third theorem) constructs an immersed Lie subgroup such that share the same Lie algebra; thus, they are locally isomorphic. Hence, G satisfies the above topological definition. Conversely, let G be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group that is locally isomorphic to G. Then, by a version of the closed subgroup theorem, is a real analytic manifold and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series; so the group operations are real analytic and G itself is a real analytic manifold. The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group.

Lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.

The only connected Lie groups with dimension one are the real line (with the group operation being addition) and the circle group of complex numbers with absolute value one (with the group operation being multiplication). The group is often denoted as , the group of unitary matrices. In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples".

The group SU(2) is the group of unitary matrices with determinant Topologically, is the sphere ; as a group, it may be identified with the group of unit quaternions. The Heisenberg group is a connected nilpotent Lie group of dimension , playing a key role in quantum mechanics. The Lorentz group is a 6 dimensional Lie group of linear isometries of the Minkowski space. The Poincaré group is a 10 dimensional Lie group of affine isometries of the Minkowski space. The exceptional Lie groups of types G2, F4, E6, E7, E8 have dimensions 14, 52, 78, 133, and 248. Along with the A B C D series of simple Lie groups, the exceptional groups complete the list of simple Lie groups. The symplectic group consists of all matrices preserving a symplectic form on It is a connected Lie group of dimension .

If G and H are Lie groups, then a Lie group homomorphism f : G → H is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic. The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let be a Lie group homomorphism and let be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements, then is a map between the corresponding Lie algebras:

which turns out to be a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity. Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group to a Lie group is an isomorphism of Lie groups if and only if it is bijective.

Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this direction is Lie's third theorem, which states that every finite dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the table of Lie groups for examples). An example of importance in physics are the groups SU(2) and SO(3). These two groups have isomorphic Lie algebras, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not. On the other hand, if we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one to one correspondence between isomorphism classes of finite dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.

A Lie group is said to be simply connected if every loop in can be shrunk continuously to a point in This notion is important because of the following result that has simple connectedness as a hypothesis:
Theorem: Suppose and are Lie groups with Lie algebras and and that is a Lie algebra homomorphism. If is simply connected, then there is a unique Lie group homomorphism such that , where is the differential of at the identity. Lie's third theorem says that every finite dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group. An example of a simply connected group is the special unitary group SU(2), which as a manifold is the 3 sphere. The rotation group SO(3), on the other hand, is not simply connected. (See Topology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half integer spin in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group SU(n), the spin group (double cover of rotation group) Spin(n) for , and the compact symplectic group Sp(n). Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups.

A Lie subgroup of a Lie group is a Lie group that is a subset of and such that the inclusion map from to is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of admits a unique smooth structure which makes it an embedded Lie subgroup of —i. e. a Lie subgroup such that the inclusion map is a smooth embedding. Examples of non closed subgroups are plentiful; for example take to be a torus of dimension 2 or greater, and let be a one parameter subgroup of irrational slope, i. e. one that winds around in G. Then there is a Lie group homomorphism with The closure of will be a sub torus in
The exponential map gives a one to one correspondence between the connected Lie subgroups of a connected Lie group and the subalgebras of the Lie algebra of Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of which determines which subalgebras correspond to closed subgroups.

One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time independent Schrödinger equation in quantum mechanics, Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator commutes with the action of SO(3) on the wave function (One important example of such a system is the Hydrogen atom, which has a single spherical orbital.) This assumption does not necessarily mean that the solutions are rotationally invariant functions. Rather, it means that the space of solutions to is invariant under rotations (for each fixed value of ). This space, therefore, constitutes a representation of SO(3). These representations have been classified and the classification leads to a substantial simplification of the problem, essentially converting a three dimensional partial differential equation to a one dimensional ordinary differential equation. The case of a connected compact Lie group K (including the just mentioned case of SO(3)) is particularly tractable. In that case, every finite dimensional representation of K decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified by Hermann Weyl. The classification is in terms of the "highest weight" of the representation. The classification is closely related to the classification of representations of a semisimple Lie algebra. One can also study (in general infinite dimensional) unitary representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the representations of the group SL(2,R) and the representations of the Poincaré group.