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Wikipedia

Lie group

In mathematics, a Lie group (pronounced /l/ LEE) is a group that is also a differentiable manifold.

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 rotational symmetry in three dimensions (given by the special orthogonal 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.

History edit

According to the most authoritative source on the early history of Lie groups,[1] Sophus Lie himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation.[1] Some of Lie's early ideas were developed in close collaboration with Felix Klein. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years.[2] Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe.[3] In 1884 a young German mathematician, Friedrich Engel, came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume Theorie der Transformationsgruppen, published in 1888, 1890, and 1893. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse.[4]

Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany.[5] Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.

Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen (The composition of continuous finite transformation groups).[6] The work of Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights.

In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.

Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups.[7] The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.

Overview edit

 
The set of all complex numbers with absolute value 1 (corresponding to points on the circle of center 0 and radius 1 in the complex plane) is a Lie group under complex multiplication: the circle group.

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.

Definitions and examples edit

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.

First examples edit

 
This is a four-dimensional noncompact real Lie group; it is an open subset of  . This group is disconnected; it has two connected components corresponding to the positive and negative values of the determinant.
  • The rotation matrices form a subgroup of GL(2, R), denoted by SO(2, R). It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle. Using the rotation angle   as a parameter, this group can be parametrized as follows:
 
Addition of the angles corresponds to multiplication of the elements of SO(2, R), and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.
  • The affine group of one dimension is a two-dimensional matrix Lie group, consisting of   real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form
 

Non-example edit

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.[8] 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  .

 
A portion of the group   inside  . Small neighborhoods of the element   are disconnected in the subset topology on  

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.

Matrix Lie groups edit

Let   denote the group of   invertible matrices with entries in  . Any closed subgroup of   is a Lie group;[9] Lie groups of this sort are called matrix Lie groups. Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall,[10] Rossmann,[11] and Stillwell.[12] Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.

  • The special linear groups over   and  ,   and  , consisting of   matrices with determinant one and entries in   or  
  • The unitary groups and special unitary groups,   and  , consisting of   complex matrices satisfying   (and also   in the case of  )
  • The orthogonal groups and special orthogonal groups,   and  , consisting of   real matrices satisfying   (and also   in the case of  )

All of the preceding examples fall under the heading of the classical groups.

Related concepts edit

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.

Topological definition edit

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.[13] First, we define an immersely linear Lie group to be a subgroup G of the general linear group   such that

  1. 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  .
  2. 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:

  1. 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.
  2. 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;[a] 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.

More examples of Lie groups edit

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.

Dimensions one and two edit

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".

Additional examples edit

  • 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  .

Constructions edit

There are several standard ways to form new Lie groups from old ones:

  • The product of two Lie groups is a Lie group.
  • Any topologically closed subgroup of a Lie group is a Lie group. This is known as the Closed subgroup theorem or Cartan's theorem.
  • The quotient of a Lie group by a closed normal subgroup is a Lie group.
  • The universal cover of a connected Lie group is a Lie group. For example, the group   is the universal cover of the circle group  . In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying universal cover, one guarantees a group structure (compatible with its other structures).

Related notions edit

Some examples of groups that are not Lie groups (except in the trivial sense that any group having at most countably many elements can be viewed as a 0-dimensional Lie group, with the discrete topology), are:

  • Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space, or the space of smooth functions from a manifold   to a Lie group  ,  . These are not Lie groups as they are not finite-dimensional manifolds.
  • Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the p-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "p-adic Lie groups".) In general, only topological groups having similar local properties to Rn for some positive integer n can be Lie groups (of course they must also have a differentiable structure).

Basic concepts edit

The Lie algebra associated with a Lie group edit

To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples:

  • The Lie algebra of the vector space Rn is just Rn with the Lie bracket given by
        [AB] = 0.
    (In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.)
  • The Lie algebra of the general linear group GL(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by
        [AB] = AB − BA.
  • If G is a closed subgroup of GL(n, C) then the Lie algebra of G can be thought of informally as the matrices m of M(n, C) such that 1 + εm is in G, where ε is an infinitesimal positive number with ε2 = 0 (of course, no such real number ε exists). For example, the orthogonal group O(n, R) consists of matrices A with AAT = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)T = 1, which is equivalent to m + mT = 0 because ε2 = 0.
  • The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroup G of GL(n, C), may be computed as
 [15][10] where exp(tX) is defined using the matrix exponential. It can then be shown that the Lie algebra of G is a real vector space that is closed under the bracket operation,  .[16]

The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of the representation we use.[17] To get around these problems we give the general definition of the Lie algebra of a Lie group (in 4 steps):

  1. Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [XY] = XY − YX, because the Lie bracket of any two derivations is a derivation.
  2. If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
  3. We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations Lg(h) = gh. This shows that the space of left invariant vector fields (vector fields satisfying Lg*XhXgh for every h in G, where Lg* denotes the differential of Lg) on a Lie group is a Lie algebra under the Lie bracket of vector fields.
  4. Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v^g = Lg*v. This identifies the tangent space TeG at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur   Thus the Lie bracket on   is given explicitly by [vw] = [v^, w^]e.

This Lie algebra   is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.

We could also define a Lie algebra structure on Te using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space Te.

The Lie algebra structure on Te can also be described as follows: the commutator operation

(x, y) → xyx−1y−1

on G × G sends (ee) to e, so its derivative yields a bilinear operation on TeG. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.

Homomorphisms and isomorphisms edit

If G and H are Lie groups, then a Lie group homomorphism f : GH 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.[18][b]

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.

Lie group versus Lie algebra isomorphisms edit

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.[19]

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,[20] but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.[21]

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.[22] (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.

Simply connected Lie groups edit

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:[23] 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).[24]

Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups.

The exponential map edit

The exponential map from the Lie algebra   of the general linear group   to   is defined by the matrix exponential, given by the usual power series:

 

for matrices  . If   is a closed subgroup of  , then the exponential map takes the Lie algebra of   into  ; thus, we have an exponential map for all matrix groups. Every element of   that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.[25]

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

For each vector   in the Lie algebra   of   (i.e., the tangent space to   at the identity), one proves that there is a unique one-parameter subgroup   such that  . Saying that   is a one-parameter subgroup means simply that   is a smooth map into   and that

 

for all   and  . The operation on the right hand side is the group multiplication in  . The formal similarity of this formula with the one valid for the exponential function justifies the definition

 

This is called the exponential map, and it maps the Lie algebra   into the Lie group  . It provides a diffeomorphism between a neighborhood of 0 in   and a neighborhood of   in  . This exponential map is a generalization of the exponential function for real numbers (because   is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because   is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because   with the regular commutator is the Lie algebra of the Lie group   of all invertible matrices).

Because the exponential map is surjective on some neighbourhood   of  , it is common to call elements of the Lie algebra infinitesimal generators of the group  . The subgroup of   generated by   is the identity component of  .

The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood   of the zero element of  , such that for   we have

 

where the omitted terms are known and involve Lie brackets of four or more elements. In case   and   commute, this formula reduces to the familiar exponential law  

The exponential map relates Lie group homomorphisms. That is, if   is a Lie group homomorphism and   the induced map on the corresponding Lie algebras, then for all   we have

 

In other words, the following diagram commutes,[26]

 

(In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.)

The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on C Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

Lie subgroup edit

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  .[27] 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.

Representations edit

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.[28] 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.

Classification edit

Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have tangent spaces at each point.

The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and Dn, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these.

Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.

A first key result is the Levi decomposition, which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup.

  • Connected compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S1 and simple compact Lie groups (which correspond to connected Dynkin diagrams).
  • Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Solvable groups are too messy to classify except in a few small dimensions.
  • Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Like solvable groups, nilpotent groups are too messy to classify except in a few small dimensions.
  • Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, SL(2, R) is simple according to the second definition but not according to the first. They have all been classified (for either definition).
  • Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras.[29] They are central extensions of products of simple Lie groups.

The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write

Gcon for the connected component of the identity
Gsol for the largest connected normal solvable subgroup
Gnil for the largest connected normal nilpotent subgroup

so that we have a sequence of normal subgroups

1 ⊆ GnilGsolGconG.

Then

G/Gcon is discrete
Gcon/Gsol is a central extension of a product of simple connected Lie groups.
Gsol/Gnil is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S1.
Gnil/1 is nilpotent, and therefore its ascending central series has all quotients abelian.

This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.

Infinite-dimensional Lie groups edit

Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces (as opposed to Euclidean space in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold.

The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie in Lie group. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.

Some of the examples that have been studied include:

See also edit

Notes edit

Explanatory notes edit

  1. ^ This is the statement that a Lie group is a formal Lie group. For the latter concept, see Bruhat.[14]
  2. ^ Hall only claims smoothness, but the same argument shows analyticity.[citation needed]

Citations edit

  1. ^ a b Hawkins 2000, p. 1.
  2. ^ Hawkins 2000, p. 2.
  3. ^ Hawkins 2000, p. 76.
  4. ^ Tresse, Arthur (1893). "Sur les invariants différentiels des groupes continus de transformations". Acta Mathematica. 18: 1–88. doi:10.1007/bf02418270.
  5. ^ Hawkins 2000, p. 43.
  6. ^ Hawkins 2000, p. 100.
  7. ^ Borel 2001.
  8. ^ Rossmann 2001, Chapter 2.
  9. ^ Hall 2015 Corollary 3.45
  10. ^ a b Hall 2015.
  11. ^ Rossmann 2001
  12. ^ Stillwell 2008
  13. ^ Kobayashi & Oshima 2005, Definition 5.3.
  14. ^ Bruhat, F. (1958). "Lectures on Lie Groups and Representations of Locally Compact Groups" (PDF). Tata Institute of Fundamental Research, Bombay.
  15. ^ Helgason 1978, Ch. II, § 2, Proposition 2.7.
  16. ^ Hall 2015 Theorem 3.20
  17. ^ But see Hall 2015, Proposition 3.30 and Exercise 8 in Chapter 3
  18. ^ Hall 2015 Corollary 3.50.
  19. ^ Hall 2015 Theorem 5.20
  20. ^ Hall 2015 Example 3.27
  21. ^ Hall 2015 Section 1.3.4
  22. ^ Hall 2015 Corollary 5.7
  23. ^ Hall 2015 Theorem 5.6
  24. ^ Hall 2015 Section 13.2
  25. ^ Hall 2015 Theorem 3.42
  26. ^ (PDF). State University of New York at Stony Brook. 2006. Archived from the original (PDF) on 28 September 2011. Retrieved 11 October 2014.
  27. ^ Hall 2015 Theorem 5.20
  28. ^ Hall 2015 Part III
  29. ^ Helgason 1978, p. 131.
  30. ^ Bäuerle, de Kerf & ten Kroode 1997

References edit

External links edit

  •   Media related to Lie groups at Wikimedia Commons
  • Journal of Lie Theory

group, confused, with, group, type, group, this, article, includes, list, general, references, lacks, sufficient, corresponding, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, june, 2023, learn, when, remove, th. Not to be confused with Group of Lie type or Ree group This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations June 2023 Learn how and when to remove this template message In mathematics a Lie group pronounced l iː LEE is a group that is also a differentiable manifold 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 rotational symmetry in three dimensions given by the special orthogonal group SO 3 displaystyle text SO 3 Lie groups are widely used in many parts of modern mathematics and physics Lie groups were first found by studying matrix subgroups G displaystyle G contained in GL n R displaystyle text GL n mathbb R or GL n C displaystyle text GL n mathbb C the groups of n n displaystyle n times n invertible matrices over R displaystyle mathbb R or C displaystyle mathbb C 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 Contents 1 History 2 Overview 3 Definitions and examples 3 1 First examples 3 2 Non example 3 3 Matrix Lie groups 3 4 Related concepts 3 5 Topological definition 4 More examples of Lie groups 4 1 Dimensions one and two 4 2 Additional examples 4 3 Constructions 4 4 Related notions 5 Basic concepts 5 1 The Lie algebra associated with a Lie group 5 2 Homomorphisms and isomorphisms 5 3 Lie group versus Lie algebra isomorphisms 5 4 Simply connected Lie groups 5 5 The exponential map 5 6 Lie subgroup 6 Representations 7 Classification 8 Infinite dimensional Lie groups 9 See also 10 Notes 10 1 Explanatory notes 10 2 Citations 11 References 12 External linksHistory editAccording to the most authoritative source on the early history of Lie groups 1 Sophus Lie himself considered the winter of 1873 1874 as the birth date of his theory of continuous groups Hawkins however suggests that it was Lie s prodigious research activity during the four year period from the fall of 1869 to the fall of 1873 that led to the theory s creation 1 Some of Lie s early ideas were developed in close collaboration with Felix Klein Lie met with Klein every day from October 1869 through 1872 in Berlin from the end of October 1869 to the end of February 1870 and in Paris Gottingen and Erlangen in the subsequent two years 2 Lie stated that all of the principal results were obtained by 1884 But during the 1870s all his papers except the very first note were published in Norwegian journals which impeded recognition of the work throughout the rest of Europe 3 In 1884 a young German mathematician Friedrich Engel came to work with Lie on a systematic treatise to expose his theory of continuous groups From this effort resulted the three volume Theorie der Transformationsgruppen published in 1888 1890 and 1893 The term groupes de Lie first appeared in French in 1893 in the thesis of Lie s student Arthur Tresse 4 Lie s ideas did not stand in isolation from the rest of mathematics In fact his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi on the theory of partial differential equations of first order and on the equations of classical mechanics Much of Jacobi s work was published posthumously in the 1860s generating enormous interest in France and Germany 5 Lie s idee fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Evariste Galois had done for algebraic equations namely to classify them in terms of group theory Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries In Lie s early work the idea was to construct a theory of continuous groups to complement the theory of discrete groups that had developed in the theory of modular forms in the hands of Felix Klein and Henri Poincare The initial application that Lie had in mind was to the theory of differential equations On the model of Galois theory and polynomial equations the driving conception was of a theory capable of unifying by the study of symmetry the whole area of ordinary differential equations However the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled Symmetry methods for ODEs continue to be studied but do not dominate the subject There is a differential Galois theory but it was developed by others such as Picard and Vessiot and it provides a theory of quadratures the indefinite integrals required to express solutions Additional impetus to consider continuous groups came from ideas of Bernhard Riemann on the foundations of geometry and their further development in the hands of Klein Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory the idea of symmetry as exemplified by Galois through the algebraic notion of a group geometric theory and the explicit solutions of differential equations of mechanics worked out by Poisson and Jacobi and the new understanding of geometry that emerged in the works of Plucker Mobius Grassmann and others and culminated in Riemann s revolutionary vision of the subject Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups a major stride in the development of their structure theory which was to have a profound influence on subsequent development of mathematics was made by Wilhelm Killing who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen The composition of continuous finite transformation groups 6 The work of Killing later refined and generalized by Elie Cartan led to classification of semisimple Lie algebras Cartan s theory of symmetric spaces and Hermann Weyl s description of representations of compact and semisimple Lie groups using highest weights In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris Weyl brought the early period of the development of the theory of Lie groups to fruition for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics but he also put Lie s theory itself on firmer footing by clearly enunciating the distinction between Lie s infinitesimal groups i e Lie algebras and the Lie groups proper and began investigations of topology of Lie groups 7 The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley Overview edit nbsp The set of all complex numbers with absolute value 1 corresponding to points on the circle of center 0 and radius 1 in the complex plane is a Lie group under complex multiplication the circle group 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 R 3 displaystyle mathbb R 3 nbsp 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 Poincare 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 Definitions and examples editA 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 m G G G m x y x y displaystyle mu G times G to G quad mu x y xy nbsp means that m 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 x y x 1 y displaystyle x y mapsto x 1 y nbsp be a smooth mapping of the product manifold into G First examples edit The 2 2 real invertible matrices form a group under multiplication denoted by GL 2 R or by GL2 R GL 2 R A a b c d det A a d b c 0 displaystyle operatorname GL 2 mathbf R left A begin pmatrix a amp b c amp d end pmatrix det A ad bc neq 0 right nbsp dd This is a four dimensional noncompact real Lie group it is an open subset of R 4 displaystyle mathbb R 4 nbsp This group is disconnected it has two connected components corresponding to the positive and negative values of the determinant The rotation matrices form a subgroup of GL 2 R denoted by SO 2 R It is a Lie group in its own right specifically a one dimensional compact connected Lie group which is diffeomorphic to the circle Using the rotation angle f displaystyle varphi nbsp as a parameter this group can be parametrized as follows SO 2 R cos f sin f sin f cos f f R 2 p Z displaystyle operatorname SO 2 mathbf R left begin pmatrix cos varphi amp sin varphi sin varphi amp cos varphi end pmatrix varphi in mathbf R 2 pi mathbf Z right nbsp dd Addition of the angles corresponds to multiplication of the elements of SO 2 R and taking the opposite angle corresponds to inversion Thus both multiplication and inversion are differentiable maps The affine group of one dimension is a two dimensional matrix Lie group consisting of 2 2 displaystyle 2 times 2 nbsp real upper triangular matrices with the first diagonal entry being positive and the second diagonal entry being 1 Thus the group consists of matrices of the formA a b 0 1 a gt 0 b R displaystyle A left begin array cc a amp b 0 amp 1 end array right quad a gt 0 b in mathbb R nbsp dd Non example edit 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 H e 2 p i 8 0 0 e 2 p i a 8 8 R T 2 e 2 p i 8 0 0 e 2 p i ϕ 8 ϕ R displaystyle H left left begin matrix e 2 pi i theta amp 0 0 amp e 2 pi ia theta end matrix right theta in mathbb R right subset mathbb T 2 left left begin matrix e 2 pi i theta amp 0 0 amp e 2 pi i phi end matrix right theta phi in mathbb R right nbsp with a R Q displaystyle a in mathbb R setminus mathbb Q nbsp a fixed irrational number is a subgroup of the torus T 2 displaystyle mathbb T 2 nbsp that is not a Lie group when given the subspace topology 8 If we take any small neighborhood U displaystyle U nbsp of a point h displaystyle h nbsp in H displaystyle H nbsp for example the portion of H displaystyle H nbsp in U displaystyle U nbsp is disconnected The group H displaystyle H nbsp winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of T 2 displaystyle mathbb T 2 nbsp nbsp A portion of the group H displaystyle H nbsp inside T 2 displaystyle mathbb T 2 nbsp Small neighborhoods of the element h H displaystyle h in H nbsp are disconnected in the subset topology on H displaystyle H nbsp The group H displaystyle H nbsp can however be given a different topology in which the distance between two points h 1 h 2 H displaystyle h 1 h 2 in H nbsp is defined as the length of the shortest path in the group H displaystyle H nbsp joining h 1 displaystyle h 1 nbsp to h 2 displaystyle h 2 nbsp In this topology H displaystyle H nbsp is identified homeomorphically with the real line by identifying each element with the number 8 displaystyle theta nbsp in the definition of H displaystyle H nbsp With this topology H displaystyle H nbsp is just the group of real numbers under addition and is therefore a Lie group The group H displaystyle H nbsp 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 Matrix Lie groups edit Let GL n C displaystyle operatorname GL n mathbb C nbsp denote the group of n n displaystyle n times n nbsp invertible matrices with entries in C displaystyle mathbb C nbsp Any closed subgroup of GL n C displaystyle operatorname GL n mathbb C nbsp is a Lie group 9 Lie groups of this sort are called matrix Lie groups Since most of the interesting examples of Lie groups can be realized as matrix Lie groups some textbooks restrict attention to this class including those of Hall 10 Rossmann 11 and Stillwell 12 Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map The following are standard examples of matrix Lie groups The special linear groups over R displaystyle mathbb R nbsp and C displaystyle mathbb C nbsp SL n R displaystyle operatorname SL n mathbb R nbsp and SL n C displaystyle operatorname SL n mathbb C nbsp consisting of n n displaystyle n times n nbsp matrices with determinant one and entries in R displaystyle mathbb R nbsp or C displaystyle mathbb C nbsp The unitary groups and special unitary groups U n displaystyle text U n nbsp and SU n displaystyle text SU n nbsp consisting of n n displaystyle n times n nbsp complex matrices satisfying U U 1 displaystyle U U 1 nbsp and also det U 1 displaystyle det U 1 nbsp in the case of SU n displaystyle text SU n nbsp The orthogonal groups and special orthogonal groups O n displaystyle text O n nbsp and SO n displaystyle text SO n nbsp consisting of n n displaystyle n times n nbsp real matrices satisfying R T R 1 displaystyle R mathrm T R 1 nbsp and also det R 1 displaystyle det R 1 nbsp in the case of SO n displaystyle text SO n nbsp All of the preceding examples fall under the heading of the classical groups Related concepts edit A complex Lie group is defined in the same way using complex manifolds rather than real ones example SL 2 C displaystyle operatorname SL 2 mathbb C nbsp and holomorphic maps Similarly using an alternate metric completion of Q displaystyle mathbb Q nbsp 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 Topological definition edit 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 13 First we define an immersely linear Lie group to be a subgroup G of the general linear group GL n C displaystyle operatorname GL n mathbb C nbsp such that for some neighborhood V of the identity element e in G the topology on V is the subspace topology of GL n C displaystyle operatorname GL n mathbb C nbsp and V is closed in GL n C displaystyle operatorname GL n mathbb C nbsp G has at most countably many connected components For example a closed subgroup of GL n C displaystyle operatorname GL n mathbb C nbsp 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 G GL n C displaystyle G subset operatorname GL n mathbb C nbsp such that G G displaystyle G G nbsp 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 G displaystyle G nbsp that is locally isomorphic to G Then by a version of the closed subgroup theorem G displaystyle G nbsp 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 a 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 More examples of Lie groups editSee also Table of Lie groups and List of simple Lie groups 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 Dimensions one and two edit The only connected Lie groups with dimension one are the real line R displaystyle mathbb R nbsp with the group operation being addition and the circle group S 1 displaystyle S 1 nbsp of complex numbers with absolute value one with the group operation being multiplication The S 1 displaystyle S 1 nbsp group is often denoted as U 1 displaystyle U 1 nbsp the group of 1 1 displaystyle 1 times 1 nbsp 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 R 2 displaystyle mathbb R 2 nbsp with the group operation being vector addition and the affine group in dimension one described in the previous subsection under first examples Additional examples edit The group SU 2 is the group of 2 2 displaystyle 2 times 2 nbsp unitary matrices with determinant 1 displaystyle 1 nbsp Topologically SU 2 displaystyle text SU 2 nbsp is the 3 displaystyle 3 nbsp sphere S 3 displaystyle S 3 nbsp as a group it may be identified with the group of unit quaternions The Heisenberg group is a connected nilpotent Lie group of dimension 3 displaystyle 3 nbsp playing a key role in quantum mechanics The Lorentz group is a 6 dimensional Lie group of linear isometries of the Minkowski space The Poincare 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 Sp 2 n R displaystyle text Sp 2n mathbb R nbsp consists of all 2 n 2 n displaystyle 2n times 2n nbsp matrices preserving a symplectic form on R 2 n displaystyle mathbb R 2n nbsp It is a connected Lie group of dimension 2 n 2 n displaystyle 2n 2 n nbsp Constructions edit There are several standard ways to form new Lie groups from old ones The product of two Lie groups is a Lie group Any topologically closed subgroup of a Lie group is a Lie group This is known as the Closed subgroup theorem or Cartan s theorem The quotient of a Lie group by a closed normal subgroup is a Lie group The universal cover of a connected Lie group is a Lie group For example the group R displaystyle mathbb R nbsp is the universal cover of the circle group S 1 displaystyle S 1 nbsp In fact any covering of a differentiable manifold is also a differentiable manifold but by specifying universal cover one guarantees a group structure compatible with its other structures Related notions edit Some examples of groups that are not Lie groups except in the trivial sense that any group having at most countably many elements can be viewed as a 0 dimensional Lie group with the discrete topology are Infinite dimensional groups such as the additive group of an infinite dimensional real vector space or the space of smooth functions from a manifold X displaystyle X nbsp to a Lie group G displaystyle G nbsp C X G displaystyle C infty X G nbsp These are not Lie groups as they are not finite dimensional manifolds Some totally disconnected groups such as the Galois group of an infinite extension of fields or the additive group of the p adic numbers These are not Lie groups because their underlying spaces are not real manifolds Some of these groups are p adic Lie groups In general only topological groups having similar local properties to Rn for some positive integer n can be Lie groups of course they must also have a differentiable structure Basic concepts editThe Lie algebra associated with a Lie group edit Main article Lie group Lie algebra correspondence To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group Informally we can think of elements of the Lie algebra as elements of the group that are infinitesimally close to the identity and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements Before giving the abstract definition we give a few examples The Lie algebra of the vector space Rn is just Rn with the Lie bracket given by A B 0 In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian The Lie algebra of the general linear group GL n C of invertible matrices is the vector space M n C of square matrices with the Lie bracket given by A B AB BA If G is a closed subgroup of GL n C then the Lie algebra of G can be thought of informally as the matrices m of M n C such that 1 em is in G where e is an infinitesimal positive number with e2 0 of course no such real number e exists For example the orthogonal group O n R consists of matrices A with AAT 1 so the Lie algebra consists of the matrices m with 1 em 1 em T 1 which is equivalent to m mT 0 because e2 0 The preceding description can be made more rigorous as follows The Lie algebra of a closed subgroup G of GL n C may be computed asLie G X M n C exp t X G for all t in R displaystyle operatorname Lie G X in M n mathbb C operatorname exp tX in G text for all t text in mathbb mathbb R nbsp 15 10 where exp tX is defined using the matrix exponential It can then be shown that the Lie algebra of G is a real vector space that is closed under the bracket operation X Y X Y Y X displaystyle X Y XY YX nbsp 16 The concrete definition given above for matrix groups is easy to work with but has some minor problems to use it we first need to represent a Lie group as a group of matrices but not all Lie groups can be represented in this way and it is not even obvious that the Lie algebra is independent of the representation we use 17 To get around these problems we give the general definition of the Lie algebra of a Lie group in 4 steps Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold and therefore form a Lie algebra under the Lie bracket X Y XY YX because the Lie bracket of any two derivations is a derivation If G is any group acting smoothly on the manifold M then it acts on the vector fields and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra We apply this construction to the case when the manifold M is the underlying space of a Lie group G with G acting on G M by left translations Lg h gh This shows that the space of left invariant vector fields vector fields satisfying Lg Xh Xgh for every h in G where Lg denotes the differential of Lg on a Lie group is a Lie algebra under the Lie bracket of vector fields Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold Specifically the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v g Lg v This identifies the tangent space TeG at the identity with the space of left invariant vector fields and therefore makes the tangent space at the identity into a Lie algebra called the Lie algebra of G usually denoted by a Fraktur g displaystyle mathfrak g nbsp Thus the Lie bracket on g displaystyle mathfrak g nbsp is given explicitly by v w v w e This Lie algebra g displaystyle mathfrak g nbsp is finite dimensional and it has the same dimension as the manifold G The Lie algebra of G determines G up to local isomorphism where two Lie groups are called locally isomorphic if they look the same near the identity element Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras and the result for groups then usually follows easily For example simple Lie groups are usually classified by first classifying the corresponding Lie algebras We could also define a Lie algebra structure on Te using right invariant vector fields instead of left invariant vector fields This leads to the same Lie algebra because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields and acts as 1 on the tangent space Te The Lie algebra structure on Te can also be described as follows the commutator operation x y xyx 1y 1on G G sends e e to e so its derivative yields a bilinear operation on TeG This bilinear operation is actually the zero map but the second derivative under the proper identification of tangent spaces yields an operation that satisfies the axioms of a Lie bracket and it is equal to twice the one defined through left invariant vector fields Homomorphisms and isomorphisms edit 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 18 b 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 ϕ G H displaystyle phi colon G to H nbsp be a Lie group homomorphism and let ϕ displaystyle phi nbsp 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 ϕ displaystyle phi nbsp is a map between the corresponding Lie algebras ϕ g h displaystyle phi colon mathfrak g to mathfrak h nbsp 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 G displaystyle G nbsp to a Lie group H displaystyle H nbsp is an isomorphism of Lie groups if and only if it is bijective Lie group versus Lie algebra isomorphisms edit 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 19 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 20 but the groups themselves are not isomorphic because SU 2 is simply connected but SO 3 is not 21 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 22 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 Simply connected Lie groups edit See also Lie group Lie algebra correspondence and Fundamental group Lie groups A Lie group G displaystyle G nbsp is said to be simply connected if every loop in G displaystyle G nbsp can be shrunk continuously to a point in G displaystyle G nbsp This notion is important because of the following result that has simple connectedness as a hypothesis Theorem 23 Suppose G displaystyle G nbsp and H displaystyle H nbsp are Lie groups with Lie algebras g displaystyle mathfrak g nbsp and h displaystyle mathfrak h nbsp and that f g h displaystyle f mathfrak g rightarrow mathfrak h nbsp is a Lie algebra homomorphism If G displaystyle G nbsp is simply connected then there is a unique Lie group homomorphism ϕ G H displaystyle phi G rightarrow H nbsp such that ϕ f displaystyle phi f nbsp where ϕ displaystyle phi nbsp is the differential of ϕ displaystyle phi nbsp 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 n 3 displaystyle n geq 3 nbsp and the compact symplectic group Sp n 24 Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups The exponential map edit Main article Exponential map Lie theory See also derivative of the exponential map and normal coordinates The exponential map from the Lie algebra M n C displaystyle mathrm M n mathbb C nbsp of the general linear group G L n C displaystyle mathrm GL n mathbb C nbsp to G L n C displaystyle mathrm GL n mathbb C nbsp is defined by the matrix exponential given by the usual power series exp X 1 X X 2 2 X 3 3 displaystyle exp X 1 X frac X 2 2 frac X 3 3 cdots nbsp for matrices X displaystyle X nbsp If G displaystyle G nbsp is a closed subgroup of G L n C displaystyle mathrm GL n mathbb C nbsp then the exponential map takes the Lie algebra of G displaystyle G nbsp into G displaystyle G nbsp thus we have an exponential map for all matrix groups Every element of G displaystyle G nbsp that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra 25 The definition above is easy to use but it is not defined for Lie groups that are not matrix groups and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups as follows For each vector X displaystyle X nbsp in the Lie algebra g displaystyle mathfrak g nbsp of G displaystyle G nbsp i e the tangent space to G displaystyle G nbsp at the identity one proves that there is a unique one parameter subgroup c R G displaystyle c mathbb R rightarrow G nbsp such that c 0 X displaystyle c 0 X nbsp Saying that c displaystyle c nbsp is a one parameter subgroup means simply that c displaystyle c nbsp is a smooth map into G displaystyle G nbsp and that c s t c s c t displaystyle c s t c s c t nbsp for all s displaystyle s nbsp and t displaystyle t nbsp The operation on the right hand side is the group multiplication in G displaystyle G nbsp The formal similarity of this formula with the one valid for the exponential function justifies the definition exp X c 1 displaystyle exp X c 1 nbsp This is called the exponential map and it maps the Lie algebra g displaystyle mathfrak g nbsp into the Lie group G displaystyle G nbsp It provides a diffeomorphism between a neighborhood of 0 in g displaystyle mathfrak g nbsp and a neighborhood of e displaystyle e nbsp in G displaystyle G nbsp This exponential map is a generalization of the exponential function for real numbers because R displaystyle mathbb R nbsp is the Lie algebra of the Lie group of positive real numbers with multiplication for complex numbers because C displaystyle mathbb C nbsp is the Lie algebra of the Lie group of non zero complex numbers with multiplication and for matrices because M n R displaystyle M n mathbb R nbsp with the regular commutator is the Lie algebra of the Lie group G L n R displaystyle mathrm GL n mathbb R nbsp of all invertible matrices Because the exponential map is surjective on some neighbourhood N displaystyle N nbsp of e displaystyle e nbsp it is common to call elements of the Lie algebra infinitesimal generators of the group G displaystyle G nbsp The subgroup of G displaystyle G nbsp generated by N displaystyle N nbsp is the identity component of G displaystyle G nbsp The exponential map and the Lie algebra determine the local group structure of every connected Lie group because of the Baker Campbell Hausdorff formula there exists a neighborhood U displaystyle U nbsp of the zero element of g displaystyle mathfrak g nbsp such that for X Y U displaystyle X Y in U nbsp we have exp X exp Y exp X Y 1 2 X Y 1 12 X Y Y 1 12 X Y X displaystyle exp X exp Y exp left X Y tfrac 1 2 X Y tfrac 1 12 X Y Y tfrac 1 12 X Y X cdots right nbsp where the omitted terms are known and involve Lie brackets of four or more elements In case X displaystyle X nbsp and Y displaystyle Y nbsp commute this formula reduces to the familiar exponential law exp X exp Y exp X Y displaystyle exp X exp Y exp X Y nbsp The exponential map relates Lie group homomorphisms That is if ϕ G H displaystyle phi G to H nbsp is a Lie group homomorphism and ϕ g h displaystyle phi mathfrak g to mathfrak h nbsp the induced map on the corresponding Lie algebras then for all x g displaystyle x in mathfrak g nbsp we have ϕ exp x exp ϕ x displaystyle phi exp x exp phi x nbsp In other words the following diagram commutes 26 nbsp In short exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups The exponential map from the Lie algebra to the Lie group is not always onto even if the group is connected though it does map onto the Lie group for connected groups that are either compact or nilpotent For example the exponential map of SL 2 R is not surjective Also the exponential map is neither surjective nor injective for infinite dimensional see below Lie groups modelled on C Frechet space even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1 Lie subgroup edit A Lie subgroup H displaystyle H nbsp of a Lie group G displaystyle G nbsp is a Lie group that is a subset of G displaystyle G nbsp and such that the inclusion map from H displaystyle H nbsp to G displaystyle G nbsp is an injective immersion and group homomorphism According to Cartan s theorem a closed subgroup of G displaystyle G nbsp admits a unique smooth structure which makes it an embedded Lie subgroup of G displaystyle G nbsp i e a Lie subgroup such that the inclusion map is a smooth embedding Examples of non closed subgroups are plentiful for example take G displaystyle G nbsp to be a torus of dimension 2 or greater and let H displaystyle H nbsp be a one parameter subgroup of irrational slope i e one that winds around in G Then there is a Lie group homomorphism f R G displaystyle varphi mathbb R to G nbsp with i m f H displaystyle mathrm im varphi H nbsp The closure of H displaystyle H nbsp will be a sub torus in G displaystyle G nbsp The exponential map gives a one to one correspondence between the connected Lie subgroups of a connected Lie group G displaystyle G nbsp and the subalgebras of the Lie algebra of G displaystyle G nbsp 27 Typically the subgroup corresponding to a subalgebra is not a closed subgroup There is no criterion solely based on the structure of G displaystyle G nbsp which determines which subalgebras correspond to closed subgroups Representations editMain article Representation of a Lie group See also Compact group Representation theory of a connected compact Lie group and Lie algebra representation 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 Schrodinger equation in quantum mechanics H ps E ps displaystyle hat H psi E psi nbsp Assume the system in question has the rotation group SO 3 as a symmetry meaning that the Hamiltonian operator H displaystyle hat H nbsp commutes with the action of SO 3 on the wave function ps displaystyle psi nbsp 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 ps displaystyle psi nbsp are rotationally invariant functions Rather it means that the space of solutions to H ps E ps displaystyle hat H psi E psi nbsp is invariant under rotations for each fixed value of E displaystyle E nbsp 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 28 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 Poincare group Classification editLie groups may be thought of as smoothly varying families of symmetries Examples of symmetries include rotation about an axis What must be understood is the nature of small transformations for example rotations through tiny angles that link nearby transformations The mathematical object capturing this structure is called a Lie algebra Lie himself called them infinitesimal groups It can be defined because Lie groups are smooth manifolds so have tangent spaces at each point The Lie algebra of any compact Lie group very roughly one for which the symmetries form a bounded set can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones The structure of an abelian Lie algebra is mathematically uninteresting since the Lie bracket is identically zero the interest is in the simple summands Hence the question arises what are the simple Lie algebras of compact groups It turns out that they mostly fall into four infinite families the classical Lie algebras An Bn Cn and Dn which have simple descriptions in terms of symmetries of Euclidean space But there are also just five exceptional Lie algebras that do not fall into any of these families E8 is the largest of these Lie groups are classified according to their algebraic properties simple semisimple solvable nilpotent abelian their connectedness connected or simply connected and their compactness A first key result is the Levi decomposition which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup Connected compact Lie groups are all known they are finite central quotients of a product of copies of the circle group S1 and simple compact Lie groups which correspond to connected Dynkin diagrams Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank and any finite dimensional irreducible representation of such a group is 1 dimensional Solvable groups are too messy to classify except in a few small dimensions Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1 s on the diagonal of some rank and any finite dimensional irreducible representation of such a group is 1 dimensional Like solvable groups nilpotent groups are too messy to classify except in a few small dimensions Simple Lie groups are sometimes defined to be those that are simple as abstract groups and sometimes defined to be connected Lie groups with a simple Lie algebra For example SL 2 R is simple according to the second definition but not according to the first They have all been classified for either definition Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras 29 They are central extensions of products of simple Lie groups The identity component of any Lie group is an open normal subgroup and the quotient group is a discrete group The universal cover of any connected Lie group is a simply connected Lie group and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center Any Lie group G can be decomposed into discrete simple and abelian groups in a canonical way as follows Write Gcon for the connected component of the identity Gsol for the largest connected normal solvable subgroup Gnil for the largest connected normal nilpotent subgroupso that we have a sequence of normal subgroups 1 Gnil Gsol Gcon G Then G Gcon is discrete Gcon Gsol is a central extension of a product of simple connected Lie groups Gsol Gnil is abelian A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S1 Gnil 1 is nilpotent and therefore its ascending central series has all quotients abelian This can be used to reduce some problems about Lie groups such as finding their unitary representations to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension The diffeomorphism group of a Lie group acts transitively on the Lie group Every Lie group is parallelizable and hence an orientable manifold there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity Infinite dimensional Lie groups editLie groups are often defined to be finite dimensional but there are many groups that resemble Lie groups except for being infinite dimensional The simplest way to define infinite dimensional Lie groups is to model them locally on Banach spaces as opposed to Euclidean space in the finite dimensional case and in this case much of the basic theory is similar to that of finite dimensional Lie groups However this is inadequate for many applications because many natural examples of infinite dimensional Lie groups are not Banach manifolds Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces In this case the relation between the Lie algebra and the Lie group becomes rather subtle and several results about finite dimensional Lie groups no longer hold The literature is not entirely uniform in its terminology as to exactly which properties of infinite dimensional groups qualify the group for the prefix Lie in Lie group On the Lie algebra side of affairs things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic For example an infinite dimensional Lie algebra may or may not have a corresponding Lie group That is there may be a group corresponding to the Lie algebra but it might not be nice enough to be called a Lie group or the connection between the group and the Lie algebra might not be nice enough for example failure of the exponential map to be onto a neighborhood of the identity It is the nice enough that is not universally defined Some of the examples that have been studied include The group of diffeomorphisms of a manifold Quite a lot is known about the group of diffeomorphisms of the circle Its Lie algebra is more or less the Witt algebra whose central extension the Virasoro algebra see Virasoro algebra from Witt algebra for a derivation of this fact is the symmetry algebra of two dimensional conformal field theory Diffeomorphism groups of compact manifolds of larger dimension are regular Frechet Lie groups very little about their structure is known The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity The group of smooth maps from a manifold to a finite dimensional Lie group is an example of a gauge group with operation of pointwise multiplication and is used in quantum field theory and Donaldson theory If the manifold is a circle these are called loop groups and have central extensions whose Lie algebras are more or less Kac Moody algebras There are infinite dimensional analogues of general linear groups orthogonal groups and so on 30 One important aspect is that these may have simpler topological properties see for example Kuiper s theorem In M theory for example a 10 dimensional SU N gauge theory becomes an 11 dimensional theory when N becomes infinite See also editAdjoint representation of a Lie group Haar measure Homogeneous space List of Lie group topics Representations of Lie groups Symmetry in quantum mechanics Lie point symmetry about the application of Lie groups to the study of differential equations Notes editExplanatory notes edit This is the statement that a Lie group is a formal Lie group For the latter concept see Bruhat 14 Hall only claims smoothness but the same argument shows analyticity citation needed Citations edit a b Hawkins 2000 p 1 Hawkins 2000 p 2 Hawkins 2000 p 76 Tresse Arthur 1893 Sur les invariants differentiels des groupes continus de transformations Acta Mathematica 18 1 88 doi 10 1007 bf02418270 Hawkins 2000 p 43 Hawkins 2000 p 100 Borel 2001 Rossmann 2001 Chapter 2 Hall 2015 Corollary 3 45 a b Hall 2015 Rossmann 2001 Stillwell 2008 Kobayashi amp Oshima 2005 Definition 5 3 Bruhat F 1958 Lectures on Lie Groups and Representations of Locally Compact Groups PDF Tata Institute of Fundamental Research Bombay Helgason 1978 Ch II 2 Proposition 2 7 Hall 2015 Theorem 3 20 But see Hall 2015 Proposition 3 30 and Exercise 8 in Chapter 3 Hall 2015 Corollary 3 50 Hall 2015 Theorem 5 20 Hall 2015 Example 3 27 Hall 2015 Section 1 3 4 Hall 2015 Corollary 5 7 Hall 2015 Theorem 5 6 Hall 2015 Section 13 2 Hall 2015 Theorem 3 42 Introduction to Lie groups and algebras Definitions examples and problems PDF State University of New York at Stony Brook 2006 Archived from the original PDF on 28 September 2011 Retrieved 11 October 2014 Hall 2015 Theorem 5 20 Hall 2015 Part III Helgason 1978 p 131 Bauerle de Kerf amp ten Kroode 1997References editAdams John Frank 1969 Lectures on Lie Groups Chicago Lectures in Mathematics Chicago Univ of Chicago Press ISBN 978 0 226 00527 0 MR 0252560 Bauerle G G A de Kerf E A ten Kroode A P E 1997 A van Groesen E M de Jager eds Finite and infinite dimensional Lie algebras and their application in physics Studies in mathematical physics Vol 7 North Holland ISBN 978 0 444 82836 1 via ScienceDirect Borel Armand 2001 Essays in the history of Lie groups and algebraic groups History of Mathematics vol 21 Providence Rhode Island American Mathematical Society ISBN 978 0 8218 0288 5 MR 1847105 Bourbaki Nicolas Elements of mathematics Lie groups and Lie algebras Chapters 1 3 ISBN 3 540 64242 0 Chapters 4 6 ISBN 3 540 42650 7 Chapters 7 9 ISBN 3 540 43405 4 Chevalley Claude 1946 Theory of Lie groups Princeton Princeton University Press ISBN 978 0 691 04990 8 Cohn Paul Moritz 1957 Lie Groups Cambridge Tracts in Mathematical Physics Cambridge University Press OCLC 529830 Coolidge Julian Lowell 2003 A History of Geometrical Methods Dover Publications pp 304 317 ISBN 978 0 486 49524 8 Fulton William Harris Joe 1991 Representation theory A first course Graduate Texts in Mathematics Readings in Mathematics Vol 129 New York Springer Verlag doi 10 1007 978 1 4612 0979 9 ISBN 978 0 387 97495 8 MR 1153249 OCLC 246650103 Gilmore Robert 2008 Lie groups physics and geometry an introduction for physicists engineers and chemists Cambridge University Press doi 10 1017 CBO9780511791390 ISBN 978 0 521 88400 6 Hall Brian C 2015 Lie Groups Lie Algebras and Representations An Elementary Introduction Graduate Texts in Mathematics vol 222 2nd ed Springer doi 10 1007 978 3 319 13467 3 ISBN 978 3319134666 Harvey F Reese 1990 Spinors and calibrations Academic Press ISBN 0 12 329650 1 Hawkins Thomas 2000 Emergence of the theory of Lie groups Sources and Studies in the History of Mathematics and Physical Sciences Berlin New York Springer Verlag doi 10 1007 978 1 4612 1202 7 ISBN 978 0 387 98963 1 MR 1771134 Borel s review Helgason Sigurdur 1978 Differential Geometry Lie Groups and Symmetric Spaces New York Academic Press p 131 ISBN 978 0 12 338460 7 Knapp Anthony W 2002 Lie Groups Beyond an Introduction Progress in Mathematics vol 140 2nd ed Boston Birkhauser ISBN 978 0 8176 4259 4 Kobayashi Toshiyuki Oshima Toshio 2005 Lie Groups and Representation Theory in Japanese Iwanami ISBN 4 00 006142 9 Lie Sophus 1876 Theorie der Transformations Gruppen I II Archiv for Mathematik og Naturvidenskab 1 19 57 152 193 Nijenhuis Albert 1959 Review Lie groups by P M Cohn Bulletin of the American Mathematical Society 65 6 338 341 doi 10 1090 s0002 9904 1959 10358 x Rossmann Wulf 2001 Lie Groups An Introduction Through Linear Groups Oxford Graduate Texts in Mathematics Oxford University Press ISBN 978 0 19 859683 7 The 2003 reprint corrects several typographical mistakes Sattinger David H Weaver O L 1986 Lie groups and algebras with applications to physics geometry and mechanics Springer Verlag doi 10 1007 978 1 4757 1910 9 ISBN 978 3 540 96240 3 MR 0835009 Serre Jean Pierre 1965 Lie Algebras and Lie Groups 1964 Lectures given at Harvard University Lecture notes in mathematics vol 1500 Springer ISBN 978 3 540 55008 2 Steeb Willi Hans 2007 Continuous Symmetries Lie algebras Differential Equations and Computer Algebra second edition World Scientific Publishing doi 10 1142 6515 ISBN 978 981 270 809 0 MR 2382250 Stillwell John 2008 Naive Lie Theory Undergraduate Texts in Mathematics Springer doi 10 1007 978 0 387 78214 0 ISBN 978 0387782140 Warner Frank W 1983 Foundations of differentiable manifolds and Lie groups Graduate Texts in Mathematics vol 94 New York Berlin Heidelberg Springer Verlag doi 10 1007 978 1 4757 1799 0 ISBN 978 0 387 90894 6 MR 0722297 Ziller Wolfgang 2010 Lie Groups Representation Theory and Symmetric Spaces PDF University of Pennsylvania External links edit nbsp Media related to Lie groups at Wikimedia Commons Journal of Lie Theory Retrieved from https en wikipedia org w index php title Lie group amp oldid 1189392997, wikipedia, wiki, book, books, library,

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