Lie group

In, a Lie group (pronounced "Lee") is a  whose elements are organized continuously and smoothly, as opposed to s, where the elements are separated—this makes Lie groups s. Lie groups are named after Norwegian mathematician , who laid the foundations of the theory of continuous s.

In rough terms, a Lie group is a : it is a group whose elements are described by several real parameters. As such, Lie groups provide a natural model for the concept of continuous symmetry, such as rotational symmetry in three dimensions. Lie groups are widely used in many parts of modern mathematics and. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of, in much the same way that finite groups are used in to model the discrete symmetries of s.

Overview
Lie groups are s and as such can be studied using, in contrast with the case of more general s. 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 groups play an enormous role in modern, on several different levels. argued in his that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties. Thus corresponds to the choice of the group  of distance-preserving transformations of the Euclidean space R3,  corresponds to enlarging the group to the, whereas in  one is interested in the properties invariant under the. This idea later led to the notion of a, 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 of the Lie group (or of its ) are especially important. Representation theory. Groups whose representations are of particular importance include (or its ),   and the.

On a "global" level, whenever a Lie group on a geometric object, such as a  or a, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a on a manifold places strong constraints on its geometry and facilitates  on the manifold. Linear actions of Lie groups are especially important, and are studied in.

In the 1940s–1950s,, , and realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of s defined over an arbitrary. This insight opened new possibilities in pure algebra, by providing a uniform construction for most s, as well as in. The theory of s, an important branch of modern, deals extensively with analogues of Lie groups over s; Lie groups play an important role, via their connections with Galois representations in number theory.

Definitions and examples
A real Lie group is a that is also a finite-dimensional real, in which the group operations of  and inversion are s.  Smoothness of the group multiplication


 * $$ \mu:G\times G\to G\quad \mu(x,y)=xy$$

means that μ is a smooth mapping of the G × G into G. These two requirements can be combined to the single requirement that the mapping
 * $$(x,y)\mapsto x^{-1}y$$

be a smooth mapping of the product manifold into G.

First examples

 * The 2×2  form a group under multiplication, denoted by  or by GL2(R):


 * $$ \operatorname{GL}(2, \mathbf{R}) = \left\{A=\begin{pmatrix}a&b\\c&d\end{pmatrix}: \det A=ad-bc \ne 0\right\}. $$


 * This is a four-dimensional real Lie group; it is an open subset of $$\mathbb R^4$$. This group is ; it has two connected components corresponding to the positive and negative values of the.


 * The matrices form a  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  to the . Using the rotation angle $$\varphi$$ as a parameter, this group can be  as follows:
 * $$ \operatorname{SO}(2, \mathbf{R}) =\left\{\begin{pmatrix} \cos\varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{pmatrix}: \varphi\in\mathbf{R}/2\pi\mathbf{Z}\right\}. $$


 * 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 is a two-dimensional matrix Lie group, consisting of $$2\times 2$$ 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
 * $$ A= \left( \begin{array}{cc} a & b\\ 0 & 1 \end{array}\right),\quad a>0,\, b\in\mathbb{R}.$$

Non-example
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


 * $$H = \left\{\left .\left(\begin{matrix}e^{2\pi i\theta} & 0\\0 & e^{2\pi ia\theta}\end{matrix}\right) \right| \theta \in \mathbb{R}\right\} \subset \mathbb{T}^2 = \left\{\left .\left(\begin{matrix}e^{2\pi i\theta} & 0\\0 & e^{2\pi i\phi}\end{matrix}\right)\right| \theta, \phi \in \mathbb{R}\right\},$$

with $$a \in \mathbb R\setminus\mathbb Q$$ a fixed, is a subgroup of the $$\mathbb T^2$$ that is not a Lie group when given the. If we take any small neighborhood $$U$$ of a point $$h$$ in $$H$$, for example, the portion of $$H$$ in $$U$$ is disconnected. The group $$H$$ winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a subgroup of $$\mathbb T^2$$.

The group $$H$$ can, however, be given a different topology, in which the distance between two points $$h_1,h_2\in H$$ is defined as the length of the shortest path in the group  $$H$$ joining $$h_1$$ to $$h_2$$. In this topology, $$H$$ is identified homeomorphically with the real line by identifying each element with the number $$\theta$$ in the definition of $$H$$. With this topology, $$H$$ is just the group of real numbers under addition and is therefore a Lie group.

The group $$H$$ is an example of a "" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.

Matrix Lie groups
Let GL(n; C) denote the group of n × n invertible matrices with entries in C. Any of GL(n; C) is a Lie group; 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 and Rossmann. 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. All of the preceding examples fall under the heading of the s.
 * The s over R and C, SL(n, R) and SL(n, C), consisting of n × n matrices with determinant one and entries in R or C
 * The s and special unitary groups, U(n) and SU(n), consisting of n × n complex matrices satisfying $$U^*=U^{-1}$$ (and also $$\det(U)=1$$ in the case of SU(n))
 * The s and special orthogonal groups, O(n) and SO(n), consisting of n × n real matrices satisfying $$R^\mathrm{T}=R^{-1}$$ (and also $$\det(R)=1$$ in the case of SO(n))

Related concepts
A ' is defined in the same way using s rather than real ones (example: SL(2, C)), and similarly, using an alternate of Q, one can define a p-adic Lie group' over the, a topological group in which each point has a p-adic neighborhood.

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,, and  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 ). If the underlying manifold is allowed to be infinite-dimensional (for example, a ), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many, and these give most of the examples of s.

The language of provides a concise definition for Lie groups: a Lie group is a  in the  of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to.

Topological definition
A Lie group can be defined as a (Hausdorff) that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup G of the general linear group $$\operatorname{GL}(n, \mathbb{C})$$ such that (For example, a closed subgroup of $$\operatorname{GL}(n, \mathbb{C})$$; that is, a matrix Lie group satisfies the above conditions.)
 * 1) for some neighborhood V of the identity element e in G, the topology on V is the subspace topology of $$\operatorname{GL}(n, \mathbb{C})$$ and V is closed in $$\operatorname{GL}(n, \mathbb{C})$$.
 * 2) G has at most countably many connected components.

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  (or a version of ) constructs an immersed Lie subgroup $$G' \subset \operatorname{GL}(n, \mathbb{C})$$ such that $$G, G'$$ 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 $$G'$$ that is locally isomorphic to G. Then, by a version of the, $$G'$$ is a  and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series;  so the group operations are real-analytic and G itself is a real-analytic manifold.

The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group.

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

Dimensions one and two
The only connected Lie groups with dimension one are the real line $$\mathbb{R}$$ (with the group operation being addition) and the $$S^1$$ of complex numbers with absolute value one (with the group operation being multiplication). The $$S^1$$ group is often denoted as $$U(1)$$, the group of $$1\times 1$$ 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 $$\mathbb{R}^2$$ (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples."

Additional examples

 * The is the group of $$2\times 2$$ unitary matrices with determinant 1. Topologically, SU(2) is the 3-sphere $$S^3$$; as a group, it may be identified with the group of unit s.
 * The is a connected  Lie group of dimension 3, playing a key role in.
 * The is a 6-dimensional Lie group of linear  of the.
 * The is a 10-dimensional Lie group of  isometries of the Minkowski space.
 * The s of types, , , , have dimensions 14, 52, 78, 133, and 248. Along with the A-B-C-D series of s, the exceptional groups complete the list of simple Lie groups.
 * The Sp(2n, R) consists of all 2n × 2n matrices preserving a ' on R'2n. It is a connected Lie group of dimension 2n2 + n''.

Constructions
There are several standard ways to form new Lie groups from old ones:
 * The product of two Lie groups is a Lie group.
 * Any subgroup of a Lie group is a Lie group. This is known as the  or Cartan's theorem.
 * The quotient of a Lie group by a closed normal subgroup is a Lie group.
 * The of a connected Lie group is a Lie group. For example, the group R is the universal cover of the circle group S1. 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
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 ), are:


 * Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space. These are not Lie groups as they are not finite-dimensional manifolds.
 * Some s, such as the 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  to Rn for some positive integer n can be Lie groups (of course they must also have a differentiable structure).

The Lie algebra associated with a Lie group
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 "ly close" to the identity, and the Lie bracket of the Lie algebra is related to the of two such infinitesimal elements. Before giving the abstract definition we give a few examples: [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.) [A, B] = AB &minus; BA.
 * The Lie algebra of the vector space Rn is just Rn with the Lie bracket given by
 * The Lie algebra of the GL(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by
 * 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, R) 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
 * $$\operatorname{Lie}(G) = \{ X \in M(n;\mathbb{C}) | \operatorname{exp}(tX) \in G \text{ for all } t \text{ in } \mathbb{\mathbb{R}} \},$$ where exp(tX) is defined using the . It can then be shown that the Lie algebra of G is a real vector space that is closed under the bracket operation, $$[X,Y]=XY-YX$$.

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 even it is not obvious that the Lie algebra is independent of the representation we use. 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 X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [X, Y] = XY &minus; YX, because the  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*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.
 * 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 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  $$\mathfrak{g}.$$ Thus the Lie bracket on $$\mathfrak{g}$$ is given explicitly by [v, w] = [v^, w^]e.

This Lie algebra $$\mathfrak{g}$$ 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 &minus;1 on the tangent space Te.

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


 * (x, y) → xyx&minus;1y&minus;1

on G &times; G sends (e, e) to e, so its derivative yields a 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, and it is equal to twice the one defined through left-invariant vector fields.

Homomorphisms and isomorphisms
If G and H are Lie groups, then a Lie group homomorphism f : G → H is a smooth. In the case of complex Lie groups, such a homomorphism is required to be a. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real).

The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let $$\phi\colon G \to H$$ be a Lie group homomorphism and let $$\phi_{*}$$ be its at the identity. If we identify the Lie algebras of G and H with their s at the identity elements then $$\phi_{*}$$ is a map between the corresponding Lie algebras:
 * $$\phi_{*}\colon\mathfrak g \to \mathfrak h$$

One can show that $$\phi_{*}$$ is actually a (meaning that it is a  which preserves the ). In the language of, we then have a covariant 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 homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a which is also a group homomorphism.

Lie group versus Lie algebra isomorphisms
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, 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, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.

On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the for examples). An example of importance in physics are the groups and. These two groups have isomorphic Lie algebras, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.

On the other hand, if we require that the Lie group be, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.

Simply connected Lie groups
A Lie group $$G$$ is said to be  if every loop in $$G$$ can be shrunk continuously to a point in $$G$$. This notion is important because of the following result that has simple connectedness as a hypothesis:
 * Theorem: Suppose $$G$$ and $$H$$ are Lie groups with Lie algebras $$\mathfrak g$$ and $$\mathfrak h$$ and that $$f:\mathfrak{g}\rightarrow\mathfrak{h}$$ is a Lie algebra homomorphism. If $$G$$ is simply connected, then there is a unique Lie group homomorphism $$\phi:G\rightarrow H$$ such that $$\phi_*=f$$, where $$\phi_*$$ is the differential of $$\phi$$ at the identity.

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, which as a manifold is the 3-sphere. The, on the other hand, is not simply connected. (See .) The failure of SO(3) to be simply connected is intimately connected to the distinction between and  in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group, the spin group (double cover of rotation group) for $$n\geq 3$$, and the compact symplectic group.

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

The exponential map
The from the Lie algebra $$M(n;\mathbb C)$$  of the  $$GL(n;\mathbb C)$$ to $$GL(n;\mathbb C)$$ is defined by the, given by the usual power series:


 * $$\exp(X) = 1 + X + \frac{X^2}{2!} + \frac{X^3}{3!} + \cdots $$

for matrices $$X$$. If $$G$$ is a closed subgroup of $$GL(n;\mathbb C)$$, then the exponential map takes the Lie algebra of $$G$$ into $$G$$; thus, we have an exponential map for all matrix groups. Every element of $$G$$ that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.

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 $$X$$ in the Lie algebra $$\mathfrak{g}$$ of $$G$$ (i.e., the tangent space to $$G$$ at the identity), one proves that there is a unique one-parameter subgroup $$c:\mathbb R\rightarrow G$$ such that $$c'(0)=X$$. Saying that $$c$$ is a one-parameter subgroup means simply that $$c$$ is a smooth map into $$G$$ and that


 * $$c(s + t) = c(s) c(t)\ $$

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


 * $$\exp(X) = c(1).\ $$

This is called the exponential map, and it maps the Lie algebra $$\mathfrak{g}$$ into the Lie group $$G$$ It provides a between a  of 0 in $$\mathfrak{g}$$ and a neighborhood of $$e$$ in $$G$$. This exponential map is a generalization of the exponential function for real numbers (because $$\mathbb{R}$$ is the Lie algebra of the Lie group of with multiplication), for complex numbers (because $$\mathbb{C}$$ is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for  (because $$M(n, \mathbb{R})$$ with the regular commutator is the Lie algebra of the Lie group $$GL(n, \mathbb{R})$$ of all invertible matrices).

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

The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the : there exists a neighborhood $$U$$ of the zero element of $$\mathfrak{g}$$, such that for $$X,Y\in U$$ we have


 * $$ \exp(X)\,\exp(Y) = \exp\left(X + Y + \tfrac{1}{2}[X,Y] + \tfrac{1}{12}[\,[X,Y],Y] - \tfrac{1}{12}[\,[X,Y],X] - \cdots \right),$$

where the omitted terms are known and involve Lie brackets of four or more elements. In case $$X$$ and $$Y$$ commute, this formula reduces to the familiar exponential law $$\exp(X)\exp(Y)=\exp(X+Y)$$

The exponential map relates Lie group homomorphisms. That is, if $$\phi: G \to H$$ is a Lie group homomorphism and $$\phi_*: \mathfrak{g} \to \mathfrak{h}$$ the induced map on the corresponding Lie algebras, then for all $$x\in\mathfrak g$$ we have
 * $$\phi(\exp(x)) = \exp(\phi_{*}(x)).\,$$

In other words, the following diagram ,

(In short, exp is a 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, 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 is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

Lie subgroup
A Lie subgroup $$H$$ of a Lie group $$G$$ is a Lie group that is a of $$G$$ and such that the  from $$H$$ to $$G$$ is an   and. According to, a closed of $$G$$ admits a unique smooth structure which makes it an  Lie subgroup of $$G$$—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.

Examples of non-closed subgroups are plentiful; for example take $$G$$ to be a torus of dimension 2 or greater, and let $$H$$ be a of irrational slope, i.e. one that winds around in G. Then there is a Lie group $$\varphi:\mathbb{R}\to G$$ with $$ \mathrm{im}(\varphi) = H$$. The of $$H$$ will be a sub-torus in $$G$$.

The gives a  between the connected Lie subgroups of a connected Lie group $$G$$ and the subalgebras of the Lie algebra of $$G$$. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of $$G$$ which determines which subalgebras correspond to closed subgroups.

Representations
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 in quantum mechanics, $$\hat{H}\psi = E\psi$$. Assume the system in question has the as a symmetry, meaning that the Hamiltonian operator $$\hat{H}$$ commutes with the action of SO(3) on the wave function $$\psi$$. (One important example of such a system is the .) This assumption does not necessarily mean that the solutions $$\psi$$ are rotationally invariant functions. Rather, it means that the space of solutions to $$\hat{H}\psi = E\psi$$ is invariant under rotations (for each fixed value of $$E$$). This space, therefore, constitutes a representation of SO(3). These representations have been and the classification leads to a substantial, essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation.

The case of a connected compact Lie group K (including the just-mentioned case of SO(3)) is particularly tractable. In that case, every finite-dimensional representation of K decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified by. is in terms of the "highest weight" of the representation. The classification is closely related to the.

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 and the.

Early history
According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1), 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 (ibid). Some of Lie's early ideas were developed in close collaboration with. 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 (ibid, p. 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 (ibid, p. 76). In 1884 a young German mathematician,, 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.

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, on the theory of s of first order and on the equations of. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany (Hawkins, p. 43). Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what 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 s and 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 s that had developed in the theory of s, in the hands of and. The initial application that Lie had in mind was to the theory of s. On the model of and s, the driving conception was of a theory capable of unifying, by the study of, the whole area of s. 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, but it was developed by others, such as Picard and Vessiot, and it provides a theory of s, the s required to express solutions.

Additional impetus to consider continuous groups came from ideas of, 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 ; geometric theory and the explicit solutions of s of mechanics, worked out by and Jacobi; and the new understanding of  that emerged in the works of, ,  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, who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen (The composition of continuous finite transformation groups) (Hawkins, p. 100). The work of Killing, later refined and generalized by, led to classification of s, Cartan's theory of , and 's description of of compact and semisimple Lie groups using s.

In 1900 challenged Lie theorists with his  presented at the  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. The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by.

The concept of a Lie group, and possibilities of classification
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 ( himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have s 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 of an  and some number of  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 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, their ( or ) and their.

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


 * Connected s 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 s).
 * 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.
 * s 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, is simple according to the second definition but not according to the first. They have all been  (for either definition).
 * Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups.

The of any Lie group is an open, and the  is a. 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 ⊆ Gnil ⊆ Gsol ⊆ Gcon ⊆ G.

Then
 * G/Gcon is discrete
 * Gcon/Gsol is a of a product of.
 * Gsol/Gnil is abelian. A connected is isomorphic to a product of copies of  R and the  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 of a Lie group acts transitively on the Lie group
 * Every Lie group is, and hence an (there is a  between its  and the product of itself with the  at the identity)

Infinite-dimensional Lie groups
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 s (as opposed to  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 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 diffeomorphism group of spacetime sometimes appears in attempts to gravity.
 * The group of s of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the, whose the  (see  for a derivation of this fact) is the symmetry algebra of . Diffeomorphism groups of compact manifolds of larger dimension are ; very little about their structure is known.
 * The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a (with operation of ), and is used  in  and . If the manifold is a circle these are called s, and have central extensions whose Lie algebras are (more or less) s.
 * There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see for example . In, for example, a 10 dimensional SU(N) gauge theory becomes an 11 dimensional theory when N becomes infinite.