Lie sphere geometry

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lie_sphere_geometry 

 

Sophus Lie, the originator of Lie sphere geometry and the line-sphere correspondence.

Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.

The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).

To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.

Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous "line-sphere correspondence" between the space of lines and the space of spheres in 3-dimensional space.

Basic concepts

The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometry in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp. spheres, planes and points) are treated on an equal footing. This is achieved in three steps. First an ideal point at infinity is added to Euclidean space so that lines (or planes) can be regarded as circles (or spheres) passing through the point at infinity (i.e., having infinite radius). This extension is known as inversive geometry with automorphisms known as "Mobius transformations". Second, points are regarded as circles (or spheres) of zero radius. Finally, for technical reasons, the circles (or spheres), including the lines (or planes) are given orientations.

These objects, i.e., the points, oriented circles and oriented lines in the plane, or the points, oriented spheres and oriented planes in space, are sometimes called cycles or Lie cycles. It turns out that they form a quadric hypersurface in a projective space of dimension 4 or 5, which is known as the Lie quadric. The natural symmetries of this quadric form a group of transformations known as the Lie transformations. These transformations do not preserve points in general: they are transforms of the Lie quadric, not of the plane/sphere plus point at infinity. The point-preserving transformations are precisely the Möbius transformations. The Lie transformations which fix the ideal point at infinity are the Laguerre transformations of Laguerre geometry. These two subgroups generate the group of Lie transformations, and their intersection are the Möbius transforms that fix the ideal point at infinity, namely the affine conformal maps.

These groups also have a direct physical interpretation: As pointed out by Harry Bateman, the Lie sphere transformations are identical with the spherical wave transformations that leave the form of Maxwell's equations invariant. In addition, Élie Cartan, Henri Poincaré and Wilhelm Blaschke pointed out that the Laguerre group is simply isomorphic to the Lorentz group of special relativity (see Laguerre group isomorphic to Lorentz group). Eventually, there is also an isomorphism between the Möbius group and the Lorentz group (see Möbius group#Lorentz transformation).

Lie sphere geometry in the plane

The Lie quadric

The Lie quadric of the plane is defined as follows. Let R^3,2 denote the space R⁵ of 5-tuples of real numbers, equipped with the signature (3,2) symmetric bilinear form defined by

${\displaystyle (x_0,x_1,x_2,x_3,x_4)\cdot (y_0,y_1,y_2,y_3,y_4)=-x_0{y}_0-x_1{y}_1+x_2{y}_2+x_3{y}_4+x_4{y}_3.}$

A ruled hyperboloid is a 2-dimensional analogue of the Lie quadric.

The projective space RP⁴ is the space of lines through the origin in R⁵ and is the space of nonzero vectors x in R⁵ up to scale, where x= (x₀,x₁,x₂,x₃,x₄). The planar Lie quadric Q consists of the points [x] in projective space represented by vectors x with x · x = 0.

To relate this to planar geometry it is necessary to fix an oriented timelike line. The chosen coordinates suggest using the point [1,0,0,0,0] ∈ RP⁴. Any point in the Lie quadric Q can then be represented by a vector x = λ(1,0,0,0,0) + v, where v is orthogonal to (1,0,0,0,0). Since [x] ∈ Q, v · v = λ² ≥ 0.

The orthogonal space to (1,0,0,0,0), intersected with the Lie quadric, is the two dimensional celestial sphere S in Minkowski space-time. This is the Euclidean plane with an ideal point at infinity, which we take to be [0,0,0,0,1]: the finite points (x,y) in the plane are then represented by the points [v] = [0,x,y, −1, (x²+y²)/2]; note that v · v = 0, v · (1,0,0,0,0) = 0 and v · (0,0,0,0,1) = −1.

Hence points x = λ(1,0,0,0,0) + v on the Lie quadric with λ = 0 correspond to points in the Euclidean plane with an ideal point at infinity. On the other hand, points x with λ nonzero correspond to oriented circles (or oriented lines, which are circles through infinity) in the Euclidean plane. This is easier to see in terms of the celestial sphere S: the circle corresponding to [λ(1,0,0,0,0) + v] ∈ Q (with λ ≠ 0) is the set of points y ∈ S with y · v = 0. The circle is oriented because v/λ has a definite sign; [−λ(1,0,0,0,0) + v] represents the same circle with the opposite orientation. Thus the isometric reflection map x → x + 2 (x · (1,0,0,0,0)) (1,0,0,0,0) induces an involution ρ of the Lie quadric which reverses the orientation of circles and lines, and fixes the points of the plane (including infinity).

To summarize: there is a one-to-one correspondence between points on the Lie quadric and cycles in the plane, where a cycle is either an oriented circle (or straight line) or a point in the plane (or the point at infinity); the points can be thought of as circles of radius zero, but they are not oriented.

Incidence of cycles

Suppose two cycles are represented by points [x], [y] ∈ Q. Then x · y = 0 if and only if the corresponding cycles "kiss", that is they meet each other with oriented first order contact. If [x] ∈ S ≅ R² ∪ {∞}, then this just means that [x] lies on the circle corresponding to [y]; this case is immediate from the definition of this circle (if [y] corresponds to a point circle then x · y = 0 if and only if [x] = [y]).

It therefore remains to consider the case that neither [x] nor [y] are in S. Without loss of generality, we can then take x= (1,0,0,0,0) + v and y = (1,0,0,0,0) + w, where v and w are spacelike unit vectors in (1,0,0,0,0)^⊥. Thus v^⊥ ∩ (1,0,0,0,0)^⊥ and w^⊥ ∩ (1,0,0,0,0)^⊥ are signature (2,1) subspaces of (1,0,0,0,0)^⊥. They therefore either coincide or intersect in a 2-dimensional subspace. In the latter case, the 2-dimensional subspace can either have signature (2,0), (1,0), (1,1), in which case the corresponding two circles in S intersect in zero, one or two points respectively. Hence they have first order contact if and only if the 2-dimensional subspace is degenerate (signature (1,0)), which holds if and only if the span of v and w is degenerate. By Lagrange's identity, this holds if and only if (v · w)² = (v · v)(w · w) = 1, i.e., if and only if v · w = ± 1, i.e., x · y = 1 ± 1. The contact is oriented if and only if v · w = – 1, i.e., x · y = 0.

The problem of Apollonius

The eight solutions of the generic Apollonian problem. The three given circles are labeled C1, C2 and C3 and colored red, green and blue, respectively. The solutions are arranged in four pairs, with one pink and one black solution circle each, labeled as 1A/1B, 2A/2B, 3A/3B, and 4A/4B. Each pair makes oriented contact with C1, C2, and C3, for a suitable choice of orientations; there are four such choices up to an overall orientation reversal.

The incidence of cycles in Lie sphere geometry provides a simple solution to the problem of Apollonius. This problem concerns a configuration of three distinct circles (which may be points or lines): the aim is to find every other circle (including points or lines) which is tangent to all three of the original circles. For a generic configuration of circles, there are at most eight such tangent circles.

The solution, using Lie sphere geometry, proceeds as follows. Choose an orientation for each of the three circles (there are eight ways to do this, but there are only four up to reversing the orientation of all three). This defines three points [x], [y], [z] on the Lie quadric Q. By the incidence of cycles, a solution to the Apollonian problem compatible with the chosen orientations is given by a point [q] ∈ Q such that q is orthogonal to x, y and z. If these three vectors are linearly dependent, then the corresponding points [x], [y], [z] lie on a line in projective space. Since a nontrivial quadratic equation has at most two solutions, this line actually lies in the Lie quadric, and any point [q] on this line defines a cycle incident with [x], [y] and [z]. Thus there are infinitely many solutions in this case.

If instead x, y and z are linearly independent then the subspace V orthogonal to all three is 2-dimensional. It can have signature (2,0), (1,0), or (1,1), in which case there are zero, one or two solutions for [q] respectively. (The signature cannot be (0,1) or (0,2) because it is orthogonal to a space containing more than one null line.) In the case that the subspace has signature (1,0), the unique solution q lies in the span of x, y and z.

The general solution to the Apollonian problem is obtained by reversing orientations of some of the circles, or equivalently, by considering the triples (x,ρ(y),z), (x,y,ρ(z)) and (x,ρ(y),ρ(z)).

Note that the triple (ρ(x),ρ(y),ρ(z)) yields the same solutions as (x,y,z), but with an overall reversal of orientation. Thus there are at most 8 solution circles to the Apollonian problem unless all three circles meet tangentially at a single point, when there are infinitely many solutions.

Lie transformations

Any element of the group O(3,2) of orthogonal transformations of R^3,2 maps any one-dimensional subspace of null vectors in R^3,2 to another such subspace. Hence the group O(3,2) acts on the Lie quadric. These transformations of cycles are called "Lie transformations". They preserve the incidence relation between cycles. The action is transitive and so all cycles are Lie equivalent. In particular, points are not preserved by general Lie transformations. The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which preserve the chosen timelike direction. This subgroup is isomorphic to the group O(3,1) of Möbius transformations of the sphere. It can also be characterized as the centralizer of the involution ρ, which is itself a Lie transformation.

Lie transformations can often be used to simplify a geometrical problem, by transforming circles into lines or points.

Contact elements and contact lifts

The fact that Lie transformations do not preserve points in general can also be a hindrance to understanding Lie sphere geometry. In particular, the notion of a curve is not Lie invariant. This difficulty can be mitigated by the observation that there is a Lie invariant notion of contact element.

An oriented contact element in the plane is a pair consisting of a point and an oriented (i.e., directed) line through that point. The point and the line are incident cycles. The key observation is that the set of all cycles incident with both the point and the line is a Lie invariant object: in addition to the point and the line, it consists of all the circles which make oriented contact with the line at the given point. It is called a pencil of Lie cycles, or simply a contact element.

Note that the cycles are all incident with each other as well. In terms of the Lie quadric, this means that a pencil of cycles is a (projective) line lying entirely on the Lie quadric, i.e., it is the projectivization of a totally null two dimensional subspace of R^3,2: the representative vectors for the cycles in the pencil are all orthogonal to each other.

The set of all lines on the Lie quadric is a 3-dimensional manifold called the space of contact elements Z³. The Lie transformations preserve the contact elements, and act transitively on Z³. For a given choice of point cycles (the points orthogonal to a chosen timelike vector v), every contact element contains a unique point. This defines a map from Z³ to the 2-sphere S² whose fibres are circles. This map is not Lie invariant, as points are not Lie invariant.

Let γ:[a,b] → R² be an oriented curve. Then γ determines a map λ from the interval [a,b] to Z³ by sending t to the contact element corresponding to the point γ(t) and the oriented line tangent to the curve at that point (the line in the direction γ '(t)). This map λ is called the contact lift of γ.

In fact Z³ is a contact manifold, and the contact structure is Lie invariant. It follows that oriented curves can be studied in a Lie invariant way via their contact lifts, which may be characterized, generically as Legendrian curves in Z³. More precisely, the tangent space to Z³ at the point corresponding to a null 2-dimensional subspace π of R^3,2 is the subspace of those linear maps (A mod π):π → R^3,2/π with

A(x) · y + x · A(y) = 0

and the contact distribution is the subspace Hom(π,π^⊥/π) of this tangent space in the space Hom(π,R^3,2/π) of linear maps.

It follows that an immersed Legendrian curve λ in Z³ has a preferred Lie cycle associated to each point on the curve: the derivative of the immersion at t is a 1-dimensional subspace of Hom(π,π^⊥/π) where π=λ(t); the kernel of any nonzero element of this subspace is a well defined 1-dimensional subspace of π, i.e., a point on the Lie quadric.

In more familiar terms, if λ is the contact lift of a curve γ in the plane, then the preferred cycle at each point is the osculating circle. In other words, after taking contact lifts, much of the basic theory of curves in the plane is Lie invariant.

Lie sphere geometry in space and higher dimensions

General theory

Lie sphere geometry in n-dimensions is obtained by replacing R^3,2 (corresponding to the Lie quadric in n = 2 dimensions) by R^{n + 1, 2}. This is R^{n + 3} equipped with the symmetric bilinear form

${\displaystyle (x_0,x_1,\dots x_n,x_{n+1},x_{n+2})\cdot (y_0,y_1,\dots y_n,y_{n+1},y_{n+2})}$

${\displaystyle =-x_0{y}_0+x_1{y}_1+\cdots +x_n{y}_n+x_{n+1}{y}_{n+2}+x_{n+2}{y}_{n+1}.}$

The Lie quadric Q_n is again defined as the set of [x] ∈ RPⁿ⁺² = P(R^n+1,2) with x · x = 0. The quadric parameterizes oriented (n – 1)-spheres in n-dimensional space, including hyperplanes and point spheres as limiting cases. Note that Q_n is an (n + 1)-dimensional manifold (spheres are parameterized by their center and radius).

The incidence relation carries over without change: the spheres corresponding to points [x], [y] ∈ Q_n have oriented first order contact if and only if x · y = 0. The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles.

The space of contact elements is a (2n – 1)-dimensional contact manifold Z^{2n – 1}: in terms of the given choice of point spheres, these contact elements correspond to pairs consisting of a point in n-dimensional space (which may be the point at infinity) together with an oriented hyperplane passing through that point. The space Z^{2n – 1} is therefore isomorphic to the projectivized cotangent bundle of the n-sphere. This identification is not invariant under Lie transformations: in Lie invariant terms, Z^{2n – 1} is the space of (projective) lines on the Lie quadric.

Any immersed oriented hypersurface in n-dimensional space has a contact lift to Z^{2n – 1} determined by its oriented tangent spaces. There is no longer a preferred Lie cycle associated to each point: instead, there are n – 1 such cycles, corresponding to the curvature spheres in Euclidean geometry.

The problem of Apollonius has a natural generalization involving n + 1 hyperspheres in n dimensions.

Three dimensions and the line-sphere correspondence

In the case n=3, the quadric Q₃ in P(R^4,2) describes the (Lie) geometry of spheres in Euclidean 3-space. Lie noticed a remarkable similarity with the Klein correspondence for lines in 3-dimensional space (more precisely in RP³).

Suppose [x], [y] ∈ RP³, with homogeneous coordinates (x₀,x₁,x₂,x₃) and (y₀,y₁,y₂,y₃). Put p_ij = x_iy_j - x_jy_i. These are the homogeneous coordinates of the projective line joining x and y. There are six independent coordinates and they satisfy a single relation, the Plücker relation

p₀₁ p₂₃ + p₀₂ p₃₁ + p₀₃ p₁₂ = 0.

It follows that there is a one-to-one correspondence between lines in RP³ and points on the Klein quadric, which is the quadric hypersurface of points [p₀₁, p₂₃, p₀₂, p₃₁, p₀₃, p₁₂] in RP⁵ satisfying the Plücker relation.

The quadratic form defining the Plücker relation comes from a symmetric bilinear form of signature (3,3). In other words, the space of lines in RP³ is the quadric in P(R^3,3). Although this is not the same as the Lie quadric, a "correspondence" can be defined between lines and spheres using the complex numbers: if x = (x₀,x₁,x₂,x₃,x₄,x₅) is a point on the (complexified) Lie quadric (i.e., the x_i are taken to be complex numbers), then

p₀₁ = x₀ + x₁, p₂₃ = –x₀ + x₁

p₀₂ = x₂ + ix₃, p₃₁ = x₂ – ix₁

p₀₃ = x₄ , p₁₂ = x₅

defines a point on the complexified Klein quadric (where i² = –1).

Dupin cyclides

A Dupin cyclide.

Lie sphere geometry provides a natural description of Dupin cyclides. These are characterized as the common envelope of two one parameter families of spheres S(s) and T(t), where S and T are maps from intervals into the Lie quadric. In order for a common envelope to exist, S(s) and T(t) must be incident for all s and t, i.e., their representative vectors must span a null 2-dimensional subspace of R^4,2. Hence they define a map into the space of contact elements Z⁵. This map is Legendrian if and only if the derivatives of S (or T) are orthogonal to T (or S), i.e., if and only if there is an orthogonal decomposition of R^4,2 into a direct sum of 3-dimensional subspaces σ and τ of signature (2,1), such that S takes values in σ and T takes values in τ. Conversely such a decomposition uniquely determines a contact lift of a surface which envelops two one parameter families of spheres; the image of this contact lift is given by the null 2-dimensional subspaces which intersect σ and τ in a pair of null lines.

Such a decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of R^4,2 whose square is the identity and whose ±1 eigenspaces are σ and τ. Using the inner product on R^4,2, this is determined by a quadratic form on R^4,2.

To summarize, Dupin cyclides are determined by quadratic forms on R^4,2 such that the associated symmetric endomorphism has square equal to the identity and eigenspaces of signature (2,1).

This provides one way to see that Dupin cyclides are cyclides, in the sense that they are zero-sets of quartics of a particular form. For this, note that as in the planar case, 3-dimensional Euclidean space embeds into the Lie quadric Q₃ as the set of point spheres apart from the ideal point at infinity. Explicitly, the point (x,y,z) in Euclidean space corresponds to the point

[0, x, y, z, –1, (x² + y² + z²)/2]

in Q₃. A cyclide consists of the points [0,x₁,x₂,x₃,x₄,x₅] ∈ Q₃ which satisfy an additional quadratic relation

${\displaystyle \sum _{i,j=1}^5{a}_{ij}{x}_i{x}_j=0}$

for some symmetric 5 ×; 5 matrix A = (a_ij). The class of cyclides is a natural family of surfaces in Lie sphere geometry, and the Dupin cyclides form a natural subfamily.

Lie group

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lie_group 

 

Lie groups and Lie algebras

Algebraic structure → Group theory Group theory

In mathematics, a Lie group (pronounced /liː/ 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 ${\displaystyle \text{SO}(3)}$). Lie groups are widely used in many parts of modern mathematics and physics.

Lie groups were first found by studying matrix subgroups ${\displaystyle G}$ contained in ${\displaystyle {\text{GL}}_n(\mathbb{R})}$ or ${\displaystyle {\text{GL}}_n(\mathbb{C})}$, the groups of ${\displaystyle n\times n}$ invertible matrices over ${\displaystyle \mathbb{R}}$ or ${\displaystyle \mathbb{C}}$. 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.

History

According to the most authoritative source on the early history of Lie groups (Hawkins, p. 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 (ibid). 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 (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, 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.

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 (Hawkins, p. 43). 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) (Hawkins, p. 100). 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.^[2] The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.

Overview

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 R³, 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

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

${\displaystyle \mu :G\times G\to G\mu (x,y)=xy}$

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

${\displaystyle (x,y)\mapsto x^{-1}y}$

be a smooth mapping of the product manifold into G.

First examples

• The 2×2 real invertible matrices form a group under multiplication, denoted by GL(2, R) or by GL₂(R):

${\displaystyle \mathrm{GL}(2,\mathbf{R})=\left\{A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):detA=ad-bc\ne 0\right\}.}$

This is a four-dimensional noncompact real Lie group; it is an open subset of ${\displaystyle {\mathbb{R}}^4}$. 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 ${\displaystyle \phi }$ as a parameter, this group can be parametrized as follows:

${\displaystyle \mathrm{SO}(2,\mathbf{R})=\left\{\left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right):\phi \in \mathbf{R}/2\pi \mathbf{Z}\right\}.}$

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 affine group of one dimension is a two-dimensional matrix Lie group, consisting of ${\displaystyle 2\times 2}$ 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

${\displaystyle A=\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right),a>0,b\in \mathbb{R}.}$

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

${\displaystyle H=\left\{\left(\begin{array}{cc}{e}^{2\pi i\theta }& 0\\ 0& e^{2\pi ia\theta }\end{array}\right):\theta \in \mathbb{R}\right\}\subset {\mathbb{T}}^2=\left\{\left(\begin{array}{cc}{e}^{2\pi i\theta }& 0\\ 0& e^{2\pi i\varphi }\end{array}\right):\theta ,\varphi \in \mathbb{R}\right\},}$

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

A portion of the group ${\displaystyle H}$ inside ${\displaystyle {\mathbb{T}}^2}$. Small neighborhoods of the element ${\displaystyle h\in H}$ are disconnected in the subset topology on ${\displaystyle H}$

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

The group ${\displaystyle H}$ 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

Let ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ denote the group of ${\displaystyle n\times n}$ invertible matrices with entries in ${\displaystyle \mathbb{C}}$. Any closed subgroup of ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ 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, Rossmann, and Stillwell. 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 ${\displaystyle \mathbb{R}}$ and ${\displaystyle \mathbb{C}}$, ${\displaystyle \mathrm{SL}(n,\mathbb{R})}$ and ${\displaystyle \mathrm{SL}(n,\mathbb{C})}$, consisting of ${\displaystyle n\times n}$ matrices with determinant one and entries in ${\displaystyle \mathbb{R}}$ or ${\displaystyle \mathbb{C}}$
• The unitary groups and special unitary groups, ${\displaystyle \text{U}(n)}$ and ${\displaystyle \text{SU}(n)}$, consisting of ${\displaystyle n\times n}$ complex matrices satisfying ${\displaystyle U^{\ast }=U^{-1}}$ (and also ${\displaystyle det(U)=1}$ in the case of ${\displaystyle \text{SU}(n)}$)
• The orthogonal groups and special orthogonal groups, ${\displaystyle \text{O}(n)}$ and ${\displaystyle \text{SO}(n)}$, consisting of ${\displaystyle n\times n}$ real matrices satisfying ${\displaystyle R^{\mathrm{T}}=R^{-1}}$ (and also ${\displaystyle det(R)=1}$ in the case of ${\displaystyle \text{SO}(n)}$)

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

Related concepts

A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: ${\displaystyle \mathrm{SL}(2,\mathbb{C})}$), and holomorphic maps. Similarly, using an alternate metric completion of ${\displaystyle \mathbb{Q}}$, 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

A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup G of the general linear group ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ such that

1. for some neighborhood V of the identity element e in G, the topology on V is the subspace topology of ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ and V is closed in ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$.
2. G has at most countably many connected components.

(For example, a closed subgroup of ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$; 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 ${\displaystyle G^{\prime }\subset \mathrm{GL}(n,\mathbb{C})}$ such that ${\displaystyle G,G^{\prime }}$ 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 ${\displaystyle G^{\prime }}$ that is locally isomorphic to G. Then, by a version of the closed subgroup theorem, ${\displaystyle G^{\prime }}$ is a real-analytic manifold and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series; so the group operations are real-analytic and G itself is a real-analytic manifold.

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

More examples of Lie groups

See 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

The only connected Lie groups with dimension one are the real line ${\displaystyle \mathbb{R}}$ (with the group operation being addition) and the circle group ${\displaystyle S^1}$ of complex numbers with absolute value one (with the group operation being multiplication). The ${\displaystyle S^1}$ group is often denoted as ${\displaystyle U(1)}$, the group of ${\displaystyle 1\times 1}$ 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 ${\displaystyle {\mathbb{R}}^2}$ (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 group SU(2) is the group of ${\displaystyle 2\times 2}$ unitary matrices with determinant ${\displaystyle 1}$. Topologically, ${\displaystyle \text{SU}(2)}$ is the ${\displaystyle 3}$-sphere ${\displaystyle S^3}$; as a group, it may be identified with the group of unit quaternions.
• The Heisenberg group is a connected nilpotent Lie group of dimension ${\displaystyle 3}$, 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 ${\displaystyle \text{Sp}(2n,\mathbb{R})}$ consists of all ${\displaystyle 2n\times 2n}$ matrices preserving a symplectic form on ${\displaystyle {\mathbb{R}}^{2n}}$. It is a connected Lie group of dimension ${\displaystyle 2n^2+n}$.

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 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 ${\displaystyle \mathbb{R}}$ is the universal cover of the circle group ${\displaystyle S^1}$. 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 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 ${\displaystyle X}$ to a Lie group ${\displaystyle G}$, ${\displaystyle C^{\mathrm{\infty }}(X,G)}$. 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

The Lie algebra associated with a Lie group

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 + 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 AA^T = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)^T = 1, which is equivalent to m + m^T = 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

${\displaystyle \mathrm{Lie}(G)=\{X\in M(n;\mathbb{C})|\exp (tX)\in G\text{ for all }t\text{ in }\mathbb{R}\},}$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, ${\displaystyle [X,Y]=XY-YX}$.

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 it is not even 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 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.
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 L_g(h) = gh. This shows that the space of left invariant vector fields (vector fields satisfying L_g*X_h = X_gh for every h in G, where L_g* denotes the differential of L_g) 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 = L_g*v. This identifies the tangent space T_eG 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 ${\displaystyle \mathfrak{g}.}$ Thus the Lie bracket on ${\displaystyle \mathfrak{g}}$ is given explicitly by [v, w] = [v^, w^]_e.

This Lie algebra ${\displaystyle \mathfrak{g}}$ 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 T_e 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 T_e.

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

(x, y) → xyx⁻¹y⁻¹

on G × G sends (e, e) to e, so its derivative yields a bilinear operation on T_eG. 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

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

The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let ${\displaystyle \varphi :G\to H}$ be a Lie group homomorphism and let ${\displaystyle {\varphi }_{\ast }}$ 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 ${\displaystyle {\varphi }_{\ast }}$ is a map between the corresponding Lie algebras:

${\displaystyle {\varphi }_{\ast }:\mathfrak{g}\to \mathfrak{h},}$

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 ${\displaystyle G}$ to a Lie group ${\displaystyle H}$ is an isomorphism of Lie groups if and only if it is bijective.

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 Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.

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

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

Simply connected Lie groups

See also: Lie group–Lie algebra correspondence and Fundamental group § Lie groups

A Lie group ${\displaystyle G}$ is said to be simply connected if every loop in ${\displaystyle G}$ can be shrunk continuously to a point in ${\displaystyle G}$. This notion is important because of the following result that has simple connectedness as a hypothesis:

Theorem: Suppose ${\displaystyle G}$ and ${\displaystyle H}$ are Lie groups with Lie algebras ${\displaystyle \mathfrak{g}}$ and ${\displaystyle \mathfrak{h}}$ and that ${\displaystyle f:\mathfrak{g}\to \mathfrak{h}}$ is a Lie algebra homomorphism. If ${\displaystyle G}$ is simply connected, then there is a unique Lie group homomorphism ${\displaystyle \varphi :G\to H}$ such that ${\displaystyle {\varphi }_{\ast }=f}$, where ${\displaystyle {\varphi }_{\ast }}$ is the differential of ${\displaystyle \varphi }$ 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 ${\displaystyle n\ge 3}$, and the compact symplectic group Sp(n).

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

The exponential map

Main article: Exponential map (Lie theory)

See also: derivative of the exponential map and normal coordinates

The exponential map from the Lie algebra ${\displaystyle \mathrm{M}(n;\mathbb{C})}$ of the general linear group ${\displaystyle \mathrm{G}\mathrm{L}(n;\mathbb{C})}$ to ${\displaystyle \mathrm{G}\mathrm{L}(n;\mathbb{C})}$ is defined by the matrix exponential, given by the usual power series:

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

for matrices ${\displaystyle X}$. If ${\displaystyle G}$ is a closed subgroup of ${\displaystyle \mathrm{G}\mathrm{L}(n;\mathbb{C})}$, then the exponential map takes the Lie algebra of ${\displaystyle G}$ into ${\displaystyle G}$; thus, we have an exponential map for all matrix groups. Every element of ${\displaystyle G}$ 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 ${\displaystyle X}$ in the Lie algebra ${\displaystyle \mathfrak{g}}$ of ${\displaystyle G}$ (i.e., the tangent space to ${\displaystyle G}$ at the identity), one proves that there is a unique one-parameter subgroup ${\displaystyle c:\mathbb{R}\to G}$ such that ${\displaystyle c^{\prime }(0)=X}$. Saying that ${\displaystyle c}$ is a one-parameter subgroup means simply that ${\displaystyle c}$ is a smooth map into ${\displaystyle G}$ and that

${\displaystyle c(s+t)=c(s)c(t)}$

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

${\displaystyle \exp (X)=c(1).}$

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

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

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 ${\displaystyle U}$ of the zero element of ${\displaystyle \mathfrak{g}}$, such that for ${\displaystyle X,Y\in U}$ we have

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

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

The exponential map relates Lie group homomorphisms. That is, if ${\displaystyle \varphi :G\to H}$ is a Lie group homomorphism and ${\displaystyle {\varphi }_{\ast }:\mathfrak{g}\to \mathfrak{h}}$ the induced map on the corresponding Lie algebras, then for all ${\displaystyle x\in \mathfrak{g}}$ we have

${\displaystyle \varphi (\exp (x))=\exp ({\varphi }_{\ast }(x)).}$

In other words, the following diagram commutes,

(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

A Lie subgroup ${\displaystyle H}$ of a Lie group ${\displaystyle G}$ is a Lie group that is a subset of ${\displaystyle G}$ and such that the inclusion map from ${\displaystyle H}$ to ${\displaystyle G}$ is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of ${\displaystyle G}$ admits a unique smooth structure which makes it an embedded Lie subgroup of ${\displaystyle G}$—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.

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

The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group ${\displaystyle G}$ and the subalgebras of the Lie algebra of ${\displaystyle G}$. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of ${\displaystyle G}$ which determines which subalgebras correspond to closed subgroups.

Representations

Main 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 Schrödinger equation in quantum mechanics, ${\displaystyle \hat{H}\psi =E\psi }$. Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator ${\displaystyle \hat{H}}$ commutes with the action of SO(3) on the wave function ${\displaystyle \psi }$. (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 ${\displaystyle \psi }$ are rotationally invariant functions. Rather, it means that the space of solutions to ${\displaystyle \hat{H}\psi =E\psi }$ is invariant under rotations (for each fixed value of ${\displaystyle E}$). 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.^[22] 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

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" A_n, B_n, C_n and D_n, 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. 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

G_con for the connected component of the identity

G_sol for the largest connected normal solvable subgroup

G_nil for the largest connected normal nilpotent subgroup

so that we have a sequence of normal subgroups

1 ⊆ G_nil ⊆ G_sol ⊆ G_con ⊆ G.

Then

G/G_con is discrete

G_con/G_sol is a central extension of a product of simple connected Lie groups.

G_sol/G_nil is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S¹.

G_nil/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

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:

• 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 Fréchet 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. 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.