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In mathematics, a Lie group (pronounced /ˈliː/, sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential They are named after the nineteenth century Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is

Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. In Mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetries as motions as opposed to e Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world They provide a natural framework for analysing the continuous symmetries of differential equations (Differential Galois theory), much in the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, the Antiderivatives of certain Elementary functions cannot themselves be expressed as elementary functions In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations, his idée fixe.

The circle of center 0 and radius 1 in the complex plane is a Lie group with complex multiplication.
The circle of center 0 and radius 1 in the complex plane is a Lie group with complex multiplication. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis

Contents

Overview

Since Lie groups are manifolds, they can be studied using differential calculus, in contrast with the case of more general topological groups. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the One of the key ideas in the theory of Lie groups, due to Sophus Lie, is to replace the global object, the group, with its local or linearised version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

Lie groups play an enormous role in modern geometry, on several different levels. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position 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. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen 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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold. In Differential geometry, a G -structure on an n - Manifold M, for a given Structure group G, is a G 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. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the 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. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Linear actions of Lie groups are especially important, and are studied in representation theory. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

In the 1950s, 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. Claude Chevalley ( 11 February 1909, Johannesburg, South Africa - 28 June 1984, Paris) was a French In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings. In Mathematics, the general notion of automorphic form is the extension to Analytic functions perhaps of Several complex variables, of the theory of Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Number theory, the adele ring is a Topological ring which is built on the field of Rational numbers (or more generally any Algebraic

Examples of Lie groups

Lie groups occur in abundance throughout mathematics and physics.

Examples

See also: Table of Lie groups and List of simple Lie groups

Constructions

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

Related notions

Some examples of groups that are not Lie groups are:

Early 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. Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. 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. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group 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. Friedrich Engel ( December 26, 1861 &ndash September 29, 1941) was a German Mathematician. From this effort resulted the three-volume Theorie der Transformationsgruppen, published in 1888, 1890, and 1893.

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. Carl Gustav Jacob Jacobi ( December 10, 1804 - February 18, 1851) was a Prussian Mathematician, widely considered to be In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. 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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Siméon-Denis Poisson (21 June 1781 &ndash 25 April 1840 was a French Mathematician, Geometer, and Physicist. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Julius Plücker ( June 16, 1801 &ndash May 22, 1868) was a German Mathematician and Physicist. August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a

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. Wilhelm Karl Joseph Killing ( May 10 1847 &ndash February 11 1923) was a German Mathematician who made important contributions 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. Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental In Mathematics, a Lie algebra is semisimple if it is a Direct sum of Simple Lie algebras i Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In the mathematical field of Representation theory, a weight of an algebra A over a field F is an Algebra homomorphism

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 (Borel (2001),). The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley. Claude Chevalley ( 11 February 1909, Johannesburg, South Africa - 28 June 1984, Paris) was a French

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, here 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"). In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. It can be defined because Lie groups are 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 symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups The structure of an abelian Lie algebra is mathematically uninteresting; the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and Dn, which have simple descriptions in terms of symmetries of Euclidean space. In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these.

Example

For example, the 2×2 real invertible matrices,

\begin{bmatrix}a&b\\c&d\end{bmatrix} , \qquad ad-bc \ne 0 ,

form a multiplicative group, denoted by GL2(R), which is a classic example of a Lie group; its manifold is 4-dimensional. In Mathematics, the real numbers may be described informally in several different ways In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak Further restricting to 2×2 rotation matrices gives a subgroup, denoted by SO2(R), which is also a Lie group; its manifold is 1-dimensional, a circle, with rotation angle as parameter. In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In this latter example we can write a group element as

\begin{bmatrix} \cos \lambda & -\sin \lambda \\ \sin \lambda & \cos \lambda \end{bmatrix} ,

and observe that the inverse for the element given by λ is that given by −λ, while the product of the elements given by λ and μ is that given by λ+μ; thus both group operations are continuous, as required.

Definitions

A (real) Lie group is a mathematical group which is also a finite-dimensional real smooth manifold, and in which the group operations of multiplication and inversion are smooth maps. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability

There are several closely related concepts. A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: SL2(C)), and similarly one can define a p-adic Lie group over the p-adic numbers. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 An Infinite dimensional Lie group is defined in the same way except that one allows the underlying manifold to be infinite dimensional. 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. In Mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Mathematics, the name symplectic group can refer to two different but closely related types of mathematical groups.

It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups. In Mathematics, a group of Lie type G(k is a (not necessarily finite group of rational points of a reductive Linear algebraic group G with In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families One could also try varying the definition by using topological or analytic manifolds instead of smooth ones, but it turns out that this gives nothing new: Gleason, Montgomery and Zippin showed in 1952 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 Hilbert's fifth problem and Hilbert-Smith conjecture). Andrew Mattei Gleason (born November 4 1921 in Fresno, California, U Deane Montgomery (1909–1992 was a topologist who served as President of the American Mathematical Society from 1961 to 1962 Hilbert's fifth problem, from the Hilbert problems list promulgated in 1900 by David Hilbert, concerns the characterization of Lie groups The theory of In Mathematics, the Hilbert-Smith conjecture is concerned with the Transformation groups of Manifolds and in particular with the limitations on Topological

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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships This is important, since it allows generalization of the notion of a Lie group to Lie supergroups. The concept of supergroup is a Generalization of that of group.

Properties

Every Lie group is parallelizable, and hence an orientable manifold. In Mathematics, a parallelizable manifold M is a Smooth manifold of dimension n having Vector fields V 1 A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back

Types of Lie groups

Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In mathematics the term semisimple is used in a number of related ways within different subjects In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be

Structure of a Lie group

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 central extension of a product of simple connected Lie groups. In Mathematics, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group. In Mathematics, the Simple Lie groups were classified by Élie Cartan.
Gsol/Gnil is abelian (and a product of copies of R and 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.

The Lie algebra associated to a Lie group

To every Lie group, we can associate a Lie algebra, whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie 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 is something to do with the commutator of two such infinitesimal elements. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have 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 − BA

The concrete definition given above 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 obvious that the Lie algebra is independent of the representation we use. To get round these problems we give the general definition of the Lie algebra of any 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. In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator Lie bracket can refer to Lie algebra Lie bracket of vector fields
  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 tangent space Te 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 \mathfrak{g}. Thus the Lie bracket on \mathfrak{g} is given explicitly by [v,w]=[v^,w^]e. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since The German word Fraktur () refers to a specific sub-group of Blackletter Typefaces The word derives from the past participle fractus (“broken”

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 −1 on the tangent space Te.

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

(x, y) → xyx−1y−1

on G × G sends (e, e) to e, so its derivative yields a bilinear operation on TeG. In Mathematics, a bilinear map is a function of two arguments that is linear in each 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. Lie bracket can refer to Lie algebra Lie bracket of 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 Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function (It is equivalent to require only that f be continuous rather than smooth. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function ) The composition of two such homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a homomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Isomorphic Lie groups are essentially the same; they only differ in the notation for their elements.

Every homomorphism f : G → H of Lie groups induces a homomorphism between the corresponding Lie algebras \mathfrak{g} and \mathfrak{h}. The association G \mapsto\mathfrak{g} is a functor. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

One version of Ado's theorem is that every finite dimensional Lie algebra is isomorphic to a matrix Lie algebra. In Mathematics, Ado's theorem states that every finite-dimensional Lie algebra L over a field K of Characteristic zero For every finite dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. So every abstract Lie algebra is the Lie algebra of some (linear) Lie group.

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). This article gives a table of some common Lie groups and their associated Lie algebras The following are noted the topological properties of the group ( Dimension A connected Lie group is simple, semisimple, solvable, nilpotent, or abelian if and only if its Lie algebra has the corresponding property. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In mathematics the term semisimple is used in a number of related ways within different subjects In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra \mathfrak{g} over F there is a simply connected Lie group G with \mathfrak{g} as Lie algebra, unique up to isomorphism. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be Moreover every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups.

The exponential map

The exponential map from the Lie algebra Mn(R) of the general linear group GLn(R) to GLn(R) is defined by the usual power series:

\exp(A) = 1 + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots

for matrices A. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation If G is any subgroup of GLn(R), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups.

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.

Every vector v in \mathfrak{g} determines a linear map from R to \mathfrak{g} taking 1 to v, which can be thought of as a Lie algebra homomorphism. Since R is the Lie algebra of the simply connected Lie group R, this induces a Lie group homomorphism c : R → G so 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 exponential function justifies the definition

exp(v) = c(1)

This is called the exponential map, and it maps the Lie algebra \mathfrak{g} into the Lie group G. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine It provides a diffeomorphism between a neighborhood of 0 in \mathfrak{g} and a neighborhood of e in G. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. This exponential map is a generalization of the exponential function for real numbers (since R is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (since C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (since Mn(R) with the regular commutator is the Lie algebra of the Lie group GLn(R) of all invertible matrices). In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

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 Baker-Campbell-Hausdorff formula: there exists a neighborhood U of the zero element of \mathfrak{g}, such that for u, v in U we have

exp(u) exp(v) = exp(u + v + 1/2 [u, v] + 1/12 [[u, v], v] − 1/12 [[u, v], u] − . In Mathematics, the Baker-Campbell-Hausdorff formula is the solution to Z = \log(e^X e^Y\ for non- commuting X . . )

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

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 SL2(R) is not surjective.

Infinite dimensional Lie groups

Lie groups are finite dimensional by definition, but there are many groups that resemble Lie groups, except for being infinite dimensional. There is very little "general theory" of such groups, but some of the examples that have been studied include:

See also

References

Notes

  1. ^ Andrew Pressley and Graeme Segal, Loop Groups, Oxford Science Publications, 1986, page 26.

Dictionary

Lie group

-noun

  1. (topology) Any of many analytic groups that are also a smooth manifold; they arise as groups of rotational symmetries
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