In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: [liː], 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. Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures. This makes Lie groups tools for nearly all parts of contemporary mathematics, as well as for modern theoretical physics, especially particle physics. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... Not to be confused with the NATO phonetic alphabet, which has also informally been called the “International Phonetic Alphabet”. For information on how to read IPA transcriptions of English words, see IPA chart for English. ... This picture illustrates how the hours in a clock form a group. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold. ... 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, as opposed to experimental processes, in an attempt to understand nature. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

Since Lie groups are manifolds, they 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, due to Sophus Lie, is to replace the global object, the group, with its local or linearized version, which Lie himself called an infinitesimal group and which has since become known as its Lie algebra. Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... 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 groups provide a natural framework to analyse continuous symmetries of differential equations (Picard-Vessiot theory), much in the same way as permutation groups are used in Galois theory to analyse discrete symmetries of algebraic equations. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Motivation and Basic Idea 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 operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

Early history

According the most authoritative source on the early history of Lie groups (Hawkins, p.1), Sophus Lie himself considered winter of 1873—1874 as the birth date of his theory of continuous groups. However, he goes on to prove 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 its creation (ibid). Lie developed some of his early ideas in close collaboration with Felix Klein, with whom he saw on a daily basis in Berlin from the end of October 1869 to end of February 1870, and in Paris, Gőttingen and Erlangen in subsequent two years, until October 1872 (ibid, p.2). Lie stated that all the principal results were obtained by the year 1884. However, during the 1870s he chose to publish all papers, except the very first note, exclusively in Norwegian journals, which impeded the recognition of his 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 exposing his theory of continuous groups. The result of this effort was the publication of three volumes of Theorie der Transformationsgruppen, which appeared in 1888, 1890, and 1893. Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... Friedrich Engel (December 26, 1861 - September 29, 1941) was born in Lugau, Germany as the son of a Lutheran pastor. ...

Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest to geometry of differential equations was first motivated by work of Carl Gustav Jacobi on the theory of partial differential equations of first order and on equations of classical mechanics, much of which was published posthumously in 1860s and generated 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 Evarist Galois has done for algebraic equations, namely, classify them in terms of group theory. Additional impetus to consider continuous groups came from ideas of Bernhard Riemann on 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. Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Galois at the age of fifteen from the pencil of a classmate. ... Bernhard Riemann. ... This picture illustrates how the hours in a clock form a group. ... An illustration of a differential equation. ... Simeon Poisson. ... Table of Geometry, from the 1728 Cyclopaedia. ... Julius Plücker. ... August Ferdinand Möbius (pronounced , November 17, 1790 - September 26, 1868) was a German mathematician and theoretical astronomer. ... Hermann Günther Grassmann (April 15, 1809 - September 26, 1877) was a mathematician, physicist, linguist, scholar, and neohumanist. ...

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 Elie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights. Wilhelm Karl Joseph Killing (1847 May 10 – 1923 February 11) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i. ... In mathematics, including applications to general relativity, a (Riemannian) symmetric space in differential geometry is a certain kind of homogeneous space in the theory of Lie groups. ... Hermann Weyl Hermann Weyl (November 9, 1885 – December 8, 1955) was a German mathematician. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... Weight is a concept arising often in representation theory of Lie groups and Lie algebras, a branch of mathematics. ...

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 connected 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),^{[citation needed]}). The theory of Lie groups was systematically reworked in modern mathematical language in a monograph of Claude Chevalley. Claude Chevalley (11 February 1909 - 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry and the theory of algebraic groups. ...

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".) It can be defined because Lie groups are manifolds, so have tangent spaces at each point. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ...

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; 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. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... Abelian, in mathematics, is used in many different definitions: In group theory: Abelian group, a group in which the binary operation is commutative Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms Metabelian group is a group where the commutator subgroup is contained in... In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ...

Example

For example, the 2×2 real invertible matrices, In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

form a multiplicative group, denoted by GL₂(R), which is a classic example of a Lie group; its manifold is 4-dimensional. 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 this latter example we can write a group element as In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ... In the three-dimensional space, the possible moves of a rigid body are rotations and translations. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

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. This picture illustrates how the hours in a clock form a group. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, multiplication is an elementary arithmetic operation. ... InVersion are a heavy metal band from Essex who came together with the aim to blend the melody of old school metal with the aggression and rhythm of more modern bands. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

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. 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 differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... 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. 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 the 1950s 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). In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field. ... For a complete list see list of finite simple groups. ... Andrew Mattei Gleason is an American mathematician. ... Deane Montgomery (1909–1992) was a topologist who served as President of the American Mathematical Society from 1961 to 1962. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, the Hilbert-Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M...

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, since it allows generalization of the notion of a Lie group to Lie supergroups. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... The concept of supergroup is a generalization of a that of group. ...

Examples of Lie groups

Here are a few examples of Lie groups and their relations to other areas of mathematics and physics.

• Euclidean space Rⁿ is an abelian Lie group (with ordinary vector addition as the group operation).
• The group GL_n(R) of invertible matrices (under matrix multiplication) is a Lie group of dimension n², called the general linear group. It has a subgroup SL_n(R) of matrices of determinant 1 which is also a Lie group, called the special linear group.
• The orthogonal group O_n(R) is a Lie group represented by orthogonal matrices. It consists of all rotations and reflections of an n-dimensional vector space. It has a subgroup SO_n(R) of elements of determinant 1, called the special orthogonal group or rotation group.
• The unitary group U(n) is a compact group of dimension n² represented by unitary matrices. It has a subgroup SU(n) of elements of determinant 1, called the special unitary group.
• Spin groups are double covers of the special orthogonal groups, used for studying fermions in quantum field theory (among other things).
• The group Sp_2n(R) of all matrices preserving a symplectic form is a Lie group called the symplectic group.
• The spheres S⁰, S¹, and S³ can be made into Lie groups by identifying them with the real numbers, complex numbers, or quaternions of absolute value 1 respectively. No other spheres are Lie groups. The Lie group S¹ is sometimes called the circle group.
• The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2.
• The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
• The Heisenberg group is a Lie group of dimension 3, used in quantum mechanics.
• The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the standard model, whose dimension corresponds to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
• The metaplectic group is a 3 dimensional Lie group that is a double cover of SL₂(R) and is used in the theory of modular forms. It cannot be represented as finite matrices.
• The exceptional Lie groups of types G₂, F₄, E₆, E₇, E₈ have dimensions 14, 52, 78, 133, and 248. There is even a group E_7½ of dimension 190.

For many more examples see the table of Lie groups and list of simple Lie groups and article on matrix groups. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... This article gives an overview of the various ways to perform matrix multiplication. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... A sphere rotating around its axis. ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In particle physics, fermions are particles with half-integer spin. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ... In physics, spacetime is a mathematical model that combines space and time into a single construct called the space-time continuum. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ... Fig. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... A vector boson is a boson with spin equal to one. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. ... Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, G2 is the name of a Lie group and also its Lie algebra . ... In mathematics, F4 is the name of a Lie group and also its Lie algebra . ... In mathematics, E6 is the name of some Lie groups and also their Lie algebras . ... In mathematics, E7 is the name of a Lie group and also its Lie algebra . ... In mathematics, E8 is the name given to a family of closely related structures. ... In mathematics, the Lie algebra E7½ is a subalgebra of E8 containing E7 defined by Landsberg and Manivel in order to fill the hole in the exceptional series of simple Lie algebras observed by Cvitanovic, Deligne, Cohen and de Man, (although it is not itself simple). ... This article gives a table of some common Lie groups and their associated Lie algebras. ... In mathematics, the simple Lie groups were classified by Élie Cartan. ... In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. ...

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

• The product of two Lie groups is a Lie group.
• Any closed subgroup of a Lie group is a Lie group.
• The quotient of a Lie group by a closed normal subgroup is a Lie group.
• The universal cover of a connected Lie group is a Lie group. For example, the group R is the universal cover of the circle group S¹.

Some examples of groups that are not Lie groups are: In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...

• Infinite dimensional groups, such as the additive group of an infinite dimensional real vector space. These are not Lie groups as they are not finite dimensional manifolds.
• Some 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".)
• The image of a connected Lie group under a homomorphism of Lie groups need not be a Lie group. The usual example of this is the image of R in the group R²/Z² (≅ S¹×S¹) under the map x→(x,√2 x). The image is a dense subset of R²/Z² that is not a manifold, and so is not a Lie group. This also gives an example where a subalgebra of a Lie algebra does not correspond to a Lie subgroup of the corresponding Lie group.
• The group of rational numbers under addition, topologized as a subset of the real numbers, is not a Lie group as it is not a manifold.

In mathematics, a Galois group is a group associated with a certain type of field extension. ... The p-adic number systems were first described by Kurt Hensel in 1897. ...

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. In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ... 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 realized by Galois theory. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In topology, a geometrical object or space is called simply connected if it is path-connected and every path between two points can be continuously transformed into every other. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

• 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.

• Compact Lie groups are all known: they are finite central extensions 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, SL₂(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 connected groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups.

• Connected abelian Lie groups are all isomorphic to products of copies of R and the circle group S¹.

In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... See also Simple Lie group. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, the simple Lie groups were classified by Élie Cartan. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ...

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

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 (and a product of copies of R and 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. In group theory, a central extension of a group G is an exact sequence of groups such that A is in Z(E), the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to... In mathematics, the simple Lie groups were classified by Élie Cartan. ...

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. Informally we can think of elements of the Lie algebra as elements of the groups that are "infinitesimally close" to the identity, and the Lie bracket is something to do with the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples: In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...

• 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(R) of invertible matrices is the vector space M_n(R) of square matrices with the Lie bracket given by

[A, B] = AB − BA

• If G is a closed subgroup of GL_n(R) then the Lie algebra of G can be thought of informally as the matrices m of M_n(R) such that 1 + εm is in G, where ε is an infinitesimal positive number with ε² = 0 (of course no such real number ε exists...). For example, the orthogonal group O_n(R) consists of matrices A with 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.
  • Formally, when working over the reals, as here, this is accomplished by considering the limit as ε→0; but the "infinitesimal" language generalizes directly to Lie groups over general rings.

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 does not depend on which representation we use. To get round these problems we give the general definition of the Lie algebra of any Lie group (in 4 steps): In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

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. This shows that the space of left invariant vector fields 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. This identifies the tangent space T_e at the identity with the space of left invariant vector fields, and therefore makes the tangent space into a Lie algebra, called the Lie algebra of G, usually denoted by a lower case g or a Fraktur .

This Lie algebra is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras. In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... The German word Fraktur (pronounced in the International Phonetic Alphabet (IPA)) refers to a specific sub-group of blackletter typefaces. ...

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. In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

Homomorphisms and isomorphisms

If G and H are Lie groups, then a Lie-group homomorphism f : G → H is a smooth group homomorphism. (It is equivalent to require only that f be continuous rather than smooth.) The composition of two such homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a homomorphism. Isomorphic Lie groups are essentially the same; they only differ in the notation for their elements. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

Every homomorphism f : G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and . The association G is a functor. Did somebody just say functor? In category theory, 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. 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. In mathematics, Ados theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. ...

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). A connected Lie group is simple, semisimple, solvable, nilpotent, or abelian if and only if its Lie algebra has the corresponding property. This article gives a table of some common Lie groups and their associated Lie algebras. ... In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ... 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 realized by Galois theory. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

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 over F there is a simply connected Lie group G with as Lie algebra, unique up to isomorphism. Moreover every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

The exponential map

The exponential map from the Lie algebra M_n(R) of the group GL_n(R) to GL_n(R) is defined by the usual power series:

for matrices A. If G is any subgroup of GL_n(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 determines a linear map from R to 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 The exponential function is one of the most important functions in mathematics. ...

exp(v) = c(1)

This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in and a neighborhood of e in G. 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 M_n(R) with the regular commutator is the Lie algebra of the Lie group GL_n(R) of all invertible matrices). There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... For the square matrix section, see square matrix. ...

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 , such that for u, v in U we have In mathematics, the Baker-Campbell-Hausdorff formula is the solution to for non-commuting x and y. ...

exp(u) exp(v) = exp(u + v + 1/2 [u, v] + 1/12 [[u, v], v] − 1/12 [[u, v], u] − ...)

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:

• 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, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory. Very little is known about the diffeomorphism groups of manifolds of larger dimension. The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity.
• The group of smooth maps from a manifold to a finite dimensional group is called a gauge group, 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.
• Just as calculus in finite-dimensional real vector spaces can be extended to calculus in Banach spaces, the definition of finite-dimensional smooth manifolds can be extended to give a definition of Banach analytic manifolds. Similarly, the standard finite-dimensional definition of Lie groups can be extended to give a definition of Banach analytic Lie groups. In this case, we have a Banach analytic manifold which simultaneously has a group structure such that multiplication and inversion are analytic maps. Some of the theorems of finite-dimensional Lie groups do not carry over to the Banach analytic case, and in particular the relation between Lie groups and Lie algebras is much more subtle in the infinite dimensional case. However, it is true that "for infinite dimensional Lie groups modeled on Banach spaces there is a well-developed theory ... which is closely parallel to the theory of finite dimensional Lie groups."^[1]

In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a Witt algebra is the Lie algebra of Killing vector fields defined on the Riemann sphere. ... In mathematics, the Virasoro algebra (named after the physicist Miguel Angel Virasoro) is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in string theory. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. ... In mathematics, a Kac-Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. ... In mathematics, Kuipers theorem is a result on the topology of the operators on an infinite-dimensional complex Hilbert space H. It states that the topological space X of all linear operators L from H to itself, which are bounded operators and invertible, is such that for any finite... Calculus [from Latin, literally pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Calculus [from Latin, literally pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

See also

• E₈
• Adjoint representation
• Armand Borel
• Homogeneous space
• List of Lie group topics
• List of simple Lie groups
• Riemannian manifold
• Representations of Lie groups
• Table of Lie groups

In mathematics, E8 is the name given to a family of closely related structures. ... In mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its own Lie algebra. ... Armand Borel (21 May 1923 - 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study, Princeton from 1957 to 1993. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... This is a list of Lie group topics, by Wikipedia page. ... In mathematics, the simple Lie groups were classified by Élie Cartan. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... This article gives a table of some common Lie groups and their associated Lie algebras. ...

References

• Armand Borel, Essays in the history of Lie groups and algebraic groups, History of Mathematics 21, American Mathematical Society, 2001. ISBN 0-8218-0288-7

• Thomas Hawkins, Emergence of the theory of Lie groups, Springer, 2000. ISBN 0-387-98963-3

• Brian C. Hall Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9

• N. Bourbaki, Elements of mathematics: Lie groups and Lie algebras Chapter 1-3 ISBN 3-540-64242-0, Chapters 4-6 ISBN 3-540-42650-7, Chapters 7-9 ISBN 3-540-43405-4

• C. Chevalley, Theory of Lie groups, ISBN 0-691-04990-4.

• J.-P. Serre. Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, LNM 1500, Springer. ISBN 3-540-55008-9

• Anthony W. Knapp, Lie Groups Beyond an Introduction, Second Edition. Birkhäuser, 2002.

• J.F. Adams, Lectures on Lie Groups (Chicago Lectures in Mathematics). ISBN 0-226-00527-5

• Representation Theory : A First Course (Graduate Texts in Mathematics / Readings in Mathematics) by William Fulton, Joe Harris Publisher: Springer; 1 edition (July 30, 1999) ISBN 0-387-97495-4

• Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford University Press ISBN 0-19-859683-9. The 2003 reprinting corrects some unfortunate typos.

Notes

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