In

^{3},

_{2}(R):
:: $\backslash operatorname(2,\; \backslash mathbf)\; =\; \backslash left\backslash .$
: This is a four-dimensional noncompact real Lie group; it is an open subset of $\backslash mathbb\; R^4$. This group is disconnected; it has two connected components corresponding to the positive and negative values of the

_{2}, F4 (mathematics), ''F''_{4}, E6 (mathematics), ''E''_{6}, E7 (mathematics), ''E''_{7}, E8 (mathematics), ''E''_{8} have dimensions 14, 52, 78, 133, and 248. Along with the A-B-C-D series of simple Lie groups, the exceptional groups complete the list of simple Lie groups.
*The symplectic group $\backslash text(2n,\backslash mathbb)$ consists of all $2n\; \backslash times\; 2n$ matrices preserving a ''symplectic form'' on $\backslash mathbb^$. It is a connected Lie group of dimension $2n^2\; +\; n$.

^{''n''} for some positive integer ''n'' can be Lie groups (of course they must also have a differentiable structure).

^{''n''} is just R^{''n''} with the Lie bracket given by

[''A'', ''B''] = 0.

(In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.) * The Lie algebra of the general linear group GL(''n'', C) of invertible matrices is the vector space M(''n'', C) of square matrices with the Lie bracket given by

[''A'', ''B''] = ''AB'' − ''BA''. *If ''G'' is a closed subgroup of GL(''n'', C) then the Lie algebra of ''G'' can be thought of informally as the matrices ''m'' of M(''n'', C) such that 1 + ε''m'' is in ''G'', where ε is an infinitesimal positive number with ε^{2} = 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 + ε''m'')(1 + ε''m'')^{T} = 1, which is equivalent to ''m'' + ''m''^{T} = 0 because ε^{2} = 0.
*The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroup ''G'' of GL(''n'', C), may be computed as
:$\backslash operatorname(G)\; =\; \backslash ,$ where exp(''tX'') is defined using the matrix exponential. It can then be shown that the Lie algebra of ''G'' is a real vector space that is closed under the bracket operation, $[X,Y]=XY-YX$.
The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of the representation we use. To get around these problems we give
the general definition of the Lie algebra of a Lie group (in 4 steps):
#Vector fields on any smooth manifold ''M'' can be thought of as Derivation (abstract algebra), 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 vector fields, Lie bracket of any two derivations is a derivation.
#If ''G'' is any group acting smoothly on the manifold ''M'', then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
#We apply this construction to the case when the manifold ''M'' is the underlying space of a Lie group ''G'', with ''G'' acting on ''G'' = ''M'' by left translations ''L_{g}''(''h'') = ''gh''. This shows that the space of left invariant vector fields (vector fields satisfying ''L_{g}''_{*}''X_{h}'' = ''X_{gh}'' for every ''h'' in ''G'', where ''L_{g}''_{*} denotes the differential of ''L_{g}'') on a Lie group is a Lie algebra under the Lie bracket of vector fields.
#Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element ''v'' of the tangent space at the identity is the vector field defined by ''v''^_{''g''} = ''L_{g}''_{*}''v''. This identifies the tangent space ''T_{e}G'' 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 (typeface sub-classification), Fraktur $\backslash mathfrak.$ Thus the Lie bracket on $\backslash mathfrak$ is given explicitly by [''v'', ''w''] = [''v''^, ''w''^]_{''e''}.
This Lie algebra $\backslash mathfrak$ is finite-dimensional and it has the same dimension as the manifold ''G''. The Lie algebra of ''G'' determines ''G'' up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element.
Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily.
For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.
We could also define a Lie algebra structure on ''T_{e}'' using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on ''G'' can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space ''T_{e}''.
The Lie algebra structure on ''T_{e}'' can also be described as follows:
the commutator operation
: (''x'', ''y'') → ''xyx''^{−1}''y''^{−1}
on ''G'' × ''G'' sends (''e'', ''e'') to ''e'', so its derivative yields a bilinear operator, bilinear operation on ''T_{e}G''. 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 algebra#Definition and first properties, Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.

^{∞} Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

_{''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. E_{8} is the largest of these.
Lie groups are classified according to their algebraic properties (simple group, simple, semisimple group, semisimple, solvable group, solvable, nilpotent group, nilpotent, abelian group, abelian), their connectedness (connected space, connected or simply connected space, simply connected) and their compact space, compactness.
A first key result is the Levi decomposition, which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup.
*Connected compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S^{1} 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), SL(2, R) is simple according to the second definition but not according to the first. They have all been list of simple Lie groups, classified (for either definition).
*Semisimple group, Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups.
The identity component of any Lie group is an open normal subgroup, and the quotient group is a _{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 group extension, central extension of a product of list of simple Lie groups, simple connected Lie groups.
:''G''_{sol}/''G''_{nil} is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group ''S''^{1}.
:''G''_{nil}/1 is nilpotent, and therefore its ascending central series has all quotients abelian.
This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.
* The diffeomorphism, diffeomorphism group of a Lie group acts transitively on the Lie group
* Every Lie group is parallelizable, and hence an orientable manifold (there is a fibre bundle, bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity)

Borel's review

* * . * T. Kobayashi and T. Oshima, Lie groups and Lie algebras I, Iwanami, 1999 (in Japanese) * * . The 2003 reprint corrects several typographical mistakes. * * . * * Heldermann Verla

* * .

Lie Groups. Representation Theory and Symmetric Spaces

Wolfgang Ziller, Vorlesung 2010

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a Lie group (pronounced "Lee") is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

that is also a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...

. A manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

is a space that locally resembles Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

, whereas groups define the abstract concept of a binary operation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

along with the additional properties it must have to be a group, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth
Smooth may refer to:
Mathematics
* Smooth function
is a smooth function with compact support.
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...

(differentiable), one obtains a Lie group.
Lie groups provide a natural model for the concept of continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant un ...

, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$\backslash text(3)$). Lie groups are widely used in many parts of modern mathematics and physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

.
Lie groups were first found by studying matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

subgroups $G$ contained in $\backslash text\_n(\backslash mathbb)$ or $\backslash text\_(\backslash mathbb)$, the groups of $n\backslash times\; n$ invertible matrices over $\backslash mathbb$ or $\backslash mathbb$. These are now called the classical group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norway, Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Biography
Marius Sop ...

(1842–1899), who laid the foundations of the theory of continuous transformation group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...

, in much the same way that finite groups are used in Galois theory
In , Galois theory, originally introduced by , provides a connection between and . This connection, the , allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois introduced the ...

to model the discrete symmetries of algebraic equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s.
History

According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1),Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norway, Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Biography
Marius Sop ...

himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation (''ibid''). Some of Lie's early ideas were developed in close collaboration with Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and 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. From this effort resulted the three-volume ''Theorie der Transformationsgruppen'', published in 1888, 1890, and 1893. The term ''groupes de Lie'' first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse.
Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

, on the theory of partial differential equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany (Hawkins, p. 43). Lie's ''idée fixe'' was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ...

had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special function
Special functions are particular function (mathematics), mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
...

s and orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomials ...

tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of ''continuous groups'', to complement the theory of discrete group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s that had developed in the theory of modular form
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, in the hands of Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

and Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
France (), officially the French Repu ...

. The initial application that Lie had in mind was to the theory of differential equation
In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

s. On the model of Galois theory
In , Galois theory, originally introduced by , provides a connection between and . This connection, the , allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois introduced the ...

and polynomial equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s, the driving conception was of a theory capable of unifying, by the study of symmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

, the whole area of ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial ...

s. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This can ...

s required to express solutions.
Additional impetus to consider continuous groups came from ideas of Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics ...

, 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
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

; geometric theory and the explicit solutions of differential equation
In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

s of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

that emerged in the works of , , Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...

and others, and culminated in Riemann's revolutionary vision of the subject.
Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of ...

, who in 1888 published the first paper in a series entitled ''Die Zusammensetzung der stetigen endlichen Transformationsgruppen'' (''The composition of continuous finite transformation groups'') (Hawkins, p. 100). The work of Killing, later refined and generalized by Élie Cartan
Élie Joseph Cartan, ForMemRS
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a ...

, led to classification of semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras (non-abelian Lie algebras without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, ideals).
Throughout the art ...

s, Cartan's theory of symmetric spaces
In mathematics, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of hol ...

, and Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of ...

's description of representations
''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It co ...

of compact and semisimple Lie groups using highest weightIn the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

s.
In 1900 David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, G ...

challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...

in Paris.
Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's ''infinitesimal groups'' (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups. The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

.
Overview

Lie groups aresmooth
Smooth may refer to:
Mathematics
* Smooth function
is a smooth function with compact support.
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...

differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...

s and as such can be studied using differential calculus
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, in contrast with the case of more general topological group
350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition
In mathematics, a topological group is a group ...

s. One of the key ideas in the theory of Lie groups is to replace the ''global'' object, the group, with its ''local'' or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
Lie groups play an enormous role in modern geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, on several different levels. Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

argued in his Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"titl ...

that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic
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Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern ...

corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space Rconformal geometry
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

corresponds to enlarging the group to the conformal group
In mathematics, the conformal group of a space is the group (mathematics), group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.
...

, whereas in projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...

one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...

, where ''G'' is a Lie group of "local" symmetries of a manifold.
Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the representations
''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It co ...

of the Lie group (or of its Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

) are especially important. Representation theory is used extensively in particle physics. Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group
The Poincaré group, named after Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French
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* Something of, from, or related to ...

.
On a "global" level, whenever a Lie group acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum), often referred to simply as Acts, or formally the Book of Acts, is the fifth book of the New Testament
The New Te ...

on a geometric object, such as a Riemannian or a symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a Differentiable manifold#Definition, smooth manifold, M , equipped with a Closed and exact differential forms, closed nondegenerate form, nondegenerate Differential form ...

, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group actionIn differential geometry, a Lie group action is a Group action (mathematics), group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable map, differentiable.
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Definition and fi ...

on a manifold places strong constraints on its geometry and facilitates analysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...

on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

.
In the 1940s–1950s, Ellis Kolchin, Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...

, and Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic group
In algebraic geometry, an algebraic group (or group variety) is a Group (mathematics), group that is an algebraic variety, such that the multiplication and inversion operations are given by regular map (algebraic geometry), regular maps on the varie ...

s defined over an arbitrary field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple group
In mathematics, the classification of finite simple groups states that every Finite group, finite simple group is cyclic group, cyclic, or alternating group, alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
...

s, as well as in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. The theory of automorphic form
500px, The Dedekind eta-function is an automorphic form in the complex plane.
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) wh ...

s, an important branch of modern number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, deals extensively with analogues of Lie groups over adele ring In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the Complete metric space, com ...

s; ''p''-adic Lie groups play an important role, via their connections with Galois representations in number theory.
Definitions and examples

A real Lie group is agroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

that is also a finite-dimensional real smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

, in which the group operations of multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

and inversion are smooth map
In mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

s. Smoothness of the group multiplication
:$\backslash mu:G\backslash times\; G\backslash to\; G\backslash quad\; \backslash mu(x,y)=xy$
means that ''μ'' is a smooth mapping of the product manifold into ''G''. The two requirements can be combined to the single requirement that the mapping
:$(x,y)\backslash mapsto\; x^y$
be a smooth mapping of the product manifold into ''G''.
First examples

* The 2×2real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

invertible matrices
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

form a group under multiplication, denoted by or by GLdeterminant
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

.
* The rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

matrices form a subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

of , denoted by . It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

to the circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

. Using the rotation angle $\backslash varphi$ as a parameter, this group can be parametrized
In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...

as follows:
:: $\backslash operatorname(2,\; \backslash mathbf)\; =\; \backslash left\backslash .$
:Addition of the angles corresponds to multiplication of the elements of , and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.
* The affine group of one dimension is a two-dimensional matrix Lie group, consisting of $2\; \backslash times\; 2$ real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form
::$A=\; \backslash left(\; \backslash begin\; a\; \&\; b\backslash \backslash \; 0\; \&\; 1\; \backslash end\backslash right),\backslash quad\; a>0,\backslash ,\; b\; \backslash in\; \backslash mathbb.$
Non-example

We now present an example of a group with anuncountable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

number of elements that is not a Lie group under a certain topology. The group given by
:$H\; =\; \backslash left\backslash \; \backslash subset\; \backslash mathbb^2\; =\; \backslash left\backslash ,$
with $a\; \backslash in\; \backslash mathbb\; R\; \backslash setminus\; \backslash mathbb\; Q$ a ''fixed'' irrational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

, is a subgroup of the torus
In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle.
If the axis of revolution does not to ...

$\backslash mathbb\; T^2$ that is not a Lie group when given the subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

. If we take any small neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

$U$ of a point $h$ in $H$, for example, the portion of $H$ in $U$ is disconnected. The group $H$ winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

subgroup of $\backslash mathbb\; T^2$.
The group $H$ can, however, be given a different topology, in which the distance between two points $h\_1,h\_2\backslash in\; H$ is defined as the length of the shortest path ''in the group '' $H$ joining $h\_1$ to $h\_2$. In this topology, $H$ is identified homeomorphically with the real line by identifying each element with the number $\backslash theta$ in the definition of $H$. With this topology, $H$ is just the group of real numbers under addition and is therefore a Lie group.
The group $H$ is an example of a " Lie subgroup" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.
Matrix Lie groups

Let $\backslash operatorname(n,\; \backslash mathbb)$ denote the group of $n\backslash times\; n$ invertible matrices with entries in $\backslash mathbb$. Anyclosed subgroup
Image:Real number line.svg, 350px, The real numbers form a topological group under addition
In mathematics, a topological group is a group (mathematics), group together with a topological space, topology on such that both the group's binary op ...

of $\backslash operatorname(n,\; \backslash mathbb)$ is a Lie group; Lie groups of this sort are called matrix Lie groups. Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall, Rossmann, and Stillwell.
Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.
*The special linear group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s over $\backslash mathbb$ and $\backslash mathbb$, $\backslash operatorname(n,\; \backslash mathbb)$ and $\backslash operatorname(n,\; \backslash mathbb)$, consisting of $n\backslash times\; n$ matrices with determinant one and entries in $\backslash mathbb$ or $\backslash mathbb$
*The unitary group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s and special unitary groups, $\backslash text(n)$ and $\backslash text(n)$, consisting of $n\backslash times\; n$ complex matrices satisfying $U^*=U^$ (and also $\backslash det(U)=1$ in the case of $\backslash text(n)$)
*The orthogonal group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s and special orthogonal groups, $\backslash text(n)$ and $\backslash text(n)$, consisting of $n\backslash times\; n$ real matrices satisfying $R^\backslash mathrm=R^$ (and also $\backslash det(R)=1$ in the case of $\backslash text(n)$)
All of the preceding examples fall under the heading of the classical group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s.
Related concepts

Acomplex Lie group
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

is defined in the same way using complex manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...

s rather than real ones (example: $\backslash operatorname(2,\; \backslash mathbb)$), and holomorphic maps. Similarly, using an alternate metric completion of $\backslash mathbb$, one can define a ''p''-adic Lie group over the ''p''-adic numbers, a topological group which is also an analytic ''p''-adic manifold, such that the group operations are analytic. In particular, each point has a ''p''-adic neighborhood.
Hilbert's fifth problemHilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influenti ...

asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, Montgomery
Montgomery may refer to:
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For people with the name Montgomery, see Montgomery (name)
Places Belgium
* Montgomery Square, Brussels
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Pakistan
* Montgomery (town), British India, former name of Sahiwal, Punj ...

and Zippin showed that if ''G'' is a topological manifold with continuous group operations, then there exists exactly one analytic structure on ''G'' which turns it into a Lie group (see also Hilbert–Smith conjecture). If the underlying manifold is allowed to be infinite-dimensional (for example, a Hilbert manifold), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple group
In mathematics, the classification of finite simple groups states that every Finite group, finite simple group is cyclic group, cyclic, or alternating group, alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
...

s.
The language of category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

provides a concise definition for Lie groups: a Lie group is a group objectIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

in the category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups.
Topological definition

A Lie group can be defined as a ()topological group
350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition
In mathematics, a topological group is a group ...

that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup ''G'' of the general linear group $\backslash operatorname(n,\; \backslash mathbb)$ such that
# for some neighborhood ''V'' of the identity element ''e'' in ''G'', the topology on ''V'' is the subspace topology of $\backslash operatorname(n,\; \backslash mathbb)$ and ''V'' is closed in $\backslash operatorname(n,\; \backslash mathbb)$.
# ''G'' has at most countably many
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

connected components.
(For example, a closed subgroup of $\backslash operatorname(n,\; \backslash mathbb)$; that is, a matrix Lie group satisfies the above conditions.)
Then a ''Lie group'' is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:
# Given a Lie group ''G'' in the usual manifold sense, the Lie group–Lie algebra correspondenceIn mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are Isomorphism, isomorphic to each other have Lie algebras ...

(or a version of Lie's third theoremIn the mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

) constructs an immersed Lie subgroup $G\text{'}\; \backslash subset\; \backslash operatorname(n,\; \backslash mathbb)$ such that $G,\; G\text{'}$ share the same Lie algebra; thus, they are locally isomorphic. Hence, ''G'' satisfies the above topological definition.
# Conversely, let ''G'' be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group $G\text{'}$ that is locally isomorphic to ''G''. Then, by a version of the closed subgroup theoremIn mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie group theory, Lie groups. It states that if is a closed subgroup of a Lie group , then is an Embedding, embedded Lie group wit ...

, $G\text{'}$ is a real-analytic manifoldIn mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with Analytic function, analytic transition maps. The term usually refers to real analytic manifolds, although complex manifold, complex manifolds a ...

and then, through the local isomorphism, ''G'' acquires a structure of a manifold near the identity element. One then shows that the group law on ''G'' can be given by formal power series; so the group operations are real-analytic and ''G'' itself is a real-analytic manifold.
The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, ''the topology of a Lie group'' together with the group law determines the geometry of the group.
More examples of Lie groups

Lie groups occur in abundance throughout mathematics and physics. Matrix groups oralgebraic group
In algebraic geometry, an algebraic group (or group variety) is a Group (mathematics), group that is an algebraic variety, such that the multiplication and inversion operations are given by regular map (algebraic geometry), regular maps on the varie ...

s are (roughly) groups of matrices (for example, orthogonal group, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.
Dimensions one and two

The only connected Lie groups with dimension one are the real line $\backslash mathbb$ (with the group operation being addition) and the circle group $S^1$ of complex numbers with absolute value one (with the group operation being multiplication). The $S^1$ group is often denoted as $U(1)$, the group of $1\backslash times\; 1$ unitary matrices. In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are $\backslash mathbb^2$ (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples."Additional examples

*The Special unitary group#n_.3D_2, group SU(2) is the group of $2\backslash times\; 2$ unitary matrices with determinant $1$. Topologically, $\backslash text(2)$ is the $3$-sphere $S^3$; as a group, it may be identified with the group of unit quaternions. *The Heisenberg group is a connected nilpotent group, nilpotent Lie group of dimension $3$, playing a key role in quantum mechanics. *The Lorentz group is a 6-dimensional Lie group of linear isometry, isometries of the Minkowski space. *The Poincaré group is a 10-dimensional Lie group of affine transformation, affine isometries of the Minkowski space. *The exceptional Lie groups of types G2 (mathematics), ''G''Constructions

There are several standard ways to form new Lie groups from old ones: *The product of two Lie groups is a Lie group. *Any Closed set, topologically closed subgroup of a Lie group is a Lie group. This is known as the Closed subgroup theorem or Cartan's theorem. *The quotient of a Lie group by a closed normal subgroup is a Lie group. *The universal cover of a connected Lie group is a Lie group. For example, the group $\backslash mathbb$ is the universal cover of the circle group $S^1$. In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying ''universal'' cover, one guarantees a group structure (compatible with its other structures).Related notions

Some examples of groups that are ''not'' Lie groups (except in the trivial sense that any group having at most countably many elements can be viewed as a 0-dimensional Lie group, with the discrete topology), are: *Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space, or the space of smooth functions from a manifold $X$ to a Lie group $G$, $C^\backslash infty(X,G)$. These are not Lie groups as they are not ''finite-dimensional'' manifolds. *Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the ''p''-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "''p''-adic Lie groups".) In general, only topological groups having similar local property, local properties to RBasic concepts

The Lie algebra associated with a Lie group

To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples: * The Lie algebra of the vector space R[''A'', ''B''] = 0.

(In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.) * The Lie algebra of the general linear group GL(''n'', C) of invertible matrices is the vector space M(''n'', C) of square matrices with the Lie bracket given by

[''A'', ''B''] = ''AB'' − ''BA''. *If ''G'' is a closed subgroup of GL(''n'', C) then the Lie algebra of ''G'' can be thought of informally as the matrices ''m'' of M(''n'', C) such that 1 + ε''m'' is in ''G'', where ε is an infinitesimal positive number with ε

Homomorphisms and isomorphisms

If ''G'' and ''H'' are Lie groups, then a Lie group homomorphism ''f'' : ''G'' → ''H'' is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic map, analytic. The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category theory, category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let $\backslash phi\backslash colon\; G\; \backslash to\; H$ be a Lie group homomorphism and let $\backslash phi\_$ be its Pushforward (differential), derivative at the identity. If we identify the Lie algebras of ''G'' and ''H'' with their tangent spaces at the identity elements then $\backslash phi\_$ is a map between the corresponding Lie algebras: :$\backslash phi\_\backslash colon\backslash mathfrak\; g\; \backslash to\; \backslash mathfrak\; h$ One can show that $\backslash phi\_$ is actually a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language ofcategory theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.
Two Lie groups are called ''isomorphic'' if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group $G$ to a Lie group $H$ is an isomorphism of Lie groups if and only if it is bijective.
Lie group versus Lie algebra isomorphisms

Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this direction isLie's third theoremIn the mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.
On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the ''global'' structure of a Lie group is not determined by its Lie algebra; for example, if ''Z'' is any discrete subgroup of the center of ''G'' then ''G'' and ''G''/''Z'' have the same Lie algebra (see the table of Lie groups for examples). An example of importance in physics are the groups Special_unitary_group#The_group_SU(2), SU(2) and Rotation group SO(3), SO(3). These two groups have isomorphic Lie algebras, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.
On the other hand, if we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.
Simply connected Lie groups

A Lie group $G$ is said to be Simply connected space, simply connected if every loop in $G$ can be shrunk continuously to a point in $G$. This notion is important because of the following result that has simple connectedness as a hypothesis: :Theorem: Suppose $G$ and $H$ are Lie groups with Lie algebras $\backslash mathfrak\; g$ and $\backslash mathfrak\; h$ and that $f:\backslash mathfrak\backslash rightarrow\backslash mathfrak$ is a Lie algebra homomorphism. If $G$ is simply connected, then there is a unique Lie group homomorphism $\backslash phi:G\backslash rightarrow\; H$ such that $\backslash phi\_*=f$, where $\backslash phi\_*$ is the differential of $\backslash phi$ at the identity. Lie group–Lie algebra correspondence#The correspondence, Lie's third theorem says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a ''unique'' simply connected Lie group. An example of a simply connected group is the special unitary group Special unitary group#n_.3D_2, SU(2), which as a manifold is the 3-sphere. The rotation group SO(3), on the other hand, is not simply connected. (See Rotation group SO(3)#Topology, Topology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group SU(n), the spin group (double cover of rotation group) Spin(n) for $n\backslash geq\; 3$, and the compact symplectic group Symplectic group#Sp.28n.29, Sp(n). Methods for determining whether a Lie group is simply connected or not are discussed in the article on Fundamental group#Lie groups, fundamental groups of Lie groups.The exponential map

The exponential map (Lie theory), exponential map from the Lie algebra $M(n;\backslash mathbb\; C)$ of the general linear group $GL(n;\backslash mathbb\; C)$ to $GL(n;\backslash mathbb\; C)$ is defined by the matrix exponential, given by the usual power series: :$\backslash exp(X)\; =\; 1\; +\; X\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ for matrices $X$. If $G$ is a closed subgroup of $GL(n;\backslash mathbb\; C)$, then the exponential map takes the Lie algebra of $G$ into $G$; thus, we have an exponential map for all matrix groups. Every element of $G$ that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra. The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows. For each vector $X$ in the Lie algebra $\backslash mathfrak$ of $G$ (i.e., the tangent space to $G$ at the identity), one proves that there is a unique one-parameter subgroup $c:\backslash mathbb\; R\backslash rightarrow\; G$ such that $c\text{'}(0)=X$. Saying that $c$ is a one-parameter subgroup means simply that $c$ is a smooth map into $G$ and that :$c(s\; +\; t)\; =\; c(s)\; c(t)\backslash $ 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 :$\backslash exp(X)\; =\; c(1).\backslash $ This is called the exponential map, and it maps the Lie algebra $\backslash mathfrak$ into the Lie group $G$. It provides a diffeomorphism between a neighborhood (topology), neighborhood of 0 in $\backslash mathfrak$ and a neighborhood of $e$ in $G$. This exponential map is a generalization of the exponential function for real numbers (because $\backslash mathbb$ is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because $\backslash mathbb$ is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrix (math), matrices (because $M(n,\; \backslash mathbb)$ with the regular commutator is the Lie algebra of the Lie group $GL(n,\; \backslash mathbb)$ of all invertible matrices). Because the exponential map is surjective on some neighbourhood $N$ of $e$, it is common to call elements of the Lie algebra infinitesimal generators of the group $G$. The subgroup of $G$ generated by $N$ is the identity component of $G$. The exponential map and the Lie algebra determine the ''local group structure'' of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood $U$ of the zero element of $\backslash mathfrak$, such that for $X,Y\backslash in\; U$ we have :$\backslash exp(X)\backslash ,\backslash exp(Y)\; =\; \backslash exp\backslash left(X\; +\; Y\; +\; \backslash tfrac[X,Y]\; +\; \backslash tfrac[\backslash ,[X,Y],Y]\; -\; \backslash tfrac[\backslash ,[X,Y],X]\; -\; \backslash cdots\; \backslash right),$ where the omitted terms are known and involve Lie brackets of four or more elements. In case $X$ and $Y$ commute, this formula reduces to the familiar exponential law $\backslash exp(X)\backslash exp(Y)=\backslash exp(X+Y)$ The exponential map relates Lie group homomorphisms. That is, if $\backslash phi:\; G\; \backslash to\; H$ is a Lie group homomorphism and $\backslash phi\_*:\; \backslash mathfrak\; \backslash to\; \backslash mathfrak$ the induced map on the corresponding Lie algebras, then for all $x\backslash in\backslash mathfrak\; g$ we have :$\backslash phi(\backslash exp(x))\; =\; \backslash exp(\backslash phi\_(x)).\backslash ,$ In other words, the following diagram commutative diagram, commutes, (In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.) The exponential map from the Lie algebra to the Lie group is not always Surjective function, 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), SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on Smooth function#Differentiability classes, ''C''Lie subgroup

A Lie subgroup $H$ of a Lie group $G$ is a Lie group that is a subset of $G$ and such that the inclusion map from $H$ to $G$ is an injective Immersion (mathematics), immersion and group homomorphism. According to Closed subgroup theorem, Cartan's theorem, a closedsubgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

of $G$ admits a unique smooth structure which makes it an embedding, embedded Lie subgroup of $G$—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.
Examples of non-closed subgroups are plentiful; for example take $G$ to be a torus of dimension 2 or greater, and let $H$ be a one-parameter subgroup of ''irrational slope'', i.e. one that winds around in ''G''. Then there is a Lie group homomorphism $\backslash varphi:\backslash mathbb\backslash to\; G$ with $\backslash mathrm(\backslash varphi)\; =\; H$. The closure (topology), closure of $H$ will be a sub-torus in $G$.
The exponential map (Lie theory), exponential map gives a Lie group–Lie algebra correspondence#The correspondence, one-to-one correspondence between the connected Lie subgroups of a connected Lie group $G$ and the subalgebras of the Lie algebra of $G$. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of $G$ which determines which subalgebras correspond to closed subgroups.
Representations

One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independent Schrödinger equation in quantum mechanics, $\backslash hat\backslash psi\; =\; E\backslash psi$. Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator $\backslash hat$ commutes with the action of SO(3) on the wave function $\backslash psi$. (One important example of such a system is the Hydrogen atom, which has a single spherical orbital.) This assumption does not necessarily mean that the solutions $\backslash psi$ are rotationally invariant functions. Rather, it means that the ''space'' of solutions to $\backslash hat\backslash psi\; =\; E\backslash psi$ is invariant under rotations (for each fixed value of $E$). This space, therefore, constitutes a representation of SO(3). These representations have been Representation of a Lie group#An example: The rotation group SO.283.29, classified and the classification leads to a substantial Hydrogen-like atom, simplification of the problem, essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation. The case of a connected compact Lie group ''K'' (including the just-mentioned case of SO(3)) is particularly tractable. In that case, every finite-dimensional representation of ''K'' decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified byHermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German
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. Compact group#Representation theory of a connected compact Lie group, The classification is in terms of the "highest weight" of the representation. The classification is closely related to the Lie algebra representation#Classifying finite-dimensional representations of Lie algebras, classification of representations of a semisimple Lie algebra.
One can also study (in general infinite-dimensional) unitary representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the Representation theory of SL2(R), representations of the group SL(2,R) and the Wigner%27s classification, representations of the Poincaré group.
Classification

Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Sophus Lie, Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have tangent spaces at each point. The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a Direct sum of modules, direct sum of an abelian Lie algebra and some number of simple Lie group, simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie group, simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" Adiscrete group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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

Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces (as opposed toEuclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general Locally convex space, locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold.
The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix ''Lie'' in ''Lie group''. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix ''Lie'' in ''Lie algebra'' are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.
Some of the examples that have been studied include:
*The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, whose Lie algebra extension, central extension the Virasoro algebra (see Lie algebra extension#Virasoro algebra, Virasoro algebra from Witt algebra for a derivation of this fact) is the symmetry algebra of two-dimensional conformal field theory. Diffeomorphism groups of compact manifolds of larger dimension are Convenient vector space#Regular Lie groups, regular Fréchet Lie groups; very little about their structure is known.
*The diffeomorphism group of spacetime sometimes appears in attempts to Quantization (physics), quantize gravity.
*The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac–Moody algebras.
*There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have ''simpler'' topological properties: see for example Kuiper's theorem. In M-theory, for example, a 10 dimensional SU(N) gauge theory becomes an 11 dimensional theory when N becomes infinite.
See also

*Adjoint representation of a Lie group *Haar measure *Homogeneous space *List of Lie group topics *Representations of Lie groups *Symmetry in quantum mechanicsNotes

Explanatory notes

Citations

References

* . * * * . Chapters 1–3 , Chapters 4–6 , Chapters 7–9 * . * P. M. Cohn (1957) ''Lie Groups'', Cambridge Tracts in Mathematical Physics. * J. L. Coolidge (1940) ''A History of Geometrical Methods'', pp 304–17, Oxford University Press (Dover Publications 2003). * * Robert Gilmore (2008) ''Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists'', Cambridge University Press . * . * F. Reese Harvey (1990) ''Spinors and calibrations'', Academic Press, .Borel's review

* * . * T. Kobayashi and T. Oshima, Lie groups and Lie algebras I, Iwanami, 1999 (in Japanese) * * . The 2003 reprint corrects several typographical mistakes. * * . * * Heldermann Verla

* * .

Lie Groups. Representation Theory and Symmetric Spaces

Wolfgang Ziller, Vorlesung 2010

External links

* {{Authority control Lie groups, Manifolds Symmetry