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| {{Lie groups}}
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| {{Group theory sidebar}}
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| In [[mathematics]], a '''Lie group''' {{IPAc-en|ˈ|l|iː}} is a [[group (mathematics)|group]] that is also a [[differentiable manifold]], with the property that the group operations are compatible with the [[Differential structure|smooth structure]]. Lie groups are named after [[Sophus Lie]], who laid the foundations of the theory of continuous [[transformation group]]s.
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| Lie groups represent the best-developed theory of [[continuous symmetry]] of [[mathematical object]]s and [[mathematical structure|structures]], which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern [[theoretical physics]]. They provide a natural framework for analysing the continuous symmetries of [[differential equations]] ([[differential Galois theory]]), in much the same way as [[permutation groups]] are used in [[Galois theory]] for analysing the discrete symmetries of [[algebraic equations]]. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.
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| == Overview ==
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| [[Image:Circle as Lie group.svg|right|thumb|The [[circle]] of center 0 and radius 1 in the [[complex plane]] is a Lie group with complex multiplication.]]
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| Lie groups are [[smooth manifold]]s and as such can be studied using [[differential calculus]], in contrast with the case of more general [[topological 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]].
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| Lie groups play an enormous role in modern [[geometry]], on several different levels. [[Felix Klein]] argued in his [[Erlangen program]] that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus [[Euclidean geometry]] corresponds to the choice of the group [[Euclidean group|E(3)]] of distance-preserving transformations of the Euclidean space '''R'''<sup>3</sup>, [[conformal geometry]] corresponds to enlarging the group to the [[conformal group]], whereas in [[projective geometry]] one is interested in the properties invariant under the [[projective group]]. This idea later led to the notion of a [[G-structure]], where ''G'' is a Lie group of "local" symmetries of a manifold. On a "global" level, whenever a Lie group [[group action|acts]] on a geometric object, such as a [[Riemannian manifold|Riemannian]] or a [[symplectic manifold]], this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates [[global analysis|analysis]] on the manifold. Linear actions of Lie groups are especially important, and are studied in [[representation theory]].
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| In the 1940s–1950s, [[Ellis Kolchin]], [[Armand Borel]], and [[Claude Chevalley]] realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of [[algebraic group]]s defined over an arbitrary [[field (mathematics)|field]]. This insight opened new possibilities in pure algebra, by providing a uniform construction for most [[finite simple group]]s, as well as in [[algebraic geometry]]. The theory of [[automorphic form]]s, an important branch of modern [[number theory]], deals extensively with analogues of Lie groups over [[adele ring]]s; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.
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| == Definitions and examples ==
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| A '''real Lie group''' is a [[group (mathematics)|group]] that is also a finite-dimensional real [[Differentiable manifold#Definition|smooth manifold]], and in which the group operations of [[multiplication]] and inversion are [[smooth map]]s. Smoothness of the group multiplication
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| :<math> \mu:G\times G\to G\quad \mu(x,y)=xy</math>
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| means that μ is a smooth mapping of the product manifold ''G''×''G'' into ''G''. These two requirements can be combined to the single requirement that the mapping
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| :<math>(x,y)\mapsto x^{-1}y</math>
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| be a smooth mapping of the product manifold into ''G''.
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| === First examples ===
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| * The 2×2 [[real number|real]] [[invertible matrix|invertible matrices]] form a group under multiplication, denoted by [[general linear group|GL(2, '''R''')]]:
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| :: <math> \operatorname{GL}(2, \mathbf{R}) = \left\{A=\begin{pmatrix}a&b\\c&d\end{pmatrix}: \det A=ad-bc \ne 0\right\}. </math>
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| : This is a four-dimensional [[compact space|noncompact]] real Lie group. This group is [[connected space|disconnected]]; it has two connected components corresponding to the positive and negative values of the [[determinant]].
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| * The [[rotation (mathematics)|rotation]] matrices form a [[subgroup]] of GL(2, '''R'''), denoted by SO(2, '''R'''). It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is [[diffeomorphic]] to the [[circle]]. Using the rotation angle φ as a parameter, this group can be [[parametric equations|parametrized]] as follows:
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| :: <math> \operatorname{SO}(2, \mathbf{R}) =\left\{\begin{pmatrix} \cos\varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{pmatrix}: \varphi\in\mathbf{R}/2\pi\mathbf{Z}\right\}. </math>
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| :Addition of the angles corresponds to multiplication of the elements of SO(2, '''R'''), and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.
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| * The [[orthogonal group]] also forms an interesting example of a Lie group.
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| All of the previous examples of Lie groups fall within the class of [[classical group]]s.
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| === Related concepts ===
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| A '''[[complex Lie group]]''' is defined in the same way using [[complex manifold]]s rather than real ones (example: SL(2, '''C''')), and similarly{{clarify|date=February 2013}}<!-- what manifold structure? --> one can define a '''''p''-adic Lie group''' over the [[p-adic number|''p''-adic numbers]]. [[Hilbert's fifth problem]] asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, [[Andrew Gleason|Gleason]], [[Deane Montgomery|Montgomery]] and [[Leo Zippin|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 [[group of Lie type|Lie groups over finite fields]], and these give most of the examples of [[finite simple group]]s.
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| The language of [[category theory]] provides a concise definition for Lie groups: a Lie group is a [[group object]] in the [[category (mathematics)|category]] of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to [[supergroup (physics)|Lie supergroups]].
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| == More examples of Lie groups ==
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| {{see also|Table of Lie groups|List of simple Lie groups}}
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| Lie groups occur in abundance throughout mathematics and physics. [[Matrix group]]s or [[algebraic group]]s are (roughly) groups of matrices (for example, [[orthogonal group|orthogonal]] and [[symplectic group]]s), and these give most of the more common examples of Lie groups.
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| === Examples with a specific number of dimensions ===
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| * The [[circle group]] '''S'''<sup>1</sup> consisting of angles mod 2''π'' under addition or, alternatively, the complex numbers with absolute value 1 under multiplication. This is a one-dimensional compact connected abelian Lie group.
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| *The [[3-sphere]] '''S'''<sup>3</sup> forms a Lie group by identification with the set of [[quaternion]]s of unit norm, called [[versor]]s. The only other spheres that admit the structure of a Lie group are the 0-sphere '''S'''<sup>0</sup> (real numbers with absolute value 1) and the circle '''S'''<sup>1</sup> (complex numbers with absolute value 1). For example, for even ''n'' > 1, '''S'''<sup>''n''</sup> is not a Lie group because it does not admit a nonvanishing vector field and so [[A fortiori argument|''a fortiori'']] cannot be [[parallelizable manifold|parallelizable]] as a differentiable manifold. Of the spheres only '''S'''<sup>0</sup>, '''S'''<sup>1</sup>, '''S'''<sup>3</sup>, and '''S'''<sup>7</sup> are parallelizable. The latter carries the structure of a Lie [[quasigroup]] (a nonassociative group), which can be identified with the set of unit [[octonions]].
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| *The (3-dimensional) [[metaplectic group]] is a double cover of [[SL2(R)|SL(2, '''R''')]] playing an important role in the theory of [[modular form]]s. It is a connected Lie group that cannot be faithfully represented by matrices of finite size, i.e., a [[linear group|nonlinear group]].
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| *The [[Heisenberg group]] is a connected [[nilpotent group|nilpotent]] Lie group of dimension 3, playing a key role in [[quantum mechanics]].
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| *The [[Lorentz group]] is a 6 dimensional Lie group of linear [[isometry|isometries]] of the [[Minkowski space]].
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| *The [[Poincaré group]] is a 10 dimensional Lie group of [[affine transformation|affine]] isometries of the Minkowski space.
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| *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]] in [[particle physics]]. The dimensions of the factors correspond to the 1 [[photon]] + 3 [[vector boson]]s + 8 [[gluon]]s of the standard model
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| *The [[exceptional Lie group]]s of types [[G2 (mathematics)|''G''<sub>2</sub>]], [[F4 (mathematics)|''F''<sub>4</sub>]], [[E6 (mathematics)|''E''<sub>6</sub>]], [[E7 (mathematics)|''E''<sub>7</sub>]], [[E8 (mathematics)|''E''<sub>8</sub>]] 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. There is also a Lie group named [[E7½ (Lie algebra)|E<sub>7½</sub>]] of dimension 190, but it is not a ''simple'' Lie group.
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| === Examples with <var>n</var> dimensions ===
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| *[[Euclidean space]] '''R'''<sup>''n''</sup> with ordinary vector addition as the group operation becomes an ''n''-dimensional noncompact [[Abelian group|abelian]] Lie group.
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| *The [[Euclidean group]] E(''n'', '''R''') is the Lie group of all Euclidean motions, i.e., isometric affine maps, of ''n''-dimensional Euclidean space '''R'''<sup>''n''</sup>.
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| *The [[orthogonal group]] O(''n'', '''R'''), consisting of all ''n'' × ''n'' [[orthogonal matrix|orthogonal matrices]] with real entries is an ''n''(''n'' − 1)/2-dimensional Lie group. This group is disconnected, but it has a connected subgroup SO(''n'', '''R''') of the same dimension consisting of orthogonal matrices of determinant 1, called the special orthogonal group (for ''n'' = 3, the [[rotation group SO(3)]]).
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| *The [[unitary group]] U(''n'') consisting of ''n'' × ''n'' [[unitary matrix|unitary matrices]] (with complex entries) is a compact connected Lie group of dimension ''n''<sup>2</sup>. Unitary matrices of determinant 1 form a closed connected subgroup of dimension ''n''<sup>2</sup> − 1 denoted SU(''n''), the [[special unitary group]].
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| *[[Spin group]]s are [[Double covering group|double covers]] of the [[special orthogonal group]]s, used for studying [[fermion]]s in [[quantum field theory]] (among other things).
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| * The group GL(''n'', '''R''') of [[invertible matrix|invertible]] [[matrix (mathematics)|matrices]] (under [[matrix multiplication]]) is a Lie group of dimension ''n''<sup>2</sup>, called the [[general linear group]]. It has a closed connected subgroup SL(''n'', '''R'''), the [[special linear group]], consisting of matrices of determinant 1 which is also a Lie group.
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| *The [[symplectic group]] Sp(2''n'', '''R''') consists of all 2''n'' × 2''n'' matrices preserving a ''[[symplectic form]]'' on '''R'''<sup>2''n''</sup>. It is a connected Lie group of dimension 2''n''<sup>2</sup> + ''n''.
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| *The group of invertible upper triangular ''n'' by ''n'' matrices is a [[solvable group|solvable Lie group]] of dimension ''n''(''n'' + 1)/2. (cf. [[Borel subgroup]])
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| *The A-series, B-series, C-series and D-series, whose elements are denoted by A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub>, and D<sub>''n''</sub>, are infinite families of simple Lie groups.
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| === Constructions ===
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| There are several standard ways to form new Lie groups from old ones:
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| *The product of two Lie groups is a Lie group.
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| *Any [[Closed set|topologically closed]] subgroup of a Lie group is a Lie group. This is known as [[Cartan's theorem]].
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| *The quotient of a Lie group by a closed normal subgroup is a Lie group.
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| *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 '''S'''<sup>1</sup>. 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).
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| === Related notions ===
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| Some examples of groups that are ''not'' Lie groups (except in the trivial sense that any group can be viewed as a 0-dimensional Lie group, with the [[discrete topology]]), are:
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| *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
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| *Some [[totally disconnected group]]s, such as the [[Galois group]] of an infinite extension of fields, or the additive group of the [[p-adic number|''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 '''R'''<sup>''n''</sup> for some positive integer ''n'' can be Lie groups (of course they must also have a differentiable structure)
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| == Early history ==
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| According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1), [[Sophus Lie]] himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation (''ibid''). Some of Lie's early ideas were developed in close collaboration with [[Felix Klein]]. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years (''ibid'', p. 2). Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe (''ibid'', p. 76). In 1884 a young German mathematician, [[Friedrich Engel (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.
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| Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of [[Carl Gustav Jacobi]], on the theory of [[partial differential equation]]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]] had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for [[special functions]] and [[orthogonal polynomials]] tend to arise from group theoretical symmetries. Additional impetus to consider continuous groups came from ideas of [[Bernhard Riemann]], on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a [[group (mathematics)|group]]; geometric theory and the explicit solutions of [[differential equation]]s of mechanics, worked out by [[Siméon Denis Poisson|Poisson]] and Jacobi; and the new understanding of [[geometry]] that emerged in the works of [[Julius Plücker|Plücker]], [[August Ferdinand Möbius|Möbius]], [[Grassmann]] and others, and culminated in Riemann's revolutionary vision of the subject.
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| Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by [[Wilhelm Killing]], who in 1888 published the first paper in a series entitled ''Die Zusammensetzung der stetigen endlichen Transformationsgruppen'' (''The composition of continuous finite transformation groups'') (Hawkins, p. 100). The work of Killing, later refined and generalized by [[Élie Cartan]], led to classification of [[semisimple Lie algebra]]s, Cartan's theory of [[Riemannian symmetric space|symmetric spaces]], and [[Hermann Weyl]]'s description of [[group representation|representations]] of compact and semisimple Lie groups using [[highest weight]]s.
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| Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's ''infinitesimal groups'' (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups (Borel (2001), {{Citation needed|date=March 2007}}). The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by [[Claude Chevalley]].
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| <!--
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| Need specific reference from Borel's book to Weyl's work, in particular, distinction mentioned in the text
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| -->
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| ==The concept of a Lie group, and possibilities of classification==
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| 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, e.g., 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 manifolds, so have [[tangent space]]s at each point.
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| The Lie algebra of any [[Compact group|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" A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub> and D<sub>''n''</sub>, 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<sub>8</sub> is the largest of these.
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| == Properties ==
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| * The [[diffeomorphism|diffeomorphism group]] of a Lie group acts transitively on the Lie group
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| * 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)
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| == Types of Lie groups and structure theory ==
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| 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]].
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| *[[Compact Lie group]]s are all known: they are finite central quotients of a product of copies of the circle group '''S'''<sup>1</sup> and simple compact Lie groups (which correspond to connected [[Dynkin diagram]]s).
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| *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.
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| *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.
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| *[[Simple Lie group]]s are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, [[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).
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| *[[Semisimple group|Semisimple]] Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras.<ref>{{cite book |first=Sigurdur |last=Helgason |title=Differential Geometry, Lie Groups, and Symmetric Spaces |location=New York |publisher=Academic Press |year=1978 |page=131 |isbn=0-12-338460-5 }}</ref> They are central extensions of products of simple Lie groups.
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| The [[identity component]] of any Lie group is an open [[normal subgroup]], and the [[quotient group]] is a [[discrete group]]. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group ''G'' can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write
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| :''G''<sub>con</sub> for the connected component of the identity
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| :''G''<sub>sol</sub> for the largest connected normal solvable subgroup
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| :''G''<sub>nil</sub> for the largest connected normal nilpotent subgroup
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| so that we have a sequence of normal subgroups
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| :1 ⊆ ''G''<sub>nil</sub> ⊆ ''G''<sub>sol</sub> ⊆ ''G''<sub>con</sub> ⊆ ''G''.
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| Then
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| :''G''/''G''<sub>con</sub> is discrete
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| :''G''<sub>con</sub>/''G''<sub>sol</sub> is a [[group extension|central extension]] of a product of [[list of simple Lie groups|simple connected Lie groups]].
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| :''G''<sub>sol</sub>/''G''<sub>nil</sub> is abelian. A connected abelian Lie group is isomorphic to a product of copies of '''R''' and the [[circle group]] ''S''<sup>1</sup>.
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| :''G''<sub>nil</sub>/1 is nilpotent, and therefore its ascending central series has all quotients abelian.
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| 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.
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| == The Lie algebra associated with a Lie group ==
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| 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 "[[infinitesimal]]ly 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:
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| * The Lie algebra of the vector space '''R'''<sup>''n''</sup> is just '''R'''<sup>''n''</sup> with the Lie bracket given by <br /> [''A'', ''B''] = 0. <br /> (In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.)
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| * 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 <br /> [''A'', ''B''] = ''AB'' − ''BA''.<br />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 + ε''m'' is in ''G'', where ε is an infinitesimal positive number with ε<sup>2</sup> = 0 (of course, no such real number ε exists). For example, the orthogonal group O(''n'', '''R''') consists of matrices ''A'' with ''AA''<sup>T</sup> = 1, so the Lie algebra consists of the matrices ''m'' with (1 + ε''m'')(1 + ε''m'')<sup>T</sup> = 1, which is equivalent to ''m'' + ''m''<sup>T</sup> = 0 because ε<sup>2</sup> = 0.
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| *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 [[ring (mathematics)|ring]]s.
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| The concrete definition given above is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not obvious that the Lie algebra is independent of the representation we use. To get round these problems we give
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| the general definition of the Lie algebra of a Lie group (in 4 steps):
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| #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.
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| #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.
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| #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<sub>g</sub>''(''h'') = ''gh''. This shows that the space of left invariant vector fields (vector fields satisfying ''L<sub>g</sub>''<sub>*</sub>''X<sub>h</sub>'' = ''X<sub>gh</sub>'' for every ''h'' in ''G'', where ''L<sub>g</sub>''<sub>*</sub> denotes the differential of ''L<sub>g</sub>'') on a Lie group is a Lie algebra under the Lie bracket of vector fields.
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| #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''^<sub>''g''</sub> = ''L<sub>g</sub>''<sub>*</sub>''v''. This identifies the [[tangent space]] ''T<sub>e</sub>'' 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]] <math>\mathfrak{g}.</math> Thus the Lie bracket on <math>\mathfrak{g}</math> is given explicitly by [''v'', ''w''] = [''v''^, ''w''^]<sub>''e''</sub>.
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| This Lie algebra <math>\mathfrak{g}</math> 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.
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| 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.
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| For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.
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| We could also define a Lie algebra structure on ''T<sub>e</sub>'' 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<sub>e</sub>''.
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| The Lie algebra structure on ''T<sub>e</sub>'' can also be described as follows:
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| the commutator operation
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| : (''x'', ''y'') → ''xyx''<sup>−1</sup>''y''<sup>−1</sup>
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| on ''G'' × ''G'' sends (''e'', ''e'') to ''e'', so its derivative yields a [[bilinear operator|bilinear operation]] on ''T<sub>e</sub>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.
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| == Homomorphisms and isomorphisms ==
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| 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 function (topology)|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 theory|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.
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| Every homomorphism ''f'' : ''G'' → ''H'' of Lie groups induces a homomorphism between the corresponding Lie algebras <math>\mathfrak{g}</math> and <math>\mathfrak{h}</math>. The association ''G'' <math>\mapsto\mathfrak{g}</math> is a [[functor]] (mapping between categories satisfying certain axioms).
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| 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.
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| 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).
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| A ''connected'' Lie group is [[simple group|simple]], [[semisimple group|semisimple]], [[solvable group|solvable]], [[nilpotent group|nilpotent]], or [[abelian group|abelian]] if and only if its Lie algebra has the corresponding property.
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| 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 <math>\mathfrak{g}</math> over '''F''' there is a simply connected Lie group ''G'' with <math>\mathfrak{g}</math> 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.
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| ==The exponential map==
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| {{Main|Exponential map}}
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| The [[exponential map]] from the Lie algebra M(''n'', '''R''') of the [[general linear group]] GL(''n'', '''R''') to GL(''n'', '''R''') is defined by the usual power series:
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| :<math>\exp(A) = 1 + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots </math>
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| 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.
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| 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.
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| Every vector ''v'' in <math>\mathfrak{g}</math> determines a linear map from '''R''' to <math>\mathfrak{g}</math> taking 1 to ''v'', which can be thought of as a Lie algebra homomorphism. Because '''R''' is the Lie algebra of the simply connected Lie group '''R''', this induces a Lie group homomorphism ''c'' : '''R''' → ''G'' so that
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|
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| :<math>c(s + t) = c(s) c(t)\ </math>
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| 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
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| :<math>\exp(v) = c(1).\ </math>
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| This is called the ''[[exponential map]]'', and it maps the Lie algebra <math>\mathfrak{g}</math> into the Lie group ''G''. It provides a [[diffeomorphism]] between a [[neighborhood (topology)|neighborhood]] of 0 in <math>\mathfrak{g}</math> and a neighborhood of ''e'' in ''G''. This exponential map is a generalization of the exponential function for real numbers (because '''R''' is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because '''C''' is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for [[matrix (math)|matrices]] (because M(''n'', '''R''') with the regular commutator is the Lie algebra of the Lie group GL(''n'', '''R''') of all invertible matrices).
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| 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''.
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| 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 <math>\mathfrak{g}</math>, such that for ''u'', ''v'' in ''U'' we have
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| :<math> \exp(u)\,\exp(v) = \exp\left(u + v + \tfrac{1}{2}[u,v] + \tfrac{1}{12}[\,[u,v],v] - \tfrac{1}{12}[\,[u,v],u] - \cdots \right),</math>
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| 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 {{nowrap|exp(''u'') exp(''v'') {{=}}}} {{nowrap|exp(''u'' + ''v'')}}.
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| 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, exponential map is not surjective nor injective for infinite-dimensional (see below) Lie groups modelled on [[Smooth function#Differentiability classes|''C''<sup>∞</sup>]] [[Fréchet space]], even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.
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| ==Infinite dimensional Lie groups==
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| 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 on [[Banach space]]s, and in this case much of the basic theory is similar to that of finite dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite dimensional Lie groups no longer hold.
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| Some of the examples that have been studied include:
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| *The group of [[diffeomorphism]]s of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the [[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 [[Quantization (physics)|quantize]] gravity.
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| *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 group]]s, and have central extensions whose Lie algebras are (more or less) [[Kac–Moody algebra]]s.
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| *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]].
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| ==See also==
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| *[[Lie subgroup]]
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| *[[E8 (mathematics)|E<sub>8</sub>]]
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| *[[Adjoint representation of a Lie group]]
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| *[[Adjoint endomorphism]]
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| * [[Haar measure]]
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| *[[Homogeneous space]]
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| *[[List of Lie group topics]]
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| *[[List of simple Lie groups]]
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| *[[Moufang polygon]]
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| *[[Riemannian manifold]]
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| *[[Representations of Lie groups]]
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| *[[Table of Lie groups]]
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| *[[Lie algebra]]
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| *[[Symmetry in quantum mechanics]]
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| *[[Lie group action]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation|authorlink=John Frank Adams|first=John Frank|last= Adams|title=Lectures on Lie Groups|series=Chicago Lectures in Mathematics|isbn= 0-226-00527-5|year=1969|publisher=Univ. of Chicago Press|location=Chicago}}.
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| *{{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | title=Essays in the history of Lie groups and algebraic groups | url=http://books.google.com/books?isbn=0821802887 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-0288-5 | mr=1847105 | year=2001 | volume=21}}
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| * {{citation|first=Nicolas|last= Bourbaki|title=Elements of mathematics: Lie groups and Lie algebras}}. Chapters 1–3 ISBN 3-540-64242-0, Chapters 4–6 ISBN 3-540-42650-7, Chapters 7–9 ISBN 3-540-43405-4
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| * {{citation|last=Chevalley|first=Claude|title=Theory of Lie groups|isbn=0-691-04990-4|year=1946|publisher=Princeton University Press|location=Princeton}}.
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| * {{Fulton-Harris}}
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| * {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction |publisher=Springer|year=2003|isbn=0-387-40122-9}}.
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| *{{Citation | last1=Hawkins | first1=Thomas | title=Emergence of the theory of Lie groups | url=http://books.google.com/books?isbn=978-0-387-98963-1 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Sources and Studies in the History of Mathematics and Physical Sciences | isbn=978-0-387-98963-1 | mr=1771134 | year=2000}} [http://www.jstor.org/stable/2695575 Borel's review]
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| * {{citation|last=Knapp|first=Anthony W.|authorlink=Anthony Knapp|title=Lie Groups Beyond an Introduction|edition= 2nd|series=Progress in Mathematics|volume=140|publisher=Birkhäuser|place= Boston|year= 2002|isbn=0-8176-4259-5}}.
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| * {{citation|last=Rossmann|first= Wulf |title=Lie Groups: An Introduction Through Linear Groups|series= Oxford Graduate Texts in Mathematics|publisher= Oxford University Press|isbn= 978-0-19-859683-7|year=2001}}. The 2003 reprint corrects several typographical mistakes.
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| * {{citation|authorlink=J.-P. Serre|first=Jean-Pierre|last=Serre|title= Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University|series=Lecture notes in mathematics|volume= 1500|publisher=Springer|isbn= 3-540-55008-9|year=1965}}.
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| * {{citation|first=Willi-Hans|last=Steeb|title=Continuous Symmetries, Lie algebras, Differential Equations and Computer Algebra: second edition | publisher=World Scientific Publishing | year=2007|isbn=981-270-809-X}}.
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| *[http://www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf Lie Groups. Representation Theory and Symmetric Spaces] Wolfgang Ziller, Vorlesung 2010
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| [[Category:Lie groups]]
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| [[Category:Manifolds]]
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| [[Category:Symmetry]]
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