Lie group

Not to be confused withGroup of Lie typeorRee group.

Inmathematics,aLie group(pronounced/liː/LEE) is agroupthat is also adifferentiable manifold,such that group multiplication and taking inverses are both differentiable.

Amanifoldis a space that locally resemblesEuclidean space,whereas groups define the abstract concept of abinary operationalong with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains acontinuous groupwhere multiplying points and their inverses are continuous. If the multiplication and taking of inverses aresmooth(differentiable) as well, one obtains a Lie group.

Lie groups provide a natural model for the concept ofcontinuous symmetry,a celebrated example of which is thecircle group.Rotating a circle is an example of a continuous symmetry. For any rotation of the circle, there exists the same symmetry,and concatenation of such rotations makes them into the circle group, an archetypal example of a Lie group. Lie groups are widely used in many parts of modern mathematics andphysics.

Lie groups were first found by studyingmatrixsubgroups$G$contained in${\text{GL}}_{n}(\mathbb {R} )$or${\text{GL}}_{n}(\mathbb {C} )$,thegroups of$n\times n$invertible matricesover$\mathbb {R}$or$\mathbb {C}$.These are now called theclassical groups,as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematicianSophus Lie(1842–1899), who laid the foundations of the theory of continuoustransformation groups.Lie's original motivation for introducing Lie groups was to model the continuous symmetries ofdifferential equations,in much the same way that finite groups are used inGalois theoryto model the discrete symmetries ofalgebraic equations.

History

Sophus Lieconsidered the winter of 1873–1874 as the birth date of his theory of continuous groups.Thomas 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.Some of Lie's early ideas were developed in close collaboration withFelix 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.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.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-volumeTheorie der Transformationsgruppen,published in 1888, 1890, and 1893. The termgroupes de Liefirst 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 ofCarl Gustav Jacobi,on the theory ofpartial differential equationsof first order and on the equations ofclassical mechanics.Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany.Lie'sidée fixewas to develop a theory of symmetries of differential equations that would accomplish for them whatÉvariste Galoishad done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations forspecial functionsandorthogonal polynomialstend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory ofcontinuous groups,to complement the theory ofdiscrete groupsthat had developed in the theory ofmodular forms,in the hands ofFelix KleinandHenri Poincaré.The initial application that Lie had in mind was to the theory ofdifferential equations.On the model ofGalois theoryandpolynomial equations,the driving conception was of a theory capable of unifying, by the study ofsymmetry,the whole area ofordinary differential equations.However, the hope that Lie theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is adifferential Galois theory,but it was developed by others, such as Picard and Vessiot, and it provides a theory ofquadratures,theindefinite integralsrequired to express solutions.

Additional impetus to consider continuous groups came from ideas ofBernhard 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 agroup;
• Geometric theory and the explicit solutions ofdifferential equationsof mechanics, worked out byPoissonand Jacobi;
• The new understanding ofgeometrythat emerged in the works ofPlücker,Möbius,Grassmannand 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 byWilhelm Killing,who in 1888 published the first paper in a series entitledDie Zusammensetzung der stetigen endlichen Transformationsgruppen(The composition of continuous finite transformation groups).The work of Killing, later refined and generalized byÉlie Cartan,led to classification ofsemisimple Lie algebras,Cartan's theory ofsymmetric spaces,andHermann Weyl's description ofrepresentationsof compact and semisimple Lie groups usinghighest weights.

In 1900David Hilbertchallenged Lie theorists with hisFifth Problempresented at theInternational Congress of Mathematiciansin 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'sinfinitesimal 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 byClaude Chevalley.

Overview

The set of allcomplex numberswithabsolute value1 (corresponding to points on thecircleof center 0 and radius 1 in thecomplex plane) is a Lie group under complex multiplication: thecircle group.

Lie groups aresmoothdifferentiable manifoldsand as such can be studied usingdifferential calculus,in contrast with the case of more generaltopological groups.One of the key ideas in the theory of Lie groups is to replace theglobalobject, the group, with itslocalor linearized version, which Lie himself called its "infinitesimal group" and which has since become known as itsLie algebra.

Lie groups play an enormous role in moderngeometry,on several different levels.Felix Kleinargued in hisErlangen programthat one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric propertiesinvariant.ThusEuclidean geometrycorresponds to the choice of the groupE(3)of distance-preserving transformations of the Euclidean space$\mathbb {R} ^{3}$,conformal geometrycorresponds to enlarging the group to theconformal group,whereas inprojective geometryone is interested in the properties invariant under theprojective group.This idea later led to the notion of aG-structure,whereGis 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, therepresentationsof the Lie group (or of itsLie algebra) are especially important. Representation theoryis used extensively in particle physics.Groups whose representations are of particular importance includethe rotation group SO(3)(or itsdouble cover SU(2)),the special unitary group SU(3)and thePoincaré group.

On a "global" level, whenever a Lie groupactson a geometric object, such as aRiemannianor asymplectic manifold,this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via aLie group actionon a manifold places strong constraints on its geometry and facilitatesanalysison the manifold. Linear actions of Lie groups are especially important, and are studied inrepresentation theory.

In the 1940s–1950s,Ellis Kolchin,Armand Borel,andClaude Chevalleyrealised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory ofalgebraic groupsdefined over an arbitraryfield.This insight opened new possibilities in pure algebra, by providing a uniform construction for mostfinite simple groups,as well as inalgebraic geometry.The theory ofautomorphic forms,an important branch of modernnumber theory,deals extensively with analogues of Lie groups overadele rings;p-adicLie groups play an important role, via their connections with Galois representations in number theory.

Definitions and examples

Areal Lie groupis agroupthat is also a finite-dimensional realsmooth manifold,in which the group operations ofmultiplicationand inversion aresmooth maps.Smoothness of the group multiplication

$\mu:G\times G\to G\quad \mu (x,y)=xy$

means thatμis a smooth mapping of theproduct manifoldG×GintoG.The two requirements can be combined to the single requirement that the mapping

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

be a smooth mapping of the product manifold intoG.

First examples

• The 2×2realinvertible matricesform a group under multiplication, denoted byGL(2,R)or by GL₂(R):

$\operatorname {GL} (2,\mathbf {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\,\det A=ad-bc\neq 0\right\}.$

This is a four-dimensionalnoncompactreal Lie group; it is an open subset of$\mathbb {R} ^{4}$.This group isdisconnected;it has two connected components corresponding to the positive and negative values of thedeterminant.

• Therotationmatrices form asubgroupofGL(2,R),denoted bySO(2,R).It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which isdiffeomorphicto thecircle.Using the rotation angle$\varphi$as a parameter, this group can beparametrizedas follows:

$\operatorname {SO} (2,\mathbf {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\,\varphi \in \mathbf {R} /2\pi \mathbf {Z} \right\}.$

Addition of the angles corresponds to multiplication of the elements ofSO(2,R),and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.

• Theaffine group of one dimensionis a two-dimensional matrix Lie group, consisting of$2\times 2$real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form

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

Non-example

We now present an example of a group with anuncountablenumber of elements that is not a Lie group under a certain topology. The group given by

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

with$a\in \mathbb {R} \setminus \mathbb {Q}$afixedirrational number,is a subgroup of thetorus$\mathbb {T} ^{2}$that is not a Lie group when given thesubspace topology.If we take any smallneighborhood$U$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 adensesubgroup of$\mathbb {T} ^{2}$.

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

The group$H$can, however, be given a different topology, in which the distance between two points$h_{1},h_{2}\in H$is defined as the length of the shortest pathin 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$\theta$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$\operatorname {GL} (n,\mathbb {C} )$denote the group of$n\times n$invertible matrices with entries in$\mathbb {C}$.Anyclosed subgroupof$\operatorname {GL} (n,\mathbb {C} )$is a Lie group;Lie groups of this sort are calledmatrix 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.

• Thespecial linear groupsover$\mathbb {R}$and$\mathbb {C}$,$\operatorname {SL} (n,\mathbb {R} )$and$\operatorname {SL} (n,\mathbb {C} )$,consisting of$n\times n$matrices with determinant one and entries in$\mathbb {R}$or$\mathbb {C}$
• Theunitary groupsand special unitary groups,${\text{U}}(n)$and${\text{SU}}(n)$,consisting of$n\times n$complex matrices satisfying$U^{*}=U^{-1}$(and also$\det(U)=1$in the case of${\text{SU}}(n)$)
• Theorthogonal groupsand special orthogonal groups,${\text{O}}(n)$and${\text{SO}}(n)$,consisting of$n\times n$real matrices satisfying$R^{\mathrm {T} }=R^{-1}$(and also$\det(R)=1$in the case of${\text{SO}}(n)$)

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

Related concepts

Acomplex Lie groupis defined in the same way usingcomplex manifoldsrather than real ones (example:$\operatorname {SL} (2,\mathbb {C} )$), and holomorphic maps. Similarly, using an alternatemetric completionof$\mathbb {Q}$,one can define ap-adic Lie groupover thep-adic numbers,a topological group which is also an analyticp-adic manifold, such that the group operations are analytic. In particular, each point has ap-adic neighborhood.

Hilbert's fifth problemasked 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,MontgomeryandZippinshowed that ifGis a topological manifold with continuous group operations, then there exists exactly one analytic structure onGwhich turns it into a Lie group (see alsoHilbert–Smith conjecture). If the underlying manifold is allowed to be infinite-dimensional (for example, aHilbert manifold), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of manyLie groups over finite fields,and these give most of the examples offinite simple groups.

The language ofcategory theoryprovides a concise definition for Lie groups: a Lie group is agroup objectin thecategoryof smooth manifolds. This is important, because it allows generalization of the notion of a Lie group toLie supergroups.This categorical point of view leads also to a different generalization of Lie groups, namelyLie groupoids,which aregroupoid objectsin the category of smooth manifolds with a further requirement.

Topological definition

A Lie group can be defined as a (Hausdorff)topological groupthat, near the identity element, looks like a transformation group, with no reference to differentiable manifolds.First, we define animmersely linear Lie groupto be a subgroupGof the general linear group$\operatorname {GL} (n,\mathbb {C} )$such that

1. for some neighborhoodVof the identity elementeinG,the topology onVis the subspace topology of$\operatorname {GL} (n,\mathbb {C} )$andVis closed in$\operatorname {GL} (n,\mathbb {C} )$.
2. Ghas at mostcountably manyconnected components.

(For example, a closed subgroup of$\operatorname {GL} (n,\mathbb {C} )$;that is, a matrix Lie group satisfies the above conditions.)

Then aLie groupis defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:

1. Given a Lie groupGin the usual manifold sense, theLie group–Lie algebra correspondence(or a version ofLie's third theorem) constructs an immersed Lie subgroup$G'\subset \operatorname {GL} (n,\mathbb {C} )$such that$G,G'$share the same Lie algebra; thus, they are locally isomorphic. Hence,Gsatisfies the above topological definition.
2. Conversely, letGbe a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group$G'$that is locally isomorphic toG.Then, by a version of theclosed subgroup theorem,$G'$is areal-analytic manifoldand then, through the local isomorphism,Gacquires a structure of a manifold near the identity element. One then shows that the group law onGcan be given by formalpower series;so the group operations are real-analytic andGitself 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 grouptogether with the group law determines the geometry of the group.

More examples of Lie groups

See also:Table of Lie groupsandList of simple Lie groups

Lie groups occur in abundance throughout mathematics and physics.Matrix groupsoralgebraic groupsare (roughly) groups of matrices (for example,orthogonalandsymplectic 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$\mathbb {R}$(with the group operation being addition) and thecircle 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$1\times 1$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$\mathbb {R} ^{2}$(with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples".

Additional examples

• Thegroup SU(2)is the group of$2\times 2$unitary matrices with determinant$1$.Topologically,${\text{SU}}(2)$is the$3$-sphere$S^{3}$;as a group, it may be identified with the group ofunit quaternions.
• TheHeisenberg groupis a connectednilpotentLie group of dimension$3$,playing a key role inquantum mechanics.
• TheLorentz groupis a 6-dimensional Lie group of linearisometriesof theMinkowski space.
• ThePoincaré groupis a 10-dimensional Lie group ofaffineisometries of the Minkowski space.
• Theexceptional Lie groupsof typesG₂,F₄,E₆,E₇,E₈have dimensions 14, 52, 78, 133, and 248. Along with the A-B-C-D series ofsimple Lie groups,the exceptional groups complete the list of simple Lie groups.
• Thesymplectic group${\text{Sp}}(2n,\mathbb {R} )$consists of all$2n\times 2n$matrices preserving asymplectic formon$\mathbb {R} ^{2n}$.It is a connected Lie group of dimension$2n^{2}+n$.

Constructions

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

• The product of two Lie groups is a Lie group.
• Anytopologically closedsubgroup of a Lie group is a Lie group. This is known as theClosed subgroup theoremorCartan's theorem.
• The quotient of a Lie group by a closed normal subgroup is a Lie group.
• Theuniversal coverof a connected Lie group is a Lie group. For example, the group$\mathbb {R}$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 specifyinguniversalcover, one guarantees a group structure (compatible with its other structures).

Related notions

Some examples of groups that arenotLie 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 thediscrete 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$G$,$C^{\infty }(X,G)$.These are not Lie groups as they are notfinite-dimensionalmanifolds.
• Sometotally disconnected groups,such as theGalois groupof an infinite extension of fields, or the additive group of thep-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 similarlocal propertiestoRⁿfor some positive integerncan be Lie groups (of course they must also have a differentiable structure).

Basic concepts

The Lie algebra associated with a Lie group

Main article:Lie group–Lie algebra correspondence

To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimallyclose "to the identity, and the Lie bracket of the Lie algebra is related to thecommutatorof two such infinitesimal elements. Before giving the abstract definition we give a few examples:

• The Lie algebra of the vector spaceRⁿis justRⁿwith 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 thegeneral linear groupGL(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.
• IfGis a closed subgroup of GL(n,C) then the Lie algebra ofGcan be thought of informally as the matricesmof M(n,C) such that 1 + εmis inG,where ε is an infinitesimal positive number with ε²= 0 (of course, no such real number ε exists). For example, the orthogonal group O(n,R) consists of matricesAwithAA^T= 1, so the Lie algebra consists of the matricesmwith (1 + εm)(1 + εm)^T= 1, which is equivalent tom+m^T= 0 because ε²= 0.
• The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroupGof GL(n,C), may be computed as

$\operatorname {Lie} (G)=\{X\in M(n;\mathbb {C} )|\operatorname {exp} (tX)\in G{\text{ for all }}t{\text{ in }}\mathbb {\mathbb {R} } \},$where exp(tX) is defined using thematrix exponential.It can then be shown that the Lie algebra ofGis 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):

1. Vector fields on any smooth manifoldMcan be thought of asderivationsXof the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [X,Y] =XY−YX,because theLie bracketof any two derivations is a derivation.
2. IfGis any group acting smoothly on the manifoldM,then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
3. We apply this construction to the case when the manifoldMis the underlying space of a Lie groupG,withGacting onG=Mby left translationsL_g(h) =gh.This shows that the space of left invariant vector fields (vector fields satisfyingL_g*X_h=X_ghfor everyhinG,whereL_g*denotes the differential ofL_g) on a Lie group is a Lie algebra under the Lie bracket of vector fields.
4. Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an elementvof the tangent space at the identity is the vector field defined byv^_g=L_g*v.This identifies thetangent spaceT_eGat 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 ofG,usually denoted by aFraktur${\mathfrak {g}}.$Thus the Lie bracket on${\mathfrak {g}}$is given explicitly by [v,w] = [v^,w^]_e.

This Lie algebra${\mathfrak {g}}$is finite-dimensional and it has the same dimension as the manifoldG.The Lie algebra ofGdeterminesGup to "local isomorphism", where two Lie groups are calledlocally isomorphicif 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 onT_eusing right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map onGcan be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent spaceT_e.

The Lie algebra structure onT_ecan also be described as follows: the commutator operation

(x,y) →xyx⁻¹y⁻¹

onG×Gsends (e,e) toe,so its derivative yields abilinear operationonT_eG.This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of aLie bracket,and it is equal to twice the one defined through left-invariant vector fields.

Homomorphisms and isomorphisms

IfGandHare Lie groups, then a Lie group homomorphismf:G→His a smoothgroup homomorphism.In the case of complex Lie groups, such a homomorphism is required to be aholomorphic map.However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real)analytic.

The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms acategory.Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let$\phi \colon G\to H$be a Lie group homomorphism and let$\phi _{*}$be itsderivativeat the identity. If we identify the Lie algebras ofGandHwith theirtangent spacesat the identity elements, then$\phi _{*}$is a map between the corresponding Lie algebras:

$\phi _{*}\colon {\mathfrak {g}}\to {\mathfrak {h}},$

which turns out to be aLie algebra homomorphism(meaning that it is alinear mapwhich preserves theLie bracket). In the language ofcategory theory,we then have a covariantfunctorfrom 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 calledisomorphicif there exists abijectivehomomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is adiffeomorphismwhich 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 theorem,which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to useAdo'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, theglobalstructure of a Lie group is not determined by its Lie algebra; for example, ifZis any discrete subgroup of the center ofGthenGandG/Zhave the same Lie algebra (see thetable of Lie groupsfor examples). An example of importance in physics are the groupsSU(2)andSO(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 besimply connected,then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic.(See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.

Simply connected Lie groups

See also:Lie group–Lie algebra correspondenceandFundamental group § Lie groups

A Lie group$G$is said to besimply connectedif 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${\mathfrak {g}}$and${\mathfrak {h}}$and that$f:{\mathfrak {g}}\rightarrow {\mathfrak {h}}$is a Lie algebra homomorphism. If$G$is simply connected, then there is a unique Lie group homomorphism$\phi:G\rightarrow H$such that$\phi _{*}=f$,where$\phi _{*}$is the differential of$\phi$at the identity.

Lie's third theoremsays 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 auniquesimply connected Lie group.

An example of a simply connected group is the special unitary groupSU(2),which as a manifold is the 3-sphere. Therotation group SO(3),on the other hand, is not simply connected. (SeeTopology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction betweeninteger spinandhalf-integer spinin quantum mechanics. Other examples of simply connected Lie groups include the special unitary groupSU(n),the spin group (double cover of rotation group)Spin(n)for$n\geq 3$,and the compact symplectic groupSp(n).

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

The exponential map

Main article:Exponential map (Lie theory)

See also:derivative of the exponential mapandnormal coordinates

Theexponential mapfrom the Lie algebra$\mathrm {M} (n;\mathbb {C} )$of thegeneral linear group$\mathrm {GL} (n;\mathbb {C} )$to$\mathrm {GL} (n;\mathbb {C} )$is defined by thematrix exponential,given by the usual power series:

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

for matrices$X$.If$G$is a closed subgroup of$\mathrm {GL} (n;\mathbb {C} )$,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${\mathfrak {g}}$of$G$(i.e., the tangent space to$G$at the identity), one proves that there is a unique one-parameter subgroup$c:\mathbb {R} \rightarrow G$such that$c'(0)=X$.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)\$

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 theexponential functionjustifies the definition

$\exp(X)=c(1).\$

This is called theexponential map,and it maps the Lie algebra${\mathfrak {g}}$into the Lie group$G$.It provides adiffeomorphismbetween aneighborhoodof 0 in${\mathfrak {g}}$and a neighborhood of$e$in$G$.This exponential map is a generalization of the exponential function for real numbers (because$\mathbb {R}$is the Lie algebra of the Lie group ofpositive real numberswith multiplication), for complex numbers (because$\mathbb {C}$is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and formatrices(because$M(n,\mathbb {R} )$with the regular commutator is the Lie algebra of the Lie group$\mathrm {GL} (n,\mathbb {R} )$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 algebrainfinitesimal generatorsof the group$G$.The subgroup of$G$generated by$N$is the identity component of$G$.

The exponential map and the Lie algebra determine thelocal group structureof every connected Lie group, because of theBaker–Campbell–Hausdorff formula:there exists a neighborhood$U$of the zero element of${\mathfrak {g}}$,such that for$X,Y\in U$we have

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

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$\exp(X)\exp(Y)=\exp(X+Y)$

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

$\phi (\exp(x))=\exp(\phi _{*}(x)).\,$

In other words, the following diagramcommutes,

(In short, exp is anatural transformationfrom 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 alwaysonto,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 ofSL(2,R)is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled onC^∞Fréchet space,even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

Lie subgroup

ALie subgroup$H$of a Lie group$G$is a Lie group that is asubsetof$G$and such that theinclusion mapfrom$H$to$G$is aninjectiveimmersionandgroup homomorphism.According toCartan's theorem,a closedsubgroupof$G$admits a unique smooth structure which makes it anembeddedLie 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 aone-parameter subgroupofirrational slope,i.e. one that winds around inG.Then there is a Lie grouphomomorphism$\varphi:\mathbb {R} \to G$with$\mathrm {im} (\varphi )=H$.Theclosureof$H$will be a sub-torus in$G$.

Theexponential mapgives aone-to-one correspondencebetween 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

Main article:Representation of a Lie group

See also:Compact group § Representation theory of a connected compact Lie group,andLie algebra representation

One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independentSchrödinger equationin quantum mechanics,${\hat {H}}\psi =E\psi$.Assume the system in question has therotation group SO(3)as a symmetry, meaning that the Hamiltonian operator${\hat {H}}$commutes with the action of SO(3) on the wave function$\psi$.(One important example of such a system is theHydrogen atom,which has a single spherical orbital.) This assumption does not necessarily mean that the solutions$\psi$are rotationally invariant functions. Rather, it means that thespaceof solutions to${\hat {H}}\psi =E\psi$is invariant under rotations (for each fixed value of$E$). This space, therefore, constitutes a representation of SO(3). These representations have beenclassifiedand the classification leads to a substantialsimplification 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 groupK(including the just-mentioned case of SO(3)) is particularly tractable.In that case, every finite-dimensional representation ofKdecomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified byHermann Weyl.The classificationis in terms of the "highest weight" of the representation. The classification is closely related to theclassification 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 therepresentations of the group SL(2,R)and therepresentations 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 (Liehimself called them "infinitesimal groups" ). It can be defined because Lie groups are smooth manifolds, so havetangent spacesat 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 adirect sumof anabelian Lie algebraand some number ofsimpleones. 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 thesimple Lie algebrasof compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A_n,B_n,C_nand 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. E₈is the largest of these.

Lie groups are classified according to their algebraic properties (simple,semisimple,solvable,nilpotent,abelian), theirconnectedness(connectedorsimply connected) and theircompactness.

A first key result is theLevi decomposition,which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup.

• Connectedcompact Lie groupsare all known: they are finite central quotients of a product of copies of the circle groupS¹and simple compact Lie groups (which correspond to connectedDynkin 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 groupsare sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example,SL(2,R)is simple according to the second definition but not according to the first. They have all beenclassified(for either definition).
• SemisimpleLie groups are Lie groups whose Lie algebra is a product of simple Lie algebras.They are central extensions of products of simple Lie groups.

Theidentity componentof any Lie group is an opennormal subgroup,and thequotient groupis adiscrete 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 groupGcan be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write

G_confor the connected component of the identity

G_solfor the largest connected normal solvable subgroup

G_nilfor 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_conis discrete

G_con/G_solis acentral extensionof a product ofsimple connected Lie groups.

G_sol/G_nilis abelian. A connectedabelian Lie groupis isomorphic to a product of copies ofRand thecircle groupS¹.

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.

• Thediffeomorphism groupof a Lie group acts transitively on the Lie group
• Every Lie group isparallelizable,and hence anorientable manifold(there is abundle isomorphismbetween itstangent bundleand the product of itself with thetangent spaceat the identity)

Infinite-dimensional Lie groups

Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally onBanach spaces(as opposed toEuclidean spacein 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 notBanach manifolds.Instead one needs to define Lie groups modeled on more generallocally convextopological 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 prefixLieinLie group.On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefixLieinLie algebraare 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 ofdiffeomorphismsof a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) theWitt algebra,whosecentral extensiontheVirasoro algebra(seeVirasoro algebra from Witt algebrafor a derivation of this fact) is the symmetry algebra oftwo-dimensional conformal field theory.Diffeomorphism groups of compact manifolds of larger dimension areregular Fréchet Lie groups;very little about their structure is known.
• The diffeomorphism group of spacetime sometimes appears in attempts toquantizegravity.
• The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of agauge group(with operation ofpointwise multiplication), and is used inquantum field theoryandDonaldson theory.If the manifold is a circle these are calledloop 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 havesimplertopological properties: see for exampleKuiper's theorem.InM-theory,for example, a 10-dimensional SU(N) gauge theory becomes an 11-dimensional theory whenNbecomes 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 mechanics
• Lie point symmetry,about the application of Lie groups to the study of differential equations.

References

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External links

• Media related toLie groupsat Wikimedia Commons
• Journal of Lie Theory