Lie group

In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse.^[1]

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. 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.

Overview

Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups 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.

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 E(3) of distance-preserving transformations of the Euclidean space R³, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. 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 acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory.

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 groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.

Definitions and examples

A real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication

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

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

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

be a smooth mapping of the product manifold into G.

First examples

The 2×2 real invertible matrices form a group under multiplication, denoted by GL(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-dimensional noncompact real Lie group. This group is disconnected; it has two connected components corresponding to the positive and negative values of the determinant.

The 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 $\varphi$ as a parameter, this group can be parametrized as 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 of SO(2, R), and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.

The orthogonal group also forms an interesting example of a Lie group.

All of the previous examples of Lie groups fall within the class of classical groups.

Related concepts

A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: SL(2, C)), and similarly, using an alternate metric completion of Q, one can define a p-adic Lie group over the p-adic numbers, a topological group in which each point has a p-adic neighborhood. 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, Gleason, Montgomery and Zippin showed that if G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group (see also Hilbert–Smith conjecture). If the underlying manifold is allowed to be infinite-dimensional (for example, a Hilbert manifold), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups.

The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups.

More examples of Lie groups

Lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.

Examples with a specific number of dimensions

The circle group S¹ consisting of angles mod 2π under addition or, alternatively, the complex numbers with absolute value 1 under multiplication. This is a one-dimensional compact connected abelian Lie group.
The 3-sphere S³ forms a Lie group by identification with the set of quaternions of unit norm, called versors. The only other spheres that admit the structure of a Lie group are the 0-sphere S⁰ (real numbers with absolute value 1) and the circle S¹ (complex numbers with absolute value 1). For example, for even n > 1, Sⁿ is not a Lie group because it does not admit a nonvanishing vector field and so a fortiori cannot be parallelizable as a differentiable manifold. Of the spheres only S⁰, S¹, S³, and S⁷ are parallelizable. The last carries the structure of a Lie quasigroup (a nonassociative group), which can be identified with the set of unit octonions.
The (3-dimensional) metaplectic group is a double cover of SL(2, R) playing an important role in the theory of modular forms. It is a connected Lie group that cannot be faithfully represented by matrices of finite size, i.e., a nonlinear group.
The Heisenberg group is a connected nilpotent Lie group of dimension 3, playing a key role in quantum mechanics.
The Lorentz group is a 6-dimensional Lie group of linear isometries of the Minkowski space.
The Poincaré group is a 10-dimensional Lie group of affine isometries of the Minkowski space.
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 bosons + 8 gluons of the standard model
The exceptional Lie groups of types G₂, F₄, E₆, E₇, E₈ 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 E_7½ of dimension 190, but it is not a simple Lie group.

Examples with `n` dimensions

Euclidean space Rⁿ with ordinary vector addition as the group operation becomes an n-dimensional noncompact abelian Lie group.
The Euclidean group E(n, R) is the Lie group of all Euclidean motions, i.e., isometric affine maps, of n-dimensional Euclidean space Rⁿ.
The orthogonal group O(n, R), consisting of all n × n 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)).
The unitary group U(n) consisting of n × n unitary matrices (with complex entries) is a compact connected Lie group of dimension n². Unitary matrices of determinant 1 form a closed connected subgroup of dimension n² − 1 denoted SU(n), the special unitary group.
Spin groups are double covers of the special orthogonal groups, used for studying fermions in quantum field theory (among other things).
The group GL(n, R) of invertible matrices (under matrix multiplication) is a Lie group of dimension n², 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.
The symplectic group Sp(2n, R) consists of all 2n × 2n matrices preserving a symplectic form on R²ⁿ. It is a connected Lie group of dimension 2n² + n.
The group of invertible upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2. (cf. Borel subgroup)
The A-series, B-series, C-series and D-series, whose elements are denoted by A_n, B_n, C_n, and D_n, are infinite families of simple Lie groups.

Constructions

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

The product of two Lie groups is a Lie group.
Any topologically closed subgroup of a Lie group is a Lie group. This is known as the Closed subgroup theorem or Cartan's theorem.
The quotient of a Lie group by a closed normal subgroup is a Lie group.
The universal cover of a connected Lie group is a Lie group. For example, the group R is the universal cover of the circle group S¹. In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying universal cover, one guarantees a group structure (compatible with its other structures).

Related notions

Some examples of groups that are not Lie groups (except in the trivial sense that any group can be viewed as a 0-dimensional Lie group, with the discrete topology), are:

Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space. These are not Lie groups as they are not finite-dimensional manifolds.
Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the p-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "p-adic Lie groups".) In general, only topological groups having similar local properties to Rⁿ for some positive integer n can be Lie groups (of course they must also have a differentiable structure).

Basic concepts

The Lie algebra associated with a Lie group

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

The Lie algebra of the vector space Rⁿ is just Rⁿ with the Lie bracket given by
[A, B] = 0.
(In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.)
The Lie algebra of the general linear group GL(n, R) of invertible matrices is the vector space M(n, R) of square matrices with the Lie bracket given by
[A, B] = AB − BA.
If G is a closed subgroup of GL(n, R) then the Lie algebra of G can be thought of informally as the matrices m of M(n, R) such that 1 + εm is in G, where ε is an infinitesimal positive number with ε² = 0 (of course, no such real number ε exists). For example, the orthogonal group O(n, R) consists of matrices A with AA^T = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)^T = 1, which is equivalent to m + m^T = 0 because ε² = 0.
Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0; but the "infinitesimal" language generalizes directly to Lie groups over general rings.^{[clarification needed]}

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 around these problems we give the general definition of the Lie algebra of a Lie group (in 4 steps):

Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [X, Y] = XY − YX, because the Lie bracket of any two derivations is a derivation.
If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L_g(h) = gh. This shows that the space of left invariant vector fields (vector fields satisfying L_g_*X_h = X_gh for every h in G, where L_g_* denotes the differential of L_g) on a Lie group is a Lie algebra under the Lie bracket of vector fields.
Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v^_g = L_g_*v. This identifies the tangent space T_eG at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur ${\mathfrak {g}}.$ Thus the Lie bracket on ${\mathfrak {g}}$ is given explicitly by [v, w] = [v^, w^]_e.

This Lie algebra ${\mathfrak {g}}$ is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.

We could also define a Lie algebra structure on T_e using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space T_e.

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

(x, y) → xyx⁻¹y⁻¹

on G × G sends (e, e) to e, so its derivative yields a bilinear operation on T_eG. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.

Homomorphisms and isomorphisms

If G and H are Lie groups, then a Lie group homomorphism f : G → H is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic.

The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. 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 its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements then $\phi _{*}$ is a map between the corresponding Lie algebras:

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

One can show that $\phi _{*}$ is actually a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.

Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism.

Ado's theorem says 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.

The global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the table of Lie groups for examples). A connected Lie group is simple, semisimple, solvable, nilpotent, or abelian if and only if its Lie algebra has the corresponding property.

If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite-dimensional Lie algebra ${\mathfrak {g}}$ over F there is a simply connected Lie group G with ${\mathfrak {g}}$ as Lie algebra, unique up to isomorphism. Moreover, every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups.

The exponential map

The exponential map from the Lie algebra M(n, R) of the general linear group GL(n, R) to GL(n, R) is defined by the usual power series:

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

for matrices A. If G is any subgroup of GL(n, R), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

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

c(s+t)=c(s)c(t)\

for all s and t. The operation on the right hand side is the group multiplication in G. The formal similarity of this formula with the one valid for the exponential function justifies the definition

\exp(v)=c(1).\

This is called the exponential map, and it maps the Lie algebra ${\mathfrak {g}}$ into the Lie group G. It provides a diffeomorphism between a neighborhood of 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 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 matrices (because M(n, R) with the regular commutator is the Lie algebra of the Lie group GL(n, 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 algebra infinitesimal generators of the group G. The subgroup of G generated by N is the identity component of G.

The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood U of the zero element of ${\mathfrak {g}}$ , such that for u, v in U we have

\exp(u)\,\exp(v)=\exp \left(u+v+{\tfrac {1}{2}}[u,v]+{\tfrac {1}{12}}[\,[u,v],v]-{\tfrac {1}{12}}[\,[u,v],u]-\cdots \right),

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

The exponential map 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 diagram commutes,^{[Note 1]}

(In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.)

The exponential map from the Lie algebra to the Lie group is not always 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 SL(2, R) is not surjective. Also, exponential map is not surjective nor injective for infinite-dimensional (see below) Lie groups modelled on C^∞ Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

Lie subgroup

A Lie subgroup H of a Lie group G is a Lie group that is a subset of G and such that the inclusion map from H to G is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of G admits a unique smooth structure which makes it an embedded Lie subgroup of G—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.

Examples of non-closed subgroups are plentiful; for example take G to be a torus of dimension ≥ 2, and let H be a one-parameter subgroup of irrational slope, i.e. one that winds around in G. Then there is a Lie group homomorphism φ : R → G with H as its image. The closure of H will be a sub-torus in G.

The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group G and the subalgebras of the Lie algebra of G.^[2] 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.

Early history

According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1), Sophus Lie himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. 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, 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.

Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. 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. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary 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 a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.

Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

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 algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights.

In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.

Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups.^[3] The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.

The concept of a Lie group, and possibilities of classification

Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are manifolds, so have tangent spaces at each point.

The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A_n, B_n, C_n and D_n, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E₈ is the largest of these.

Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.

Compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S¹ and simple compact Lie groups (which correspond to connected Dynkin diagrams).
Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Solvable groups are too messy to classify except in a few small dimensions.
Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Like solvable groups, nilpotent groups are too messy to classify except in a few small dimensions.
Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, SL(2, R) is simple according to the second definition but not according to the first. They have all been classified (for either definition).
Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras.^[4] They are central extensions of products of simple Lie groups.

The identity component of any Lie group is an open normal subgroup, and the quotient group is a 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

G_con for the connected component of the identity

G_sol for the largest connected normal solvable subgroup

G_nil for the largest connected normal nilpotent subgroup

so that we have a sequence of normal subgroups

1 ⊆ G_nil ⊆ G_sol ⊆ G_con ⊆ G.

Then

G/G_con is discrete

G_con/G_sol is a central extension of a product of simple connected Lie groups.

G_sol/G_nil is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S¹.

G_nil/1 is nilpotent, and therefore its ascending central series has all quotients abelian.

This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.

The diffeomorphism group of a Lie group acts transitively on the Lie group
Every Lie group is parallelizable, and hence an orientable manifold (there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at 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 on Banach spaces, 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.

The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie in Lie group. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.

Some of the examples that have been studied include:

The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory. Diffeomorphism groups of compact manifolds of larger dimension are regular Fréchet Lie groups; very little about their structure is known.

The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity.

The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac–Moody algebras.
There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem. In M-Theory, for example, a 10 dimensional SU(N) gauge theory becomes an 11 dimensional theory when N becomes infinite.

Notes

Explanatory notes

^ http://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf

Citations

^ Arthur Tresse (1893). "Sur les invariants différentiels des groupes continus de transformations". Acta Mathematica. 18: 1–88. doi:10.1007/bf02418270.
^ Hall 2015 Theorem 5.20
^ Borel (2001).
^ Helgason, Sigurdur (1978). Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press. p. 131. ISBN 0-12-338460-5.

References

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Robert Gilmore (2008) Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists, Cambridge University Press ISBN 9780521884006 .
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 0-387-40122-9.
F. Reese Harvey (1990) Spinors and calibrations, Academic Press, ISBN 0-12-329650-1 .
Hawkins, Thomas (2000), Emergence of the theory of Lie groups, Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98963-1, MR 1771134 Borel's review
Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2848-9, MR 1834454
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Lie Groups. Representation Theory and Symmetric Spaces Wolfgang Ziller, Vorlesung 2010

[2] ttp://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf

[1] Arthur Tresse (1893). "Sur les invariants différentiels des groupes continus de transformations". Acta Mathematica. 18: 1–88. doi:10.1007/bf02418270.

[3] Hall 2015 Theorem 5.20

[FOOTNOTEBorel2001-4] Borel (2001).

[5] Helgason, Sigurdur (1978). Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press. p. 131. ISBN 0-12-338460-5.

[1]

[Note 1]

[2]

[3]

[4]

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