Definitions

# Lie group

[lee]
In mathematics, a Lie group (sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. They are named after the nineteenth century Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.

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 manifolds and, therefore, 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, from Sophus Lie, is to replace the global object, the group, with its local or linearised 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 1950s, 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.

## Definitions and examples

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

### First examples

The 2×2 real invertible matrices form a group under multiplication, denoted by GL2(R):

$GL_2\left(mathbb\left\{R\right\}\right)=left\left\{A=begin\left\{pmatrix\right\}a&bc&dend\left\{pmatrix\right\}: det A=ad-bc ne 0right\right\}.$

This is a four-dimensional noncompact 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, denoted by SO2(R), which is a Lie group in its own right: a one-dimensional compact connected Lie group diffeomorphic to a circle. Using the rotation angle $varphi$ as a parameter, this group can be parametrized as follows:

$SO_2\left(mathbb\left\{R\right\}\right)=left\left\{begin\left\{pmatrix\right\} cosvarphi & -sin varphi sin varphi & cos varphi end\left\{pmatrix\right\}:$
varphiinmathbb{R}/2pimathbb{Z}right}.

Addition of the angles corresponds to multiplication of the elements of SO2(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 are known as the classical groups

### Related concepts

A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: SL2(C)), and similarly one can define a p-adic Lie group over the p-adic numbers. 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.

### 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.
• 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.

### Related notions

Some examples of groups that are not Lie groups 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)

## 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. 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.

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), ). 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, e.g. 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; 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" An, Bn, Cn and Dn, 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.

## Types of Lie groups and structure theory

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, SL2(R) is simple according to the second definition but not according to the first. They have all been classified (for either definition).
• Semisimple Lie groups are connected groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups.

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

Gcon for the connected component of the identity
Gsol for the largest connected normal solvable subgroup
Gnil for the largest connected normal nilpotent subgroup
so that we have a sequence of normal subgroups
1 ⊆ GnilGsolGconG
Then
G/Gcon is discrete
Gcon/Gsol is a central extension of a product of simple connected Lie groups.
Gsol/Gnil is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S1.
Gnil/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 Lie algebra associated to a Lie group

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

• The Lie algebra of the vector space Rn is just Rn with the Lie bracket given by

[AB] = 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 GLn(R) of invertible matrices is the vector space Mn(R) of square matrices with the Lie bracket given by

[AB] = AB − BA

• If G is a closed subgroup of GLn(R) then the Lie algebra of G can be thought of informally as the matrices m of Mn(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 On(R) consists of matrices A with AAT = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)T = 1, which is equivalent to m + mT = 0 because ε2 = 0.
• Formally, when working over the reals, as here, this is accomplished by considering the limit as ε→0; but the "infinitesimal" language generalizes directly to Lie groups over general rings.

The concrete definition given above is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not obvious that the Lie algebra is independent of the representation we use. To get round these problems we give the general definition of the Lie algebra of any Lie group (in 4 steps):

1. Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [XY] = XY − YX, because the Lie bracket of any two derivations is a derivation.
2. If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
3. We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations Lg(h)=gh. This shows that the space of left invariant vector fields (vector fields satisfying Lg*Xh=Xgh for every h in G, where Lg* denotes the differential of Lg) 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 element v of the tangent space at the identity is the vector field defined by v^g=Lg*v. This identifies the tangent space Te 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 $mathfrak\left\{g\right\}.$ Thus the Lie bracket on $mathfrak\left\{g\right\}$ is given explicitly by [v,w]=[v^,w^]e.

This Lie algebra $mathfrak\left\{g\right\}$ 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 Te 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 Te.

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

(x, y) → xyx−1y−1

on G × G sends (ee) to e, so its derivative yields a bilinear operation on TeG. 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 : GH is a smooth group homomorphism. (It is equivalent to require only that f be continuous rather than smooth.) The composition of two such homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a homomorphism. Isomorphic Lie groups are essentially the same; they only differ in the notation for their elements.

Every homomorphism f : GH of Lie groups induces a homomorphism between the corresponding Lie algebras $mathfrak\left\{g\right\}$ and $mathfrak\left\{h\right\}$. The association G $mapstomathfrak\left\{g\right\}$ is a functor (mapping between categories satisfying certain axioms).

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.

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 $mathfrak\left\{g\right\}$ over F there is a simply connected Lie group G with $mathfrak\left\{g\right\}$ 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 Mn(R) of the general linear group GLn(R) to GLn(R) is defined by the usual power series:

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

for matrices A. If G is any subgroup of GLn(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 $mathfrak\left\{g\right\}$ determines a linear map from R to $mathfrak\left\{g\right\}$ 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 : RG 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 $mathfrak\left\{g\right\}$ into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in $mathfrak\left\{g\right\}$ 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 Mn(R) with the regular commutator is the Lie algebra of the Lie group GLn(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 $mathfrak\left\{g\right\}$, such that for u, v in U we have

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

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

The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL2(R) is not surjective.

## Infinite dimensional Lie groups

Lie groups are finite dimensional by definition, but there are many groups that resemble Lie groups, except for being infinite dimensional. There is very little "general theory" of such groups, but some of the examples that have been studied include:

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

## References

• .
• .
• . Chapters 1-3 ISBN 3-540-64242-0, Chapters 4-6 ISBN 3-540-42650-7, Chapters 7-9 ISBN 3-540-43405-4
• .
• .
• .
• . The 2003 reprint corrects several typographical mistakes.
• .

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