Lie group

In mathematics, a Lie group (IPA pronunciation: , sounds like "Lee") is a group that is also a smooth manifold; for example, the rotation group of rotations of 3 dimensional space. Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of structures. They were introduced by Sophus Lie in 1870 in order to study symmetries of differential equations.

Definition of a Lie group

A (real) Lie group is a smooth manifold (real and finite dimensional) that is also a group such that the group operations multiplication and inversion are smooth maps.

There are several closely 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 p-adic Lie groups over the p-adic numbers. Infinite dimensional Lie groups are defined in the same way except that one allows the underlying manifold to be infinite dimensional. 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.

It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups. One could also try varying the definition by using topological or analytic manifolds instead of smooth ones, but it turns out that this gives nothing new: Gleason, Montgomery and Zippin showed in the 1950s 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 Hilbert's fifth problem and Hilbert-Smith conjecture).

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.

Examples of Lie groups

Here are a few examples of Lie groups and their relations to other areas of mathematics and physics.

Euclidean space Rⁿ is an abelian Lie group (with ordinary vector addition as the group operation).
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 subgroup SL_n(R) of matrices of determinant 1 which is also a Lie group, called the special linear group.
The group O_n(R) generated by all rotations and reflections of an n-dimensional vector space is a Lie group called the orthogonal group.
Spin groups are double covers of the special orthogonal groups, used for studying fermions in quantum field theory (among other things).
The group Sp_2n(R) of all matrices preserving a symplectic form is a Lie group called the symplectic group.
The spheres S⁰, S¹, and S³ can be made into Lie groups by identifying them with the real numbers, complex numbers, or quaternions of absolute value 1. No other spheres are Lie groups. The Lie group S¹ is sometimes called the circle group.
The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2.
The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
The Heisenberg group is a Lie group of dimension 3, used in quantum mechanics.
The unitary group U(n) is a compact group of dimension n² consisting of unitary matrices. It has a subgroup SU(n) of elements of determinant 1, called the special unitary group.
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, whose dimension corresponds to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
The metaplectic group is a 3 dimensional Lie group that is a double cover of SL₂(R) and is used in the theory of modular forms. It cannot be represented as finite matrices.
The exceptional Lie groups of types G₂, F₄, E₆, E₇, E₈ have dimensions 14, 52, 78, 133, and 248. There is even a group E7½ of dimension 190.

For many more examples see the table of Lie groups and list of simple Lie groups and article on matrix groups.

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

The product of two Lie groups is a Lie group.
Any closed 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.

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. There 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".)
The image of a connected Lie group under a homomorphism of Lie groups need not be a Lie group. The usual example of this is the image of R in the group S¹×S¹ under the map x→(x,√2x). The image is a dense subset of S¹×S¹ that is not a manifold, and so is not a Lie group. This also gives an example where a subalgebra of a Lie algebra does not correspond to a Lie subgroup of the corresponding Lie group.
The group of rational numbers under addition, topologized as a subset of the reals, is not a Lie group as it is not a manifold.

Types of Lie groups

One classifies Lie groups regarding their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.

The identity component of any Lie group is a connected normal closed subgroup, and the quotient 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.

Compact Lie groups are all known: they are finite central extensions of a product of copies of the circle group S¹ and simple compact Lie groups, and simple compact Lie groups 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 aretoo 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.

Connected Abelian Lie groups are all isomorphic to products of copies of R and the circle group S¹.

Structure of a Lie group

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 (and a product of copies of R and 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.

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 groups that are "infinitely close" to the identity, and the Lie bracket is something to do with the commutator of two such "infinitely small" 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

^* = 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

^* = 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.

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 does not depend on which representation we use. To get round these problems we give the general definition of the Lie algebra of any 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 ^* = 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. This shows that the space of left invariant vector fields 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. This identifies the tangent space T_e at the identity with the space of left invariant vector fields, and therefore makes the tangent space into a Lie algebra, called the Lie algebra of G, usually denoted by a lower case g or a Gothic $\mathfrak{g}$ .

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 (both real or both complex), then a Lie-group-homomorphism f : G → H is a group homomorphism which is also an analytic map. (One can show that it is equivalent to require only that f be continuous.) The composition of two such homomorphisms is again a homomorphism, and the class of all (real or complex) Lie groups, together with these morphisms, forms a category. The two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a homomorphism. Isomorphic Lie groups do not need to be distinguished for all practical purposes; they only differ in the notation of their elements.

Every homomorphism f : G → H of Lie groups induces a homomorphism between the corresponding Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ . The association G $\mapsto\mathfrak{g}$ is a functor.

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 in general not completely 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). We can say however that 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 unique (up to isomorphism) simply connected Lie group G with $\mathfrak{g}$ as Lie algebra. 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 group GL_n(R) to GL_n(R) is defined by the usual power series:

exp(A) = 1 + A + A²/2! + A³/3! + ...

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. Since 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 (since R is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (since C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (since 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(u + v + 1/2 v + 1/12 u," target="_blank" >v, v" target="_blank" >^* − 1/12 u," target="_blank" >v, u" target="_blank" >^* − ...)

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 solvable). 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 group is called a gauge group, 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.

References

J.-P. Serre. Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, LNM 1500, Springer. ISBN 3-540-55008-9

Anthony W. Knapp, Lie Groups Beyond an Introduction, Second Edition. Birkhäuser, 2002.

J.F. Adams, Lectures on Lie Groups (Chicago Lectures in Mathematics). ISBN 0-226-00527-5

Representation Theory : A First Course (Graduate Texts in Mathematics / Readings in Mathematics) by William Fulton, Joe Harris Publisher: Springer; 1 edition (July 30, 1999) ISBN 0387974954

Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford University Press ISBN 0-19-859683-9. Be sure to get hold of the 2003 reprinting or later, which corrects some unfortunate typos. An introduction to Lie groups and their algebras through the nontrivial example of linear groups (i.e. those defined by continuous groups of finite dimensional matrices).

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