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Lie Group E-8
In mathematics, a Lie group (IPA pronunciation: [li?], sounds like "Lee") is a smooth group, in the sense that the set of group elements has topology and smooth structure of a smooth manifold, and the group operations are smooth functions of the elements. For example, the 2×2 real invertible matrices,
form a multiplicative group, denoted by GL2(R), which is a classic example of a Lie group; its manifold is 4-dimensional. Further restricting to 2×2 rotation matrices gives a subgroup, denoted by SO2(R), which is also a Lie group; its manifold is 1-dimensional, a circle, with rotation angle as parameter. In this latter example we can write a group element as
and observe that the inverse for the element given by ? is that given by -?, while the product of the elements given by ? and µ is that given by ?+µ; thus both group operations are continuous, as required.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. The theory of Lie groups has become the standard mathematical way of handling continuous symmetry, i.e. symmetries that are continuous 'motions'. They are now applied not only in mathematics, but in a major way in theoretical physics (where the quark theory, for example, came out of Lie group ideas).Definitions
A (real) Lie group is a mathematical group which is also a finite-dimensional real smooth manifold, and in which the group operations of 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: ), and similarly one can define a p-adic Lie group over the p-adic numbers. An Infinite dimensional Lie group is 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 GLn(R) of invertible matrices (under matrix multiplication) is a Lie group of dimension n2, called the general linear group. It has a subgroup SLn(R) of matrices of determinant 1 which is also a Lie group, called the special linear group.
The orthogonal group On(R) is a Lie group represented by orthogonal matrices. It consists of all rotations and reflections of an n-dimensional vector space. It has a subgroup SO(n) of elements of determinant 1, called the special orthogonal group or rotation group.
The unitary group U(n) is a compact group of dimension n2 represented by unitary matrices. It has a subgroup SU(n) of elements of determinant 1, called 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 Sp2n(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 respectively. 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 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. 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".)
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 R2/Z2 (? S1×S1) under the map x?(x,v2 x). The image is a dense subset of R2/Z2 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 real numbers, 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 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.
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 (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.
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. WriteGcon 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 subgroups1 ? Gnil ? Gsol ? Gcon ? G
ThenG/Gcon is discrete
Gcon/Gsol is a central extension of a product of simple connected Lie groups.
Gsol/Gnil is abelian (and a product of copies of R and 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.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 "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 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 GLn(R) of invertible matrices is the vector space Mn(R) of square matrices with the Lie bracket given by
[A, B] = 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 + em is in G, where e is an infinitesimal positive number with e2 = 0 (of course no such real number e 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 + em)(1 + em)T = 1, which is equivalent to m + mT = 0 because e2 = 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 [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. 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 Te 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 .
This Lie algebra 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 (e, e) 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 : G ? H 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 : G ? H of Lie groups induces a homomorphism between the corresponding Lie algebras and . The association 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 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 over F there is a simply connected Lie group G with 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 group GLn(R) to GLn(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 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 determines a linear map from R to 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 thatc(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 definitionexp(v) = c(1)
This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in 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 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 , such that for u, v in U we haveexp(u) exp(v) = exp(u + v + 1/2 [u, v] + 1/12
