In mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures. They were introduced by Sophus Lie in 1870 in order to study symmetries of differential equations.
While the Euclidean space Rn is a real Lie group (with ordinary vector addition as the group operation), more typical examples are groups of invertible matrices (under matrix multiplication), for instance the group SO(3) of all rotations in 3-dimensional space. See below for a more complete list of examples.
One classifies Lie groups regarding their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.
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 that f only 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.
To every Lie group, we can associate a Lie algebra which completely captures the local structure of the group, at least if the Lie group is connected. This is done as follows.
A vector field on a Lie group G is said to be left-invariant if it commutes with
left translation, which means the following. Define Lg[f](x)= f(gx) for any analytic function f : G -> F and all g, x in G (here F stands for the field R or C). Then the vector field X
is left-invariant if X Lg = Lg X for all g in G.
The set of all vector fields on an analytic manifold is a Lie algebra over F. On a Lie group, the left-invariant vector fields form a subalgebra, the Lie algebra associated with G, usually denoted by a gothic g. This Lie algebra g is finite-dimensional (it has the same dimension as the manifold G) which makes it susceptible to classification attempts. By classifying g, one can also get a handle on the Lie group G. The representation theory of simple Lie groups is the best and most important example.
Every element v of the tangent space Te at the identity element e of G determines a unique left-invariant vector field whose value at the element x of G will be denoted by xv; the vector space underlying g may therefore be identified with Te. The Lie algebra structure on Te can also be described as follows :
the commutator operation
Every vector v in g determines a function c : R -> G whose derivative everywhere is given by the corresponding left-invariant vector field
Types of Lie groups
Homomorphisms and isomorphisms
The Lie algebra associated to a Lie group
on G × G
sends (e,e) to e, so its derivative yields a bilinear operation on Te. It turns out that this bilinear operation satisfies the axioms of a Lie bracket, and it is equal to the one defined through left-invariant vectorfields.
and which has the property
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
This is called the exponential map, and it maps the Lie algebra g into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in 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).
The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Campbell-Hausdorff formula: there exists a neighborhood U of the zero element of g, such that for u, v in U we have
Every homomorphism f : G -> H of Lie groups induces a homomorphism between the corresponding Lie algebras g and h. The association G |-> g is a functor.
The global structure of a Lie group is in general not completely determined by its Lie algebra; see the table below for examples of different Lie groups sharing the same Lie algebra. 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 g over F there is a unique (up to isomorphism) simply connected Lie group G with g as Lie algebra. Moreover every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups.
List of real Lie groups and their Lie algebras
| Lie group | Description | Remarks | Lie algebra | Description | dim/R |
|---|---|---|---|---|---|
| Rn | Euclidean space with addition | abelian, simply connected, not compact | Rn | the Lie bracket is zero | n |
| R× | nonzero real numbers with multiplication | abelian, not connected, not compact | R | the Lie bracket is zero | 1 |
| R>0 | positive real numbers with multiplication | abelian, simply connected, not compact | R | the Lie bracket is zero | 1 |
| S1 = R/Z | complex numbers of absolute value 1, with multiplication | abelian, connected, not simply connected, compact | R | the Lie bracket is zero | 1 |
| H× | non-zero quaternions with multiplication | simply connected, not compact | H | quaternions, with Lie bracket the commutator | 4 |
| S3 | quaternions of absolute value 1, with multiplication | simply connected, compact, simple and semi-simple, isomorphic to SU(2) and to Spin(3) | R3 | real 3-vectors, with Lie bracket the cross product; isomorphic to the quaternions with zero real part, with Lie bracket the commutator; also isomorphic to su(2) and to so(3) | 3 |
| GL(n,R) | general linear group: invertible n-by-n real matrices | not connected, not compact | M(n,R) | n-by-n matrices, with Lie bracket the commutator | n2 |
| GL+(n,R) | n-by-n real matrices with positive determinant | simply connected, not compact | M(n,R) | n-by-n matrices, with Lie bracket the commutator | n2 |
| SL(n,R) | special linear group: real matrices with determinant 1 | simply connected, not compact if n>1 | sl(n,R) | square matrices with trace 0, with Lie bracket the commutator | n2-1 |
| O(n,R) | orthogonal group: real orthogonal matrices | not connected, compact | so(n,R) | skew-symmetric square real matrices, with Lie bracket the commutator; so(3,R) is isomorphic to su(2) and to R3 with the cross product | n(n-1)/2 |
| SO(n,R) | special orthogonal group: real orthogonal matrices with determinant 1 | connected, compact, for n≥ 2: not simply connected, for n=3 and n≥5: simple and semisimple | so(n,R) | skew-symmetric square real matrices, with Lie bracket the commutator | n(n-1)/2 |
| Spin(n) | spinor group | simply connected, compact, for n=3 and n≥5: simple and semisimple | so(n,R) | skew-symmetric square real matrices, with Lie bracket the commutator | n(n-1)/2 |
| U(n) | unitary group: complex unitary n-by-n matrices | isomorphic to S1 for n=1; simply connected and compact for n>1. Note: this is not a complex Lie group/algebra | u(n) | square complex matrices A satisfying A = -A*, with Lie bracket the commutator | n2 |
| SU(n) | special unitary group: complex unitary n-by-n matrices with determinant 1 | simply connected, compact, for n≥2: simple and semisimple. Note: this is not a complex Lie group/algebra | su(n) | square complex matrices A with trace 0 satisfying A = -A*, with Lie bracket the commutator | n2-1 |
The dimensions given are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
| Lie group | Description | Remarks | Lie algebra | Description | dim/C |
|---|---|---|---|---|---|
| Cn | group operation is addition | abelian, simply connected, not compact | Cn | the Lie bracket is zero | n |
| C× | nonzero complex numbers with multiplication | abelian, simply connected, not compact | C | the Lie bracket is zero | 1 |
| GL(n,C) | general linear group: invertible n-by-n complex matrices | simply connected, not compact, for n=1: isomorphic to C× | M(n,C) | n-by-n matrices, with Lie bracket the commutator | n2 |
| SL(n,C) | special linear group: complex matrices with determinant 1 | simple, semisimple, simply connected, for n≥2: not compact | sl(n,C) | square matrices with trace 0, with Lie bracket the commutator | n2-1 |
| O(n,C) | orthogonal group: complex orthogonal matrices | not connected, for n≥2: not compact | so(n,C) | skew-symmetric square complex matrices, with Lie bracket the commutator | n(n-1)/2 |
| SO(n,C) | special orthogonal group: complex orthogonal matrices with determinant 1 | simply connected, for n≥2: not compact, for n=3 and n≥5: simple and semisimple | so(n,C) | skew-symmetric square complex matrices, with Lie bracket the commutator | n(n-1)/2 |
Sometimes, real Lie groups are defined as topological manifolds with continuous group operations; this definition is equivalent to our definition given above. This is the content of Hilbert's fifth problem. The precise statement, proven by Gleason, Montgomery and Zippin in the 1950s, is as follows: If G is a topological manifold with continuous group operations, then there exists exactly one differentiable structure on G which turns it into a Lie group in our sense.
Representations of Lie groups/algebras
See Representations of Lie groups/algebras.