While the Euclidean space R^{n} is a real Lie group, more typical examples are groups of invertible matrices, for instance the group SO(3) of all rotations in 3dimensional 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 Liegrouphomomorphism 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 leftinvariant if it commutes with left translation, which means the following. Define L_{g}[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 leftinvariant if X L_{g} = L_{g} 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 leftinvariant vector fields form a subalgebra, the Lie algebra associated with G, usually denoted by a gothic g. This Lie algebra g is finitedimensional (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 T_{e} at the identity element e of G determines a unique leftinvariant 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 T_{e}. The Lie algebra structure on T_{e} 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 leftinvariant vector field
The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the CampbellHausdorff 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.
Lie group  Description  Remarks  Lie algebra  Description  dim/R 

R^{n}  Euclidean space with addition  abelian, simply connected, not compact  R^{n}  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 
S^{1} = R/Z  complex numbers of absolute value 1, with multiplication  abelian, connected, not simply connected, compact  R  the Lie bracket is zero  1 
H^{×}  nonzero quaternions with multiplication  simply connected, not compact  H  quaternions, with Lie bracket the commutator  4 
S^{3}  quaternions of absolute value 1, with multiplication  simply connected, compact, simple and semisimple, isomorphic to SU(2) and to Spin(3)  R^{3}  real 3vectors, 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 nbyn real matrices  not connected, not compact  M(n,R)  nbyn matrices, with Lie bracket the commutator  n^{2} 
GL^{+}(n,R)  nbyn real matrices with positive determinant  simply connected, not compact  M(n,R)  nbyn matrices, with Lie bracket the commutator  n^{2} 
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  n^{2}1 
O(n,R)  orthogonal group: real orthogonal matrices  not connected, compact  so(n,R)  skewsymmetric square real matrices, with Lie bracket the commutator; so(3,R) is isomorphic to su(2) and to R^{3} with the cross product  n(n1)/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)  skewsymmetric square real matrices, with Lie bracket the commutator  n(n1)/2 
Spin(n)  spinor group[?]  simply connected, compact, for n=3 and n≥5: simple and semisimple  so(n,R)  skewsymmetric square real matrices, with Lie bracket the commutator  n(n1)/2 
U(n)  unitary group: complex unitary nbyn matrices  isomorphic to S^{1} 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  n^{2} 
SU(n)  special unitary group: complex unitary nbyn 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  n^{2}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 

C^{n}  group operation is addition  abelian, simply connected, not compact  C^{n}  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 nbyn complex matrices  simply connected, not compact, for n=1: isomorphic to C^{×}  M(n,C)  nbyn matrices, with Lie bracket the commutator  n^{2} 
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  n^{2}1 
O(n,C)  orthogonal group: complex orthogonal matrices  not connected, for n≥2: not compact  so(n,C)  skewsymmetric square complex matrices, with Lie bracket the commutator  n(n1)/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)  skewsymmetric square complex matrices, with Lie bracket the commutator  n(n1)/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.
Search Encyclopedia

Featured Article

