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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 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 Map s. There are several closely related concepts. A complex Lie group is defined in the same way using . Infinite dimensional Lie groups are defined in the same way except that one allows the underlying manifold to be infinite dimensional. ''' Matrix Group s''' or ''' Algebraic Group s''' are (roughly) groups of matrices, (for example, Orthogonal and Symplectic Group s) and these give most of the more common examples of Lie groups. It is possible to define analogues of many , 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 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.
For many more examples see the Table Of Lie Groups and List Of Simple Lie Groups and article on Matrix Group s. There are several standard ways to form new Lie groups from old ones:
Some examples of groups that are not Lie groups are:
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 .
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 G G so that we have a sequence of normal subgroups :1 ⊆ ''G''nil ⊆ ''G''sol ⊆ ''G''con ⊆ ''G'' Then G G G G 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:
:: {Link without Title} = 0. (In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.)
:: {Link without Title} = ''AB'' − ''BA''
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 {Link without Title} = ''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''−1''y''−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 (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 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 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 isomorphism) simply connected Lie group ''G'' with 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'' + ''A''2/2! + ''A''3/3! + ... 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 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 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 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 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 , such that for ''u'', ''v'' in ''U'' we have :exp(''u'') exp(''v'') = exp(''u'' + ''v'' + 1/2 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 connnected 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:
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