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Since Lie groups are manifolds, they can be studied using Differential Calculus , in contrast with the case of more general Topological Group s. One of the key ideas in the theory of Lie groups, due to Sophus Lie, is to replace the global object, the ''group'', with its ''local'' or linearized version, which Lie himself called an ''infinitesimal group'' and which has since become known as its Lie Algebra . Lie groups provide a natural framework to analyse continuous symmetries of Differential Equations ( Picard-Vessiot Theory ), much in the same way as Permutation Groups are used in Galois Theory to analyse discrete symmetries of Algebraic Equations . EARLY HISTORY According to the most authoritative source on the early history of Lie groups (Hawkins, p.1), Sophus Lie himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. However, Hawkins suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation (''ibid''). Some of Lie's early ideas were developed in close collaboration with , came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume ''Theorie der Transformationsgruppen'', published in 1888, 1890, and 1893. Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of ; geometric theory and the explicit solutions of Differential Equation s of mechanics, worked out by Poisson and Jacobi; and the new understanding of Geometry that emerged in the works of Plücker , Möbius , Grassmann and others, and culminated in Riemann's revolutionary vision of the subject. Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing , who in 1888 published the first paper in a series entitled ''Die Zusammensetzung der stetigen endlichen Transformationsgruppen'' (''The composition of continuous finite transformation groups'') (Hawkins, p.100). The work of Killing, later refined and generalized by Elie Cartan , led to classification of Semisimple Lie Algebra s, Cartan's theory of Symmetric Spaces , and Hermann Weyl 's description of Representations of compact and semisimple Lie groups using Highest Weight s. Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's ''infinitesimal groups'' (i.e. Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups (Borel (2001), ). The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley . THE CONCEPT OF A LIE GROUP, AND POSSIBILITIES OF CLASSIFICATION Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, here rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie Algebra ( Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are manifolds, so have tangent spaces at each point. The Lie algebra of any of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A''n'', B''n'', C''n'' and D''n'', which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these. EXAMPLE 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 λ is that given by −λ, while the product of the elements given by λ and μ is that given by λ+μ; thus both group operations are continuous, as required. 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 Map s. There are several closely related concepts. A complex Lie group is defined in the same way using . An '''Infinite dimensional Lie group''' is 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 1952 that if 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. This is important, since it allows generalization of the notion of a Lie group to Lie Supergroups . 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 Lie groups are classified according to 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 group that are " Infinitesimal ly 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:
:: {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 Fraktur . 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, 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 M''n''('''R''') of the group GL''n''('''R''') to GL''n''('''R''') is defined by the usual power series: : 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 connected groups that are either compact or nilpotent). 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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