In mathematics, a Lie group 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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = −1. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ... In mathematics, an analytic function is one that is locally given by a convergent power series. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... A Superconductor demonstrating the Meissner Effect Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... Marius Sophus Lie (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician who largely created the theory of continuous symmetry, and applied it to the study of geometric structures and differential equations. ... 1870 was a common year starting on Saturday (see link for calendar). ...

While the Euclidean space Rⁿ is a real Lie group (with ordinary vector addition as the group operation), more typical examples are given by matrix Lie groups, i.e. groups of invertible matrices (under matrix multiplication). For instance, the group SO(3) of all rotations in 3-dimensional space is a matrix Lie group. For a more complete list of examples see the table of Lie groups and list of simple Lie groups. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... This article gives an overview of the various ways to multiply matrices. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... This article gives a table of some common Lie groups and their associated Lie algebras. ... In mathematics, the simple Lie groups were classified by Élie Cartan. ...

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. In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...

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. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

The Lie algebra associated to a Lie group

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. In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

Conventionally, one can regard any field X of tangent vectors on a Lie group as a partial differential operator, denoting by Xf the Lie derivative (the directional derivative) of the scalar field f in the direction of X. Then 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 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 left-invariant if XL_g=L_gX for all g in G. Other ways of expressing left-invariantness of X are T_xL_gX_x = X_gx and TL_gX = X. The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... In mathematics, the directional derivative of a multivariate differentiable function along a given vector intuitively represents the rate of change of the function in the direction of that vector. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

The set of all vector fields on an analytic manifold is a Lie algebra over F. On a Lie group G, the left-invariant vector fields form a subalgebra, the Lie algebra associated with G, usually denoted by a Gothic . This Lie algebra is finite-dimensional (it has the same dimension as the manifold G) which makes it susceptible to classification attempts. By classifying , 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. In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ...

Every element v of the tangent space T_e 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 may therefore be identified with T_e. The Lie algebra structure on T_e can also be described as follows : the commutator operation The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

(x, y) → xyx⁻¹y⁻¹

on G × G sends (e,e) to e, so its derivative yields a bilinear operation on T_eG. 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. In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

Every vector v in determines a function c : R → G whose derivative everywhere is given by the corresponding left-invariant vector field

c′(t) = TL_c(t) v

and which has the property

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 The exponential function is one of the most important functions in mathematics. ...

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). In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... For the square matrix section, see square matrix. ...

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 will in fact only be the whole group G when G is connected.

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 In mathematics, the Baker-Campbell-Hausdorff formula is the solution to for non-commuting x and y. ...

exp(u) exp(v) = exp(u + v + 1/2 [u, v] + 1/12 [[u, v], v] − 1/12 [[u, v], u] − ...)

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).

Every homomorphism f : G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and . The association G is a functor. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

The global structure of a Lie group is in general not completely determined by its Lie algebra; see the table of Lie groups 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. This article gives a table of some common Lie groups and their associated Lie algebras. ... In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

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 unique (up to 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. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...

Alternative definitions

Sometimes, real Lie groups are defined as topological manifolds with continuous group operations; this definition is equivalent to our definition given above. This is an interpretation of the content of Hilbert's fifth problem (see also Hilbert-Smith conjecture). 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. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, the Hilbert-Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M... Andrew Mattei Gleason is an American mathematician. ...

Therefore one can also take the definition to use smooth functions. This is probably the most common approach now, in textbooks. In mathematics, a smooth function is one that is infinitely differentiable, i. ...

An excellent and unusual introduction to Lie groups and their algebras through the nontrivial example of linear groups (i.e. those defined by continuous groups of finite dimensional matrices) is given by Prof. Wulf Rossmann (see below). This approach is nontrivial, especially given that one version of Ado's Theorem is that every finite dimensional Lie algebra is isomorphic to a matrix Lie algebra. Through standard construction, it can be shown that, for every finite dimensional matrix Lie algebra, there a linear group (matrix Lie group) with this algebra as its Lie algebra. In mathematics, Ados theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. ...

See also

• Adjoint representation
• List of Lie group topics
• Table of Lie groups
• List of simple Lie groups
• Representations of Lie groups
• Armand Borel
• Homogeneous space

The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ... This is a list of Lie group topics, by Wikipedia page. ... This article gives a table of some common Lie groups and their associated Lie algebras. ... In mathematics, the simple Lie groups were classified by Élie Cartan. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... Armand Borel (21 May 1923 - 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study, Princeton from 1957 to 1993. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

References

• J.-P. Serre. Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, LNM 1500, Springer. ISBN 3-540-55008-9

• J.F. Adams, Lectures on Lie Groups (Chicago Lectures in Mathematics). ISBN 0-226-00527-5

• Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford University Press ISBN 0-19-859683-9. Be sure to get hold of the 2003 reprinting or later, which corrects some unfortunate typos that would otherwise slightly mar an excellent learning adventure.