Lie group Totally Explained

Everything about Lie Group totally explained

In mathematics, a Lie group (sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. They are named after the nineteenth century Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (Differential Galois theory), much in the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations, his idée fixe.

Overview

   Since Lie groups are manifolds, they can be studied using differential calculus, in contrast with the case of more general topological groups. 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 linearised version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra.
   Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R³, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory.
   In the 1950s, Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings.

Examples of Lie groups

Lie groups occur in abundance throughout mathematics and physics.

Examples

Euclidean space Rⁿ with ordinary vector addition as the group operation becomes an n-dimensional noncompact abelian Lie group.
The circle group S¹ consisting of angles mod 2π under addition or complex numbers with absolute value 1 under multiplication is a one-dimensional compact connected abelian Lie group.
The group GL_n(R) of invertible matrices (under matrix multiplication) is a Lie group of dimension n², called the general linear group. It has a closed connected subgroup SL_n(R), the special linear group, consisting of matrices of determinant 1 which is also a Lie group.
The orthogonal group O_n(R), consisting of all n × n orthogonal matrices with real entries is an n(n − 1)/2-dimensional Lie group. This group is disconnected, but it has a connected subgroup SO_n(R) of the same dimension consisting of orthogonal matrices of determinant 1, called the special orthogonal group (for n = 3, the rotation group).
The Euclidean group E_n(R) is the Lie group of all Euclidean motions, for example isometric affine maps, of n-dimensional Euclidean space Rⁿ.
The unitary group U(n) consisting of n × n unitary matrices (with complex entries) is a compact connected Lie group of dimension n². Unitary matrices of determinant 1 form a closed connected subgroup of dimension n² − 1 denoted SU(n), the special unitary group.
Spin groups are double covers of the special orthogonal groups, used for studying fermions in quantum field theory (among other things).
The symplectic group Sp_2n(R) consists of all 2n × 2n matrices preserving a nondegenerate skew-symmetric bilinear form on R²ⁿ (the symplectic form). It is a connected Lie group of dimension 2n² + n. The fundamental group of the symplectic group is Z and this fact is related to the theory of Maslov index.
The 3-sphere S³ may be identified with the set of quaternions of unit norm, which form a Lie group. The only other spheres that admit the structure of a Lie group are the 0-sphere S⁰ (real numbers with absolute value 1) and the circle S¹ (complex numbers with absolute value 1).
The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2.
The Lorentz group and the Poincare group are the groups of linear and affine isometries of the Minkowski space (interpreted as the spacetime of the special relativity). They are Lie groups of dimensions 6 and 10.
The Heisenberg group is a connected nilpotent Lie group of dimension 3, playing a key role in quantum mechanics.
The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that's the gauge group of the Standard Model in particle physics. The dimensions of the factors correspond to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
The (3-dimensional) metaplectic group is a double cover of SL₂(R) playing an important role in the theory of modular forms. It is a connected Lie group that can't be faithfully represented by matrices of finite size, for example a nonlinear group.
The exceptional Lie groups of types G₂, F₄, E₆, E₇, E₈ have dimensions 14, 52, 78, 133, and 248. There is also a group E_7½ of dimension 190.

Constructions

There are several standard ways to form new Lie groups from old ones:

The product of two Lie groups is a Lie group.

Any closed subgroup of a Lie group is a Lie group.

The quotient of a Lie group by a closed normal subgroup is a Lie group.

The universal cover of a connected Lie group is a Lie group. For example, the group R is the universal cover of the circle group S¹.

Related notions

Some examples of groups that are not Lie groups are:

Infinite dimensional groups, such as the additive group of an infinite dimensional real vector space. These are not Lie groups as they're not finite dimensional manifolds.

Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the p-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "p-adic Lie groups".)

The image of a connected Lie group under a homomorphism of Lie groups need not be a Lie group. The usual example of this is the image of R in the group R²/Z² (≅ S¹×S¹) under the map x→(x,√2 x). The image is a dense subset of R²/Z² that isn't a manifold, and so isn't a Lie group. This also gives an example where a subalgebra of a Lie algebra doesn't correspond to a Lie subgroup of the corresponding Lie group.

The group of rational numbers under addition, topologized as a subset of the real numbers, isn't a Lie group as it isn't a manifold.

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. Hawkins, however, 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 Felix Klein. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years (ibid, p.2). Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe (ibid, p.76). In 1884 a young German mathematician, Friedrich Engel, 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 didn't 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 Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany (Hawkins, p.43). Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group; geometric theory and the explicit solutions of differential equations 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 Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights.
   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 (for example 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 compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting; the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras 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 don't fall into any of these families. E₈ is the largest of these.

Example

For example, the 2×2 real invertible matrices, �

egin

, such that for u, v in U we have � 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).
The exponential map from the Lie algebra to the Lie group isn't 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) isn't 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:

The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory. Very little is known about the diffeomorphism groups of manifolds of larger dimension. The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity.

The group of smooth maps from a manifold to a finite dimensional group is called a gauge group, and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac-Moody algebras.

There are infinite dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem.

Just as calculus in finite-dimensional real vector spaces can be extended to calculus in Banach spaces, the definition of finite-dimensional smooth manifolds can be extended to give a definition of Banach analytic manifolds. Similarly, the standard finite-dimensional definition of Lie groups can be extended to give a definition of Banach analytic Lie groups. In this case, we've a Banach analytic manifold which simultaneously has a group structure such that multiplication and inversion are analytic maps. Some of the theorems of finite-dimensional Lie groups don't carry over to the Banach analytic case, and in particular the relation between Lie groups and Lie algebras is much more subtle in the infinite dimensional case. However, it's true that "for infinite dimensional Lie groups modeled on Banach spaces there's a well-developed theory ... which is closely parallel to the theory of finite dimensional Lie groups."Further Information

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