Lie group

A Lie group ( Lie group also ), named after Sophus Lie, is a mathematical structure that is used to describe continuous symmetries. Lie groups are in almost all parts of today's mathematics and in theoretical physics, especially particle physics, important tools.

Formally, for a Lie group is a group that can be construed as differentiable manifold such that the group link and inverses are compatible with this smooth structure.

Lie groups and Lie algebras were introduced in 1870 by Sophus Lie in the Lie theory for the study of symmetries in differential equations. Regardless of Lie Wilhelm Killing developed similar ideas to study non- Euclidean geometries. The older group names steady or continuous group of a Lie group better describe what is today understood a topological group. Every Lie group is a topological group.

The first examples

The set of complex numbers is equal to 0 with the usual multiplication is a group. Multiplication is a differentiable map defined by, the inversion is defined by differentiable. The group structure of the complex plane (with respect to multiplication ) is therefore "with the differential calculus compatible". ( The same would also apply to the group with the addition as a shortcut: is There and. )

The unit circle in the complex plane, that is, the set of complex numbers of magnitude 1, is a subset of, called the circle group: The product of two numbers on the amount one has again Amount 1, as is the inverse. Again, you have a " differential calculus compatible with the group structure ", that is, a Lie group.

On the other hand, the set of rotation matrix ( the rotation ) a group is defined by the multiplication and by the inversion.

If we identify the set of matrices in an obvious way with that, then is a differentiable submanifold and you can check that multiplication and inversion are differentiable, so is a Lie group.

It turns out that it is in and around the "same" Lie group, ie that the two Lie groups are isomorphic. Namely, one can define a mapping by mapping the complex number, which is on the unit circle. This is a group homomorphism, since

One can check that this group homomorphism and its inverse mapping are differentiable, so is a Lie group isomorphism, from the viewpoint of Lie groups theory are the same group, the group of rotation matrices and the unit circle.

An important motivation of the Lie group theory is that one can define a Lie algebra of Lie groups and traced back many group theory or differential-geometric problems on the corresponding problem in the Lie algebra and can be solved there. ( " Linear Algebra is easier than group theory .") For the definition of the Lie algebra one needs the differentiability and the compatibility of the group operations with this.

For the Lie algebra is the imaginary axis with the trivial Lie bracket. ( Triviality of the Lie bracket in this case stems ultimately from the fact that an abelian Lie group. ) The Lie algebra is the trivial Lie bracket and it is easily seen that these two Lie algebras are isomorphic. ( General correspond to isomorphic Lie groups always isomorphic Lie algebras. )


Lie group

A Lie group is a smooth real manifold, which also has the structure of a group so that the group join and the inversion are infinitely differentiable. The dimension of the Lie group is the dimension of the underlying manifold. If this is finite, then the unsuccessful manifold is automatically analytically and the group multiplication and inversion are analytic functions.

A complex Lie group is a complex manifold with a group structure, so that the group link and the inversion are complex differentiable.

Lie algebra of the Lie group

The vector fields on a smooth manifold form an infinite-dimensional Lie algebra with the Lie bracket. The belonging to a Lie group Lie algebra consists of on the subspace of left - invariant vector fields. This vector space is isomorphic to the tangent space at the neutral element of. In particular we. With respect to the Lie bracket of the vector space is complete. Thus, the tangent space is a Lie group at the identity element a Lie algebra. This Lie algebra is called the Lie algebra of the Lie group.

There is an exponential map for each Lie group with Lie algebra. This exponential can be defined by where the flow of the left - invariant vector field and the neutral element is. If a closed subgroup of the or, the so -defined exponential map is identical to the matrix exponential.

Each dot on defines a left- invariant Riemannian metric. In the special case that this metric is additionally rechtsinvariant, agrees the exponential map of the Riemannian manifold at the point with the Lie group exponential map.

The correlation between the multiplication in the Lie group and the Lie bracket in its Lie algebra, the Baker -Campbell - Hausdorff formula ago:

Lie groups homomorphism

A homomorphism of Lie groups is a group homomorphism, which is also a smooth map. It can be shown that this is already the case, then, if it is continuous. If and are finite, even analytically.

For every Lie group homomorphism one gets by differentiation in the neutral element a Lie algebra homomorphism. It is

For everyone. If and are simply connected, corresponding to each Lie algebra homomorphism clearly a Lie group homomorphism.

An isomorphism of Lie groups is a bijective Lie groups homomorphism.

Lie subgroup

Let be a Lie group. A Lie subgroup is a subgroup of, together with a topology and a smooth structure that makes this sub-group back to a Lie group.

Lie subgroups are therefore generally embedded submanifolds, but always ized submanifolds. However, if an embedded topological group having the structure of an embedded submanifolds, then is also a Lie group.


For completed subsets can be defined as the Lie algebra, and this is equivalent to the above definition. Herein, the Matrixexponential. In this case the exponential with Matrixexponential match.

Not every Lie group is isomorphic to a subgroup of a general linear group. An example of this is the universal superposition of SL (2, R).

Early History

According to authoritative sources about the early history of Lie groups Sophus Lie regarded themselves as the winter 1873-1874 birth to his theory of continuous groups. Hawkins, however, suggests that it was "Read amazing research activity during the four year period from autumn 1869 to autumn 1873 ," which led to the creation of the theory. Many early by Lies ideas were developed in close collaboration with Felix Klein. Lie looked small from October 1869 to 1872 daily: in Berlin by the end of October 1869 to the end of February 1870 and in Paris, Göttingen and Erlangen in the next two years. Lie indicates that all main results had been achieved in 1884. However, during the 1870s all of his papers have been published (except for the very first release) in Norwegian journals, which prevented a perception in the rest of Europe. In 1884, the young German mathematician Friedrich Engel worked with Lie to a systematic treatise on the theory of continuous groups. For these efforts, the three -volume work theory of transformation groups emerged whose volumes were published in the years 1888, 1890, and 1893.

Hilbert's fifth problem asked whether any locally Euclidean topological group is a Lie group. ( " Locally Euclidean " means that the group is to be a manifold. There are topological groups that are not manifolds, for example, the Cantor group or solenoids. ) The problem was solved only in 1952 by Gleason, Montgomery and Zippin, with a positive response. The proof is closely related to the structure theory of locally compact groups, which form a wide generalization of Lie groups.

Read ideas were not isolated from the rest of mathematics. In fact, his interest in the geometry of differential equations was initially motivated by the work of Carl Gustav Jacobi on the theory of partial differential equations of first order and the equations of classical mechanics. Much of the work of Jacobi was published posthumously in the 1860s, which created an enormous interest in France and Germany. Read idée fixe was to develop a theory of symmetry of differential equations, which should accomplish this for what Évariste Galois had reached for algebraic equations: namely, to classify them with the help of group theory. Extra drive to the consideration of continuous groups created by Bernhard Riemann's ideas on the foundations of geometry and their development by Klein ( see also Erlanger program).

Thus, three main topics of the mathematics of the 19th century were united by Lie in the creation of his new theory:

  • The idea of ​​symmetry, as explained by Galois ' idea of ​​a group,
  • The geometrical theory and explicit solution of the differential equations of the mechanics of how it was worked out by Poisson and Jacobi and
  • The new understanding of the geometry, which had been created by the work Plücker, Möbius, Grassmann and others, and which reached its climax in Riemann's revolutionary vision of this article.

Even if Sophus Lie is now considered legally as the creator of the theory of continuous groups, was a major step forward in the development of the corresponding structure theory, which had a profound influence on the subsequent development of mathematics, provided by Wilhelm Killing, in 1888 the first article a series titled published the composition of steady finite transformation groups.

The work Killings, which was later refined by Élie Cartan, led to the classification of semisimple Lie algebras, Cartan's theory of symmetric spaces and Hermann Weyl's description of the representations of compact semisimple Lie groups by weights.

Weyl brought the early period in the development of the theory of Lie groups to maturity by classified not only the irreducible representations of semisimple Lie groups and brought the theory of groups and quantum mechanics newly created in conjunction, but by also Lies theory a solid foundation thereby gave that he ( the current Lie algebras ) and the real Lie groups differed clearly between infinitesimal Lie groups and began investigating the topology of Lie groups. The theory of Lie groups has been systematically worked out in contemporary mathematical language in a monograph by Claude Chevalley.

Differential geometry of Lie groups

Let be a compact Lie group with Killing form and adjoint representation. Then defines an invariant scalar product on the Lie algebra and thus a bi -invariant Riemannian metric. For this metric the following formulas, the differential geometry variables ( calculation of commutators ) are using linear algebra allow to determine:

In particular, the sectional curvature bi- invariant metrics on compact Lie groups always non-negative.

Classification options

Every Lie group is a topological group. Thus, a Lie group also has a topological structure and can be classified according to topological attributes: Lie groups, for example, connected, simply connected or be compact.

One can classify Lie groups even after their algebraic, group theoretical properties. Lie groups can be simple, semisimple, solvable, nilpotent or abelian. It should be noted that certain properties in the theory of Lie groups are defined differently than usual in group theory usual: So a coherent Lie group is called simple or semisimple if its Lie algebra is simple or semi- simple. A simple Lie group G is then the group-theoretic sense is not necessarily easy. But it is true:

If G is a simple Lie group with center Z, then the factor group G / Z is also easy in the group-theoretic sense.

The properties of nilpotent and solvable is usually defined via the corresponding Lie algebra.

Simple half complex Lie algebras are classified by their Dynkin diagrams. Because every Lie algebra is the Lie algebra of a unique simply connected Lie group, one gets from a classification of simply connected semisimple complex Lie groups (and therefore a classification of universal superpositions of Komplexifierungen any semisimple real Lie groups ).