Lie group

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

Formally, a Lie group is a group that can be understood as a differentiable manifold , so that the group connection and inverse formation are compatible with this smooth structure.

Lie groups and Lie algebras were introduced around 1870 by Sophus Lie in the Lie theory for the investigation of symmetries in differential equations . Independently of Lie, Wilhelm Killing developed similar ideas for studying non-Euclidean geometries . The older names continuous group or continuous group for a Lie group better describe what is now understood as a topological group . Each Lie group is also a topological group.

This article deals with finite-dimensional Lie groups (following common terminology). There is also a theory of infinite-dimensional Lie groups, such as Banach-Lie groups .

First examples

The circle with center 0 and radius 1 in the complex number plane is a Lie group with complex multiplication.

The set of complex numbers not equal to 0 forms a group with the usual multiplication . The multiplication is a differentiable mapping defined by ; also the inversion defined by it is differentiable. The group structure of the complex level (with regard to multiplication) is therefore "compatible with differential calculus". (The same would apply to the group with addition as a link: there is and .)

The unit circle in the complex number plane, i.e. H. the set of complex numbers of magnitude 1 is a subgroup of , the so-called circle group : The product of two numbers with magnitude 1 again has magnitude 1, as does the inverse. Here, too, one has a “group structure compatible with differential calculus”; H. a Lie group.

On the other hand, the crowd forms

of the rotary dies (rotations in ) a group; the multiplication is defined by

and the inversion through


If one identifies the set of matrices with the in an obvious way , then there is a differentiable submanifold and one can check that multiplication and inversion are differentiable, so it is a Lie group.

It turns out that it is in and are "the same" Lie group, d. that is, the two Lie groups are isomorphic . You can define a mapping by mapping to the complex number that lies on the unit circle. This is a group homomorphism because

It can be verified that this group homomorphism and its inverse mapping are differentiable. is thus a Lie group isomorphism. From the point of view of Lie group theory, the group of rotary matrices and the unit circle are the same group.

An important motivation of Lie group theory is that one can define a Lie algebra for Lie groups and that many group theoretical or differential geometric problems can be traced back to the corresponding problem in Lie algebra and solved there. (“ Linear algebra is easier than group theory ”.) To define Lie algebra, one needs the differentiability and the compatibility of the group operations with it.

For them , the Lie algebra is the imaginary axis with the trivial Lie bracket . The triviality of the Lie bracket in this case comes from the fact that it is an Abelian Lie group. The Lie algebra that is

with the trivial Lie bracket and it is easy to see that these two Lie algebras are isomorphic. (In general, isomorphic Lie groups always correspond to isomorphic Lie algebras.)


Lie group

A Lie group is a smooth, real manifold that also has the structure of a group , so that the group connection and the inversion can be differentiated as often as desired . The dimension of the Lie group is the dimension of the underlying manifold. One can show that the underlying manifold of a Lie group even has a real-analytic structure and the group multiplication and inversion are automatically (real) analytic functions .

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

Lie algebra of the Lie group

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

For every Lie group with Lie algebra there is an exponential map . This exponential mapping can be defined by , where is the flow of the left-invariant vector field and is the neutral element. If a closed subgroup is or , the exponential mapping defined in this way is identical to the matrix exponential function .

Each scalar product auf defines a left-invariant Riemannian metric auf . In the special case that this metric is additionally rechtsinvariant that true exponential of the Riemann manifold at point -group Lie exponential map with the match.

The connection between the multiplication in the Lie group and the Lie bracket in its Lie algebra is established by the Baker-Campbell-Hausdorff formula :

Lie group homomorphism

A homomorphism of Lie groups is a group homomorphism that is also a smooth map . One can show that this is already the case when is continuous and that it must then even be analytical .

For every Lie group homomorphism , a Lie algebra homomorphism is obtained by differentiation in the neutral element . It applies

for everyone . If and are simply connected , then every Lie algebra homomorphism clearly corresponds to a Lie group homomorphism.

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

Lie subgroup

Be a Lie group. A Lie subgroup is a subgroup of along with a topology and smooth structure that makes that subgroup a Lie group again.

Thus, Lie subsets are generally not embedded submanifolds , but only injectively immersed submanifolds . However, if there is an embedded topological subgroup with the structure of an embedded submanifold, then it is also a Lie group.


  1. Typical examples are the general linear group , i.e. the group of invertible matrices with matrix multiplication as a link, as well as their closed subgroups, for example the circle group or the group SO (3) of all rotations in three-dimensional space. Further examples of subsets of the general linear group are the
    • Orthogonal group and the special orthogonal group , see the treatment as a Lie group
    • General complex-linear group that is isomorphic to the closed subgroup with
    • Unitary group
    • Special unitary group
    • Special linear group or
  2. Euclidean group
  3. Poincaré group
  4. Galileo group
  5. The Euclidean space with vector addition as a group operation is a somewhat trivial real Lie group ( as a -dimensional manifold im ).

For closed subgroups , the Lie algebra can be defined as and this is equivalent to the above definition. Here the matrix exponential . In this case the exponential map corresponds to the matrix exponential.

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 the authoritative sources on the early history of the Lie groups, Sophus Lie himself regarded the winter of 1873–1874 as the date of birth of his theory of continuous groups . Hawkins suggests, however, that it was "Read's amazing research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the creation of that theory. Many of Lie's early ideas were developed in close collaboration with Felix Klein . Lie saw Klein daily from October 1869 to 1872: in Berlin from late October 1869 to late February 1870 and in Paris, Göttingen and Erlangen for the following two years. Lie states that all of the main results were obtained in 1884. However, during the 1870s, all of his treatises (except for the very first communication) were published in Norwegian journals, which prevented it from being recognized in the rest of Europe. In 1884 the young German mathematician Friedrich Engel worked with Lie on a systematic treatise on his theory of continuous groups. These efforts resulted in the three-volume work Theory of Transformation Groups, the volumes of which were published in 1888, 1890, and 1893.

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

Read's ideas weren't isolated from the rest of the math. Indeed, 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 the first order and the equations of classical mechanics . Much of Jacobi's work was published posthumously in the 1860s, generating tremendous interest in France and Germany. Lies idée fixe was to develop a theory of the symmetry of differential equations that would accomplish for them what Évariste Galois had achieved for algebraic equations: namely, to classify them with the help of group theory. Additional impetus to consider constant groups came from ideas by Bernhard Riemann on the fundamentals of geometry and their development by Klein (see also Erlangen program ).

Thus, three main themes of 19th century mathematics were brought together by Lie in the creation of his new theory:

  • the idea of ​​symmetry as explained by Galois' idea of ​​a group ,
  • the geometric theory and explicit solution of the differential equations of mechanics as worked out by Poisson and Jacobi and
  • the new understanding of geometry , which had arisen through the work of Plücker , Möbius , Graßmann and others and which culminated in Riemann's revolutionary vision of this subject.

Even if Sophus Lie is rightly regarded today as the creator of the theory of continuous groups, a great step forward in the development of the corresponding structural theory, which had a profound influence on the subsequent development of mathematics, was made by Wilhelm Killing , who wrote the first article in 1888 published a series entitled The Composition of Continuous Finite Transformation Groups.

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

Weyl brought the early period in the development of Lie group theory to maturity by not only classifying the irreducible representations of semi-simple Lie groups and relating the group theory to the newly emerged quantum mechanics, but by incorporating Lie's theory as well gave a more solid foundation by clearly distinguishing between Lie's infinitesimal groups (today's Lie algebras) and the actual Lie groups and began to study the topology of the Lie groups. The theory of Lie groups was systematically elaborated 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 . The following formulas apply to this metric, which allow differential geometric quantities to be determined using linear algebra (calculation of commutators in ):

  • Levi-Civita connection :
  • Section curvature : for orthonormal
  • Ricci curvature : for an orthonormal basis with
  • Scalar curvature : for an orthonormal basis.

In particular, the cutting curvature of bi-invariant metrics on compact Lie groups is always nonnegative.

Classification options

Each 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 can, for example, be connected, singly connected or compact.

Lie groups can also be classified according to their algebraic , group-theoretical properties. Lie groups can be simple , semi-simple , resolvable, nilpotent, or abelian . It should be noted that certain properties are defined differently in the theory of Lie groups than is otherwise usual in group theory: A connected Lie group is called simple or semi-simple if its Lie algebra is simple or semi-simple . A simple Lie group G is then not necessarily simple in the group-theoretical sense. But the following applies:

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

The properties nilpotent and solvable are also usually defined using the corresponding Lie algebra .

Semi-simple complex Lie algebras are classified using their Dynkin diagrams . Because every Lie algebra is the Lie algebra of a unique, simply connected Lie group, you get a classification of the simply connected semi-simple complex Lie groups (and thus a classification of the universal superpositions of complexifications of any semi-simple real Lie groups).

Generalizations (and related theories)

The theory of the (finite-dimensional, real or complex) Lie groups presented here can be generalized in various ways:

  • If, instead of finite-dimensional manifolds , one allows infinite-dimensional manifolds that are modeled over a Hilbert space , a Banach space , a Fréchet space or a locally convex space , then one obtains, depending on the number of Hilbert-Lie groups, Banach-Lie groups, Frechet- Lie groups or locally convex Lie groups. The theory of Hilbert-Lie groups and Banach-Lie groups are still comparatively similar to the finite-dimensional theory, but for more general spaces the matter becomes much more complicated, since the differential calculus in such spaces becomes more complicated. In particular, there are several non-equivalent theories for such differential calculations. Every infinite-dimensional Lie group has an (also infinite-dimensional) Lie algebra.
  • If one allows other topological fields instead of real and complex numbers, one obtains e.g. B. -adic Lie groups. Here, too, it is possible to assign a Lie algebra to each such Lie group; this is of course also defined using another basic field.
  • If one closes the class of (finite-dimensional, real) Lie groups with regard to projective limits , one obtains the class of Pro-Lie groups , which in particular contains all connected locally compact groups . Each such group also has a Lie algebra, which arises as a projective limit of finite-dimensional Lie algebras.
  • No generalization, but a similar concept is obtained if one does not consider smooth manifolds, but rather algebraic varieties with a compatible group structure. This leads to the theory of algebraic groups , which has much in common with the theory of Lie groups. In particular, every algebraic group also has an associated Lie algebra. The finite groups of the Lie type also belong to this category.


  1. ^ First by his doctoral student Arthur Tresse in his dissertation 1893, Acta Mathematica
  2. Roughly speaking, a Lie group is a group that forms a continuum or a continuously connected whole. A simple example of a Lie group is the totality of all rotations of a plane around a fixed point that lies in this plane: All these rotations together form a group, but also a continuum in the sense that each of these rotations clearly passes through an angle between 0 ° and 360 ° degrees or a radian measure between 0 and 2 π can be described and in the sense that rotations that differ from one another only by small angles can be continuously converted into one another. A circle that lies in the plane under consideration and that has the fixed point as its center point can then be described as symmetrical from the point of view of this Lie group, since it remains unchanged under every rotation. On the other hand, a rectangle whose center corresponds to the specified point is not symmetrical from the point of view of the Lie group. The specified Lie group can therefore be used to describe figures in the plane that have "rotational symmetry".
  3. a b Hawkins, 2000, p. 1
  4. ^ Hawkins, 2000, p. 2
  5. ^ Hawkins, 2000, p. 76
  6. ^ Hawkins, 2000, p. 43
  7. ^ Hawkins, 2000, p. 100
  8. Borel, 2001


  • John F. Adams : Lectures on exceptional Lie Groups (= Chicago Lectures in Mathematics. ). University of Chicago Press, Chicago IL a. a. 1996, ISBN 0-226-00527-5 .
  • Armand Borel : Essays in the history of Lie groups and algebraic groups (= History of Mathematics. Vol. 21). American Mathematical Society et al. a., Providence RI 2001, ISBN 0-8218-0288-7 .
  • Daniel Bump : Lie groups (= Graduate Texts in Mathematics. Volume 225). 2nd edition. Springer, New York NY a. a. 2013, ISBN 978-1-4614-8023-5 .
  • Nicolas Bourbaki : Elements of mathematics. Lie groups and Lie algebras. 3 volumes. (Vol. 1: Chapters 1-3. Vol. 2: Chapters 4-6. Vol. 3: Chapters 7-9. ). Addison-Wesley, Reading 1975-2005, ISBN 3-540-64242-0 (Vol. 1), ISBN 3-540-42650-7 (Vol. 2), ISBN 3-540-43405-4 (Vol. 3) .
  • Claude Chevalley : Theory of Lie groups (= Princeton Mathematical Series. Vol. 8). Volume 1. 15th printing. Princeton University Press, Princeton NJ 1999, ISBN 0-691-04990-4 .
  • William Fulton , Joe Harris , Representation Theory. A First Course (= Graduate Texts in Mathematics. Volume 129). Springer, New York NY a. a. 1991, ISBN 0-387-97495-4 .
  • Thomas Hawkins: Emergence of the theory of Lie groups. An essay in the history of mathematics 1869-1926. Springer, New York NY a. a. 2000. ISBN 0-387-98963-3 .
  • Brian C. Hall: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction (= Graduate Texts in Mathematics. Vol. 222). Springer, New York NY a. a. 2003, ISBN 0-387-40122-9 .
  • Anthony W. Knapp : Lie Groups Beyond an Introduction. 2nd Edition. Birkhäuser, Boston MA a. a. 2002, ISBN 3-7643-4259-5 .
  • Wulf Rossmann: Lie Groups. An Introduction Through Linear Groups (= Oxford Graduate Texts in Mathematics. Volume 5). Reprint 2003 (with corrections). Oxford University Press, Oxford u. a. 2004, ISBN 0-19-859683-9 (The 2003 new edition corrects some unfortunate misprints).
  • Jean-Pierre Serre : Lie Algebras and Lie Groups. 1964 Lectures given at Harvard University (= Lecture Notes in Mathematics. Vol. 1500). Springer, Berlin a. a. 1992, ISBN 3-540-55008-9 .
  • John Stillwell : Naive Lie Theory (= Undergraduate Texts in Mathematics. ). Springer, New York NY a. a. 2008, ISBN 978-0-387-78214-0 (from the preface: "developing .. Lie theory .. from single-variable calculus and linear algebra").

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