1. Equivalence, Invariants, and Symmetry Chapter 2
Felix Klein
( 1849 – 1925 )
PETER J. OLVER
University of Minnesota
Presented by Mehdi Nadjafikhah
webpages.iust.ac.ir\m_nadjafikhah/
Last Update: January 27, 2010

2. Chapter 2 Lie Algebras Transformation Groups
Invariant Subsets and Equations
Canonical Forms
Invariant Functions
Lie Algebras
Structure Constants
The Exponential Map
Subgroups and Subalgebras
Infinitesimal Group Actions
Classification of Group Actions
Infinitesimal Invariance
Invariant Vector Fields
Lie Derivatives and Invariant Differential Forms
The Maurer-Cartan Forms

3. The symmetry groups that arise most often in the applications to geometry and differential equations are Lie groups of transformations acting on a finite dimensional manifold.
Since Lie groups will be one of the cornerstones of our investigations, it is essential that we gain a basic familiarity with these fundamental mathematical objects.
The present chapter is devoted to a survey of a number of fundamental facts concerning Lie groups and Lie algebras, and their actions on manifolds.

4. Any smooth Lie group can be endowed with an analytic structure, [195].
Often, an r-dimensional Lie group is referred to as an r parameter group, the "group parameters" referring to a choice of local coordinates on the group manifold.

11. Transformation Groups

26. Invariant Subsets and Equations

28. Hydon, Symmetry methods for differential equations,1999, page 21.

29. An orbit of a transformation group is a minimal (nonempty) invariant subset.

31. Homogeneous space
Every global transitive group action can be identified with a homogeneous space.

33. In general, the orbits of a Lie group of transformations are all submanifolds
of the manifold M.
A group action is called semi-regular if all its orbits have the same dimension.
The action is called regular if, in addition, each point x of M has arbitrarily small neighborhoods whose intersection with each orbit is a connected subset thereof.
The condition that each orbit be a regular submanifold is necessary, but not sufficient for the regularity of the group action.

37. Theorem 2.23 demonstrates that the orbits form a foliation of the underlying manifold.
This fundament a1 result is a direct consequence of Frobenius' Theorem, to be proved in Chapter 14.
In practice, the rectifying coordinates are most readily constructed using the infinitesimal methods to be discussed below.

38. Canonical Forms

42. Invariant Functions
An invariant of a transformation group is defined as a real-valued function
whose values are unaffected by the group transformations.
The determination of a complete set of invariants of a given group action
is a problem of supreme importance for the study of equivalence and
canonical forms.
In the regular case, the orbits, and hence the canonical forms, for a group action are completely characterized by its invariants.
Consequently, a significant portion of the book will be devoted to characterizing,
classifying, analyzing, and applying invariants of group actions.

46. A fundamental problem is
“ the determination of all the invariants of a group of transformations’’.

49. Theorem 2.34 provides a complete answer to the question of local invariants of group actions.

54. Lie Algebras

64. Lie Algebras of Local Lie Groups

67. Structure Constants

69. Commutator Tables

71. A matrix Lie algebra

72. A vector field Lie algebra

73. The Exponential Map

77. Subgroups and Subalgebras

83. Infinitesimal Group Actions

92. Classification of Group Actions

96. Infinitesimal Invariance
The fundamental feature of (connected) Lie groups is the ability to work
infinitesimally, thereby effectively linearizing complicated invariance criteria.
Indeed, the practical applications of Lie groups all ultimately rely on this basic method, and its importance cannot be overestimated.
We begin by stating the infinitesimal criterion for the invariance of a real-valued function.

106. Invariant Vector Fields

113. Lie Derivatives and Invariant Differential Forms

119. The Maurer-Cartan Forms