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

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

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.

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.

Transformation Groups

Invariant Subsets and Equations

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

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

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

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.

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.

Canonical Forms

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.

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

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

Lie Algebras

Lie Algebras of Local Lie Groups

Structure Constants

Commutator Tables

A matrix Lie algebra

A vector field Lie algebra

The Exponential Map

Subgroups and Subalgebras

Infinitesimal Group Actions

Classification of Group Actions

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.

Invariant Vector Fields

Lie Derivatives and Invariant Differential Forms

The Maurer-Cartan Forms