Introduction

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The equations that Lie studied are now known as equations of Lie

type, and an example of these is the well-known Riccati equation.

Lie developed a method of solving these equations that is related to

the process of "solution by quadrature" (cf. [Fr-Uh, pp. 14, 55],

[Ku]). In Galois' terms, for a solution of a polynomial equation with

radicals, there is a corresponding finite group. Correspondingly, to a

solution of a differential equation of Lie type by quadrature, there is

a corresponding continuous group.

The term "Lie group" is generally attributed to E. Cartan (1930).

It is defined as a manifold G endowed with a group structure, such

that the maps G x G — G (x,y) i— xy and G — * G x i—

x~l

are

smooth (i.e. differentiable). The simplest examples of Lie groups are

the groups of isometries of

Rn,

C

n

or H

n

(H is the set of quaternions).

Hence, we obtain the orthogonal group 0(n), the unitary group J7(ra),

and the symplectic group Sp(n).

An algebra g can be associated with each Lie group G in a natural

way; this is called the Lie algebra of G. In the early development of

the theory, g was referred to as an "infinitesimal group". The modern

term is attributed by most people to H. Weyl (1934). A fundamental

theorem of Lie states that every Lie group G (in general, a compli-

cated non-linear object) is "almost" determined by its Lie algbera g

(a simpler, linear object). Thus, various calculations concering G are

reduced to algebraic (but often non-trivial) computations on g.

A homogeneous space is a manifold M on which a Lie group acts

transitively. As a consequense of this, M is diffeomorphic to the coset

space G/K, where K is a (closed Lie) subgroup of G. In fact, if we

fix a base point m £ M, then K is the subgroup of G that consists of

the points in G that fix m (it is called the isotropy subgroup ofm).

As mentioned above, these are the geometries according to Klein, in

the sense that they are obtained from a manifold M and a transitive

action of a Lie group G on M. The advantage is that instead of

studying a geometry with base point m as the pair (M, m) with the

group G acting on M, we could equally study the pair (G, K).

One of the fundamental properties of a homogeneous space is

that, if we know the value of a geometrical quantity (e.g. curvature)

at a given point, then we can calculate the value of this quantity at