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