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
or H
(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
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