- ALGEBRAIC STRUCTURES 455
under an infinite group of translations of the plane. Klein introduced the concept of
an automorphic function, an analytic function f(z) that is invariant under a group
of fractional-linear (Mobius) transformations
æ H-» aZ ~*~ ^ , ad — be ö 0.
cz + d
Both Klein and Lie made use of groups to unify many aspects of projective
geometry, and Klein suggested that various kinds of geometry could be classified in
relation to the groups of transformations that leave their basic objects invariant.
Lie groups. One of the most fundamental and far-reaching applications of the group
concept is due to Lie, Klein's companion from 1869, when both young geometers
felt like outsiders in the intensely analytic and algebraic world of Berlin mathe-
matics. In studying surfaces in three-dimensional space, Lie and Klein naturally
encountered the problem of solving the differential equations that lead to such sur-
faces. Lie had the idea of solving these equations using continuous transformations
that leave the differential equation invariant, in analogy with what Galois had done
for algebraic equations. Klein noticed an analogy between this early work of Lie
and Abel's work on the solution of equations and wrote to Lie about it. Lie was
very pleased at Klein's suggestion. He believed as a matter of faith in the basic
validity of this analogy and developed it into the theory of Lie groups. Lie himself
did not present a whole Lie group, only a portion of it near its identity element. He
considered a set of one-to-one transformations indexed by ç-tuples of sufficiently
small real numbers in a neighborhood of (0,0,..., 0) in such a way that the com-
position of the transformations corresponded to addition of the points that indexed
them. This subject was developed by Lie, Wilhelm Killing (1847-1923), Elie Cartan
(1869-1951), Hermann Weyl (1885-1955), Claude Chevalley (1909-1984), Harish-
Chandra (1923-1983), and others into one of the most imposing edifices of modern
mathematics. A Lie group is a manifold in Riemann's sense that also happens to be
a group, in which the group operations (multiplication and inversion) are analytic
functions of the coordinates.
Lie's work is far too complicated to summarize, but we can explain his basic
ideas with a simple example. The sphere S^3 in four-dimensional space can be
regarded as the set of quaternions of unit norm, that is, A = a + a such that
\A\^2 — a^2 + \a\^2 = 1. Because \AB\ = \A\ \B\, this set is closed under quaternion
multiplication and inverses. But this sphere is also a three-dimensional manifold
and can be parameterized by, say, the stereographic projection from (—1,0,0,0)
through the equatorial hyperplane consisting of points (0,x,y, z). This projection
maps (0, x, y, z) to
/1 - x^2 - y^2 - z^2 2x 2y 2z \
\ 1 + x
2
- y
2 - z
2
' ITx^Tj^T?' l + x
2 - y
2 - z
2
' 1 + x
2 - y
2 - z
2
) '
This parameterization covers the entire group except for the point (—1,0,0,0).
To parameterize a portion of the sphere containing this point requires a second
parametrization, which can be projection from the opposite pole (1,0,0,0). When
the points in the group with coordinates (u, v, w) and (x, y, z) are multiplied, the
result is the point whose first coordinate is
~(u^2 x + (-1 + v^2 + w^2 )x + 2wy-2vz + u{-l + x^2 + y^2 + z^2 ))
1 - 2(ux + vy + wz) + (u^2 + v^2 + w^2 )(x^2 + y^2 + z^2 )