(^456) 15. MODERN ALGEBRA
Since we have no need to do any computations, we omit the other two coordinates.
The point to be noticed is that this function is differentiable, so that the group
operation, when interpreted in terms of the parameters, is a differentiable operation.
To study a Lie group, one passes to the tangent spaces it has as a manifold: in
particular, the tangent space at the group identity. This tangent space is determined
by the directions in the parameter space around the point corresponding to the
identity. Each direction gives a directional derivative that operates on differentiable
functions. For that reason, the tangent space is defined to be the set of differential
operators of the form X = J2aj(x)alT- The composition of two such operators
involves second derivatives, so that XY is not in general an element of the tangent
space. However, the second partial derivatives cancel in the expression XY - YX
(the Lie bracket). This multiplication operation makes the tangent space into a Lie
algebra. This algebra, being a finite-dimensional vector space, is determined as an
algebra once the multiplication table for the elements of a basis is given. To take
the simplest nontrivial example, the Lie group of rotations of three-dimensional
space (represented as 3 ÷ 3 rotation matrices) has the vector algebra developed by
Gibbs (with the cross product as multiplication) as its Lie algebra.
Elements of the group can be generated from the Lie algebra by applying a
mapping called the exponential mapping from the Lie algebra into the Lie group.
Finding this mapping amounts to solving a differential equation. The resulting
combination of algebra, geometry, and analysis is both profound and beautiful.
It would have been pleasant for mathematics in general and for Lie in particular
if this beauty and profundity had been recognized immediately. Unfortunately, Lie's
work was not well understood at first. In January 1884 he wrote to his friend, the
German mathematician Adolf Mayer (1839 1908):
I am so certain, so absolutely certain, that these theories will be
acknowledged as fundamental one day in the future. If I wish to
procure such an opinion soon, it is... because I could produce ten
times more. [Engel, 1899, quoted by Parshall, 1985, p. 265]
Lie's vision was soon vindicated. By the end of his life, the potential of the
theory was being recognized, and its development has never slowed in the century
that has elapsed since that time.
Group representations. The road to abstraction is a two-way street. Once an ax-
iomatic characterization of an object is stated, a classification program starts au-
tomatically, aimed at answering two important questions: First, how abstract is
the abstract object, really? Second, how many abstractly different objects fit the
axioms?
The first question leads to the search for concrete representations of abstract
groups given only by a multiplication table. We know, for example, that every group
G can be thought of as a group of one-to-one mappings by associating with each
á e G the mapping La : G —> G given by æ <—> az. This fact was noted by Cay ley
when he introduced the abstract concept. It is also easy to show that any finite
group can be represented in a trivial way as a group of invertible matrices whose
entries are all zeros and ones. First regard the group as a group of permutations
of a set of k objects. Then make the k objects into the basis of a vector space
and associate with each permutation the matrix of the linear transformation it
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