| CONTENTS | NEXT | OCTONION HOME | MATHEMATICAL IDEAS IN SCIENCE |

**Division Algebras,
Lie Algebras, Lie Groups and Spinors**

Some familiarity
with algebraic concepts is assumed, but as this material will stress

utility over rigor, there will be no theorems, axioms, corollaries or lemmas.
So, when

we say that A and B are elements of an algebra, we understand that they can
be

added (A + B), subtracted (A - B), and multiplied (AB), and in all cases the
result is

an element of the same set we started with. And if x is a real number, then
xA is

also an element of that set. Any generalizations of these ideas will occur as
needed.

Let **R** be
the algebra of real numbers, and **C** be the algebra of complex numbers.

**C** is 2-dimensional over the reals: each complex number z = x + iy has
a real part

and imaginary part (x and y are real numbers, so x is the real part of z, and
iy the

imaginary part).

In Lie group theory
the exponential function is extremely important. The exponential

of a real number is something one can easily determine by punching the number
into

a calculator and hitting the "e^{x}" button. The answer is
a positive real number. But

a calculator figures out this answer by expanding the exponential function in
a

Taylor series - and infinite polynomial sum (the calculator uses enough of the
terms

of the infinite
sum to be as accurate as its screen allows). The Taylor expansion of

exp(A) = e^{A} is given below:

If A is a real
number, then the answer is a real number. But the righthand side of

this equation can be defined for many other kinds of algebraic objects. This

expression can be defined for any of the kinds of algebras we will deal with
here.

It uses algebraic addition, multiplication, and the multiplication by scalars
(real

numbers). Mathematicians worry about it being "well-defined", meaning
that there

is some unique algebraic answer to the infinite sum. We'll let them worry about

that: all of our exponentials will be well-defined.

| CONTENTS | NEXT | OCTONION HOME | MATHEMATICAL IDEAS IN SCIENCE |