Page 468 - Advanced Linear Algebra

452 Advanced Linear Algebra

we will not study them here. Also, in this chapter, we will assume that all
algebras are associative. Nonassociative algebras, such as Lie algebras and
Jordan algebras, are important as well.
The Center of an Algebra
Definition The center of an -algebra is the set
-
(
A²(³ ~¸ ( “ %~% for all %(¹
(
of all elements of that commute with every element of .
(
The center of an algebra is never trivial since it contains a copy of :
-
¸ “  -¹ ‹ A²(³

Definition An -algebra is central if its center is as small as possible, that
-
(
is, if
A²(³ ~ ¸ “  -¹

From a Vector Space to an Algebra
If is a vector space over a field and if 8 ~ ¸ “  0 ¹ is a basis for ,
-
=
=
then it is natural to wonder whether we can form an - -algebra simply by
defining a product for the basis elements and then using the distributive laws to
=
extend the product to . In particular, we choose a set of constants Á with the

Á
property that for each pair ² Á ³ , only finitely many of the are nonzero. Then

we set
~ Á

and make multiplication bilinear, that is,

~
8 9
~ ~

~
8 9
~ ~
and

8 9 ~
~ ~
for  - . It is easy to see that this does define a nonunital associative algebra
( provided that