Page 469 - Advanced Linear Algebra

An Introduction to Algebras 453

² ³ ~ ² ³

(
for all Á Á  0 and that is commutative if and only if
~

for all Á  0 . The constants Á are called the structure constants for the

(
algebra . To get a unital algebra, we can take for a given  0 , the structure
constants to be
Á ~ Á ~ Á

(
in which case is the multiplicative identity. An alternative is to adjoin a new

element to the basis and define its structure constants in this way.)
Examples
The following examples will make it clear why algebras are important.
,
Example 18.1 If - , are fields, then is a vector space over . This vector
-
,
space structure, along with the ring structure of , is an algebra over .
-
-
Example 18.2 The ring -´%µ of polynomials is an algebra over .
-
Example 18.3 The ring C ²-³ of all d matrices over a field is an
algebra over , where scalar multiplication is defined by
-
4 ~ ² ³Á  - ¬ 4 ~ ² ³
Á
Á
=
Example 18.4 The set B - ²= ³ of all linear operators on a vector space over a
field is an -algebra, where addition is addition of functions, multiplication is
-
-
composition of functions and scalar multiplication is given by

² ³²#³ ~ ´ #µ
The identity map B  ² = - ³ is the multiplicative identity and the zero map
²= ³ is the additive identity. This algebra is also denoted by End ²= ³,
 B - -
=
since the linear operators on are also called endomorphisms of .
=
Example 18.5 If is a group and is a field, then we can form a vector space
-
.
-´.µ over by taking all formal -linear combinations of elements of . and
-
-
treating as a basis for ´ - . µ . This vector space can be made into an -algebra
-
.
where the structure constants are determined by the group product, that is, if
~ " , then Á ~ Á" . The group identity ~ is the algebra identity

since ~ and so Á ~ Á and similarly, Á ~ Á .

-
The resulting associative algebra -´.µ is called the group algebra over .
Specifically, the elements of -´.µ have the form