Page 473 - Advanced Linear Algebra

An Introduction to Algebras 457

)
2 Conversely, if is a ring with identity and if - ‹ A ² 9 ³ is a field, then 9
9
9
is an -algebra with scalar multiplication defined by the product in .
-
One interesting consequence of this theorem is that a ring whose center does
9
not contain a field is not an algebra over any field . This happens, for example,
-
.
with the ring {
The Regular Representation of an Algebra
An algebra homomorphism B¢( ¦ - ²= ³ is called a representation of the
algebra in B ² = ³ . A representation is faithful if it is injective, that is, if
(
-
(
is an embedding. In this case, is isomorphic to a subalgebra of B - ² = . ³
Actually, the endomorphism algebras B - ²= ³ are the most general algebras
possible, in the sense that any algebra has a faithful representation in some
(
endomorphism algebra.
Theorem 18.2 Any associative -algebra is isomorphic to a subalgebra of
-
(
the endomorphism algebra B . In fact, if - ²(³ is the left multiplication map
defined by
%~ %
then the map B¢( ¦ ²(³ is an algebra embedding, called the left regular
(
representation of .
(
When dim²(³ ~  B , we can select an ordered basis for and represent
8
the elements of B - ²(³ by matrices. This gives an embedding of into the
(
matrix algebra C ²-³ , called the left regular matrix representation of (
with respect to the ordered basis .
8
Example 18.7 Let . ~ ¸ Á Á Ã Á c ¹ be a finite cyclic group. Let

8 ~² Á Á Ã Á c ³
be an ordered basis for the group algebra -´.µ . The multiplication map that
)
(
8
is multiplcation by is a shifting of with wraparound and so the matrix
is the matrix whose columns are obtained from the identity
representation of
(
matrix by shifting columns to the right with wrap around . For example,
)

v Ä y
x Ä {
x {
´ µ 8 ~ x Æ {
x {
Å Å Æ Å Å
w Ä z
These matrices are called circulant matrices .