Page 408 - Advanced Linear Algebra

392 Advanced Linear Algebra

If ~ ² ³ , then applied to ² Á ³ gives ² Á ³ and not ² Á ³ , since

permutes the two coordinate positions in ²#Á #³ .

Graded Algebras
We need to pause for a few definitions that are useful in discussing tensor
algebras. An algebra over is said to be a graded algebra if as a vector
-
(
space over , can be written in the form
-(
B
(~ (
~
for subspaces ( of , and where multiplication behaves nicely, that is,
(
(( ‹ ( b

The elements of ( are said to be homogeneous of degree . If  ( is written

~ bÄb
is called the homogeneous component of of

for  ( , £ , then
degree .

The ring of polynomials -´%µ provides a prime example of a graded algebra,
since
B
-´%µ ~ - ´%µ

~

%
where - ´%µ is the subspace of -´%µ consisting of all scalar multiples of .

More generally, the ring -´% Á Ã Á % µ of polynomials in several variables is a

graded algebra, since it is the direct sum of the subspaces of homogeneous
(
polynomials of degree . A polynomial is homogeneous of degree if each

term has degree . For example, ~ %% b %%% is homogeneous of degree

.)
The Symmetric and Antisymmetric Tensor Algebras
Our discussion of symmetric and antisymmetric tensors will benefit by a
discussion of a few definitions and setting a bit of notation at the outset.
Let -´ Á Ã Á µ denote the vector space of all homogeneous polynomials of

degree (together with the zero polynomial) in the independent variables

. As is sometimes done in this context, we denote the product in
Á Ã Á
- ´ Á Ã Á µ by v , for example, writing as v v . The algebra of

is denoted by -´ ÁÃÁ µ .
all polynomials in Á ÃÁ