Page 406 - Advanced Linear Algebra

390 Advanced Linear Algebra

tr²# n ³ ~ ²#³ ~ ²# n ³

where is the contraction map.
The Tensor Algebra of =

Consider the contravariant tensor spaces

;²= ³ ~ ;²= ³ ~ = n

For ~ we take ; ²= ³ ~ - . The external direct sum
B

;²= ³ ~ ; ²= ³
~
of these tensor spaces is a vector space with the property that

; ²=³ n ; ²=³ ~ ; b ²=³

This is an example of a graded algebra , where ;²= ³ are the elements of grade
(
;. ²The graded algebra = ³ is called the over = tensor algebra . We will
formally define graded structures a bit later in the chapter.)
Since

i
i
; ²= ³ ~ = nÄn= i ~ ; ²= ³
’••••“••••”

factors
there is no need to look separately at ;²= ³ .

Special Multilinear Maps
The following definitions describe some special types of multilinear maps.
Definition
)
1 A multilinear map ¢ = d ¦ > is symmetric if interchanging any two
coordinate positions changes nothing, that is, if
²# Á ÃÁ# ÁÃÁ# ÁÃÁ# ³ ~ ²# ÁÃÁ# ÁÃÁ# ÁÃÁ# ³

for any £ .
)
2 A multilinear map ¢ = d ¦ > is antisymmetric or skew-symmetric if
interchanging any two coordinate positions introduces a factor of c , that
is, if
²# Á ÃÁ# ÁÃÁ# ÁÃÁ# ³ ~ c ²# ÁÃÁ# ÁÃÁ# ÁÃÁ# ³

for £ .