Graphics Reference
In-Depth Information
C.6.14. Theorem.
(1) The alternation map Alt : L
k
(
V
) Æ L
k
(
V
) is a linear transformation.
(2) If aŒL
k
(
V
), then Alt(a) is an alternating multilinear map.
(3) If aŒL
k
(
V
) is an alternating multilinear, then Alt(a) =a.
Proof.
This is easy and proved like Theorem C.6.8.
Theorem C.6.14 implies that
(
)
L
k
k
()
=
()
V
Alt L
V
and is a direct summand of L
k
(
V
). We can define a product, called the
exterior
or
wedge
product
,
r
()
¥
s
()
Æ
r
+
s
()
Ÿ
: L
VV
L
L
V
(C.17)
by the same formula (C.15) that we used for the exterior algebra. Theorem C.6.10
holds verbatim for alternating multilinear maps. This is exactly how Spivak ([Spiv65])
develops the exterior algebra.
Definition.
An element of L
k
(
V
) is called an
exterior k-form
on
V
. Let L
V
denote the
direct sum of the vector spaces L
k
(
V
), k ≥ 0. The exterior product Ÿ makes L
V
into
an algebra called the
algebra of exterior forms
on
V
.
Finally, we already know from Theorem C.6.11 and Theorem C.6.13 that there are
natural isomorphisms
(
Æ
(
)
¨
k
k
k
()
E
V
*
E
V
*
L
V
,
(C.18)
so that E
k
(
V
*) and L
k
(
V
) are isomorphic.
C.6.15. Theorem.
The isomorphisms in (C.18) induce an isomorphism of algebras
(
Æ
()
F
:
E
V
*
L
V
,
(C.19a)
where
Fa
(
Ÿ
a
Ÿ◊◊◊Ÿ
a
)
=
a
Ÿ
a
Ÿ◊◊◊Ÿ
a
(C.19b)
1
2
k
1
2
k
for all a
i
Œ
V
*.
Proof.
This is another easy exercise working through the definitions. Note that the
left side of Equation (C.19b) has an exterior product of simply “elements” or symbols
a
i
that happen to belong to
V
* whereas the right side is an exterior product of
maps
as defined by (C.17).
Theorems C.6.6 and C.6.15 can be summarized compactly by saying that we have
a commutative diagram