9Dual spaces and tensor products of representations
II Representation Theory
9.6 Tensor algebra
This part is for some unknown reason included in the schedules. We do not need
it here, but it has many applications elsewhere.
Consider a field F with characteristic 0.
Definition
(Tensor algebra)
.
Let
T
n
V
=
V
⊗n
. Then the tensor algebra of
V
is
T
·
(V ) = T (V ) =
M
n≥0
T
n
V,
with T
0
V = F by convention.
This is a vector space over
F
with the obvious addition and multiplication by
scalars.
T
(
V
) is also a (non-commutative) (graded) ring with product
x·y
=
x⊗y
.
This is graded in the sense that if
x ∈ T
n
V
and
y ∈ T
m
V
,
x·y
=
x⊗y ∈ T
n+m
V
.
Definition
(Symmetric and exterior algebra)
.
We define the symmetric algebra
of V to be
S(V ) = T (V )/(ideal of T (V ) generated by all u ⊗ v − v ⊗ u).
The exterior algebra of V is
Λ(V ) = T (V )/(ideal of T (V ) generated by all v ⊗ v).
Note that
v
and
u
are not elements of
V
, but arbitrary elements of
T
n
V
for
some n.
The symmetric algebra is commutative, while the exterior algebra is graded-
commutative, i.e. if x ∈ Λ
n
(V ) and y ∈ Λ
m
(V ), then x · y = (−1)
n+m+1
y · x.