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3. Similarly, denote by I − (V ) the two-sided ideal of T (V ) that is generated by all elements
of the form x ⊗ y + y ⊗ x with x, y ∈ V . The quotient
Λ(V ) := T (V )/I − (V )
with its natural algebra structure is called the alternating algebra or exterior algebra
over V . The alternating algebra is a Z + -graded algebra, as well. If V is finite-dimensional,
n := dim V , it is finite-dimensional. The dimension of the homogeneous component is
( ) n
dim Λ r (V ) =
r
A central notion for this lecture is the one of a module:
Definition 2.1.4
Let A be a K algebra. A left A-module is a pair (M, ρ), consisting of a K-vector space M and
a map of K-algebras
ρ : A → End K (M) .
Remark 2.1.5.
1. We also write
a.m := ρ(a) m for all a ∈ A and m ∈ M
and thus obtain a K-linear map which by abuse of notation we also denote by ρ:
ρ : A ⊗ M → M
a ⊗ m ↦→ a.m
such that for all a, b ∈ A and m, n ∈ M and λ, μ ∈ K the following identities hold:
a.(λm + μn) = λ(a.m) + μ(a.n)
(λa + μb).m = λ(a.m) + μ(b.m)
(a ∙ b).m = a.(b.m)
1.m = m
(The first two lines just express that ρ is K-bilinear.) For the properties of this map, one
can again use a graphical representation and write down the two commuting diagrams:
A ⊗ A ⊗ M μ⊗id M
A ⊗ M
id⊗ρ
A ⊗ M
ρ
ρ
A
while unitality reads
K ⊗ M η⊗id M id
M ⊗η
A ⊗ M M ⊗ K
ρ
M M M
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