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Definition 2.1.7
Let A be a K-algebra and M, N be A-modules. A K-linear map ϕ : M → N is called a morphism
of A-modules or, equivalently, an A-linear map, if
ϕ(a.m) = a.ϕ(m) for all m ∈ M, a ∈ A .
As a diagram, this reads
A ⊗ M
id A⊗ϕ
A ⊗ N
ρ M
M
ϕ
ρ N
N .
One goal of this lecture is to obtain insights on representations of groups and to generalize
them to a class of algebraic structures much larger than groups.
To this end, it is convenient to have more terminology available to talk about all modules
over a given algebra A at once: they form a category.
Definition 2.1.8
1. A category C consists
(a) of a class of objects Obj(C), whose entries are called the objects of the category.
(b) a class Hom(C), whose entries are called morphisms of the category
(c) Maps
such that
(a) s(id V ) = t(id V ) = V
(b) id t(f) ◦ f = f ◦ id s(f) = f
id : Obj(C) → Hom (C)
s, t : Hom(C) → Obj(C)
o : Hom(C) × Obj (C) Hom(C) → Hom(C)
for all V ∈ Obj(C)
for all f ∈ Hom(C)
(c) for all f, g, h ∈ Hom(C) with t(f) = s(g) and t(g) = s(h) the associativity identity
(h ◦ g) ◦ f = h ◦ (g ◦ f) holds.
2. We write for V, W ∈ Obj(C)
Hom C (V, W ) = {f ∈ Hom(C) | s(f) = V, t(f) = W }
and End C (V ) for Hom C (V, V ). For any pair V, W , we require Hom C (V, W ) to be a set.
Elements of End C (V ) are called endomorphisms of V .
3. A morphism f ∈ Hom(V, W ) which we also write V f
−→ W or in the form f : V → W is
called an isomorphism, if there exists a morphism g : W → V , such that
g ◦ f = id V and f ◦ g = id W .
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