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e the functor which associates to a pair M, N of A-modules their tensor product M ⊗ K N as
vector spaces with the A-module structure given by the morphism of algebras
A
Δ
−→ A ⊗ A ρ M ⊗ρ N
−→ End(M) ⊗ End(N) −→ End(M ⊗ N) .
Then (A−mod, ⊗, (K, ɛ)), together with the canonical associativity and unit constraints of the
category vect(K) of K-vector spaces is a monoidal category, if and only if (A, μ, Δ) is a bialgebra
with counit ɛ, i.e. if and only if (A, Δ, ɛ) is a coalgebra.
Proof.
• Suppose that (A, μ, Δ) is a coalgebra. We have to show that the canonical isomorphisms
of vector spaces
(U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W )
(u ⊗ v) ⊗ w ↦→ u ⊗ (v ⊗ w)
are morphisms of A-modules. Using Sweedler notation, the element a ∈ A acts on the left
hand side by
a.(u ⊗ v) ⊗ w = a (1) .(u ⊗ v) ⊗ a (2) .w = ((a (1) ) (1) .u ⊗ (a (1) ) (2) .v) ⊗ a (2) .w
and on the right hand side
a.u ⊗ (v ⊗ w) = a (1) .u ⊗ a (2) .(v ⊗ w) = a (1) .u ⊗ ((a (2) ) (1) .v ⊗ (a (2) ) (2) .w)
Coassociativity of A implies that the right hand side of the first equation is mapped to
the right hand side of the second equation after rebracketing.
Since the standard associativity constraints in vect(K) obey the pentagon relation, this
relation also holds in A−mod. Similarly, we have to show that the two unit constraints
V ⊗ K → V and K ⊗ V → V
v ⊗ λ ↦→ λv λ ⊗ v ↦→ λv
are morphisms of A-modules. For the second isomorphism, note that
a.(λ ⊗ v) = ɛ(a (1) )λ ⊗ a (2) .v ↦→ ɛ(a (1) )a (2) .λv = a.λv
where in the last step we used one defining property of the counit. The other unit constraint
is dealt with in complete analogy.
• Conversely, suppose that (A−mod, ⊗, K) is a monoidal category. Then
(A ⊗ A) ⊗ A → A ⊗ (A ⊗ A)
(u ⊗ v) ⊗ w ↦→ u ⊗ (v ⊗ w)
is an isomorphism of A-modules, which, upon setting u = v = w = 1 A , implies by
the identities above that the given unital algebra map Δ is coassociative. Similarly, we
conclude from the fact that the canonical maps K ⊗ A → A and A ⊗ K → A are
isomorphisms of A-modules that ɛ is a counit.
✷
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