CHAPTER 3 Hopf Algebras, Algebraic, Formal, and Quantum Groups

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4. Finite reconstruction

The endomorphism ring of a vector space enjoys the following universal property. It is a vector space itself and allows a homomorphism ρ : End(V) ⊗ V ⟶ V. It is universal with respect to this property, i. e. if Z is a vector space and f : Z ⊗ V ⟶ V is a homomorphism, then there is a unique homomorphism g : Z ⟶ End(V) such that

[figure: End(V) ⊗ V; V; ρ; f; Z ⊗ V; g⊗1]

commutes.

The algebra structure of End(V) comes for free from this universal property. If we replace the vector space V by a diagram of vector spaces ω : D ⟶ Vec we

get a similar universal object End(ω). Again the universal property induces a unique algebra structure on End(ω).

Problem 3.4.1. 1. Let V be a vector space. Show that there is a universal vector space E and homomorphism ρ : E ⊗ V ⟶ V (such that for each vector space Z and each homomorphism f : Z ⊗ V ⟶ V there is a unique homomorphism g : Z ⟶ E such that

[figure: E ⊗ V; V; ρ; f; Z ⊗ V; g⊗1]

commutes). We call E and ρ : E ⊗ V ⟶ V a vector space acting universally on V.

2. Let E and ρ : E ⊗ V ⟶ V be a vector space acting universally on V. Show that E uniquely has the structure of an algebra such that V becomes a left E-module.

3. Let ω : D ⟶ Vec be a diagram of vector spaces. Show that there is a universal vector space E and natural transformation ρ : E ⊗ ω ⟶ ω (such that for each vector space Z and each natural transformation f : Z ⊗ω ⟶ ω there is a unique homomorphism g : Z ⟶ E such that

[figure: E ⊗ ω; ω; ρ; f; Z ⊗ ω; g⊗1]

commutes). We call E and ρ : E ⊗ ω ⟶ ω a vector space acting universally on ω.

4. Let E and ρ : E ⊗ ω ⟶ ω be a vector space acting universally on ω. Show that E uniquely has the structure of an algebra such that ω becomes a diagram of left E-modules.

Similar considerations can be carried out for coactions V ⟶ V ⊗C or ω ⟶ ω ⊗C and a coalgebra structure on C. There is one restriction, however. We can only use finite dimensional vector spaces V or diagrams of finite dimensional vector spaces. This will be done further down.

As we have seen, ???

We want to find a universal natural transformation δ : ω ⟶ ω ⊗ coend(ω). For this purpose we consider the isomorphisms

Mor_{C}(ω(X), ω(X) ⊗ M) ≅ Mor_{C}(ω(X)^{∗} ⊗ ω(X), M)

that are given by f ↦ (ev ⊗1)(1 ⊗ f) and as inverse g ↦ (1 ⊗ g)(db ⊗1). We first develop techniques to describe the properties of a natural transformation ϕ : ω ⟶ ω ⊗ M as properties of the associated family g(X) : ω(X)^{∗} ⊗ ω(X) ⟶ M. We will see that g : ω^{∗} ⊗ ω ⟶ M will be a cone. Then we will show that ϕ is a universal natural transformation if and only if its associated cone is universal. In the literature this is called a coend.

Throughout this section assume the following. Let D be an arbitrary diagram scheme. Let C be a cocomplete monoidal category such that the tensor product preserves colimits in both arguments. Let C_{0} be the full subcategory of those objects in C that have a left dual. Let ω : D ⟶ C be a diagram in C such that ω(X) ∈ C_{0} for all X ∈ D, i. e. ω is given by a functor ω_{0} : D ⟶ C_{0}. We call such a diagram a finite diagram in C. Finally for an object M ∈ C let ω ⊗ M : D ⟶ C be the functor with (ω ⊗ M)(X) = ω(X) ⊗ M.

Remark 3.4.1. Consider the following category D̃. For each morphism f : X ⟶ Y there is an object f̃ ∈ D̃. The object corresponding to the identity 1_{X} : X ⟶ X is denoted by X̃ ∈ D̃. For each morphism f : X ⟶ Y in D there are two morphisms f_{1} : f̃ ⟶ X̃ and f_{2} : f̃ ⟶ Ỹ in D̃. Furthermore there are the identities 1_{f} : f̃ ⟶ f̃ in D̃.

Since there are no morphisms with X̃ as domain other than (1_{X})_{i} : X̃ ⟶ X̃ and 1_{f} : f̃ ⟶ f̃ we only have to define the following compositions (1_{X})_{i} ∘ f_{j} := f_{j}. Then D̃ becomes a category and we have 1X̃ = (1_{X})_{1} = (1_{X})_{2}.

We define a diagram ω^{∗} ⊗ ω : D̃ ⟶ C as follows. If f : X ⟶ Y is given then

(ω^{∗} ⊗ ω)(f̃) := ω(Y)^{∗} ⊗ ω(X)

and

\begin{matrix}ω(f_{1}) := ω(f)^{∗} ⊗ ω(1_{X}), \\ ω(f_{2}) := ω(1_{Y})^{∗} ⊗ ω(f). \end{matrix}

The colimit of ω^{∗} ⊗ ω consists of an object coend(ω) ∈ C together with a family of morphisms ι(X, X) : ω(X)^{∗} ⊗ ω(X) ⟶ coend(ω) such that the diagrams

[figure: ω(X)^{∗} ⊗ ω(X); (ω^{∗} ⊗ ω)(f̃) = ω(Y)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; coend(ω); ι(X,X); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y)]

commute for all f : X ⟶ Y in D. Indeed, such a family ι(X̃) := ι(X, X) can be uniquely extended to a natural transformation by defining ι(f̃) := ι(X, X)(ω(f)^{∗} ⊗ ω(X)) = ι(Y, Y)(ω(Y)^{∗} ⊗ ω(f)). In addition the pair (coend(ω), ι) is universal with respect to this property.

In the literature such a universal object is called a coend of the bifunctor ω^{∗} ⊗ ω : D^{op} × D ⟶ C.

Corollary 3.4.2. The following is a coequalizer

[figure: ∐_{f∈MorD} ω(Zi(f))^{∗} ⊗ ω(Qu(f)); ∐_{X∈Ob D} ω(X)^{∗} ⊗ ω(X); p; q; coend(ω)]

Proof. This is just a reformulation of Remark A.10.11, since the colimit may also be built from the commutative squares given above.

Observe that for the construction of the colimit not all objects of the diagram have to be used but only those of the form ω(X)^{∗} ⊗ ω(X).

Theorem 3.4.3. (Tannaka-Krein)

Let ω : D ⟶ C_{0} ⊆ C be a finite diagram. Then there exists an object coend(ω) ∈ C and a natural transformation δ : ω ⟶ ω ⊗ coend(ω) such that for each object M ∈ C and each natural transformation φ : ω ⟶ ω ⊗ M there exists a unique morphism φ̃: coend(ω) ⟶ M such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; φ; ω ⊗ M; 1⊗φ̃]

commutes.

Proof. Let coend(ω) ∈ C together with morphisms ι(f̃) : ω(Y)^{∗} ⊗ ω(X) ⟶ coend(ω) be the colimit of the diagram ω^{∗} ⊗ ω : D̃ ⟶ C. So we get commutative diagrams

[figure: ω(X)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; ι(X,X); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y); ω(Y)^{∗} ⊗ ω(X); coend(ω); ι(f̃)]

for each f : X ⟶ Y in C.

For X ∈ C we define a morphism δ(X) : ω(X) ⟶ ω(X) ⊗ coend(ω) by (1 ⊗ ι(X, X))(db ⊗1) : ω(X) ⟶ ω(X) ⊗ ω(X)^{∗} ⊗ ω(X) ⟶ ω(X) ⊗ coend(ω). Then we get as in Corollary 3.3.5 ι(X, X) = (1 ⊗ ev)(1 ⊗ δ(X)).

We show that δ is a natural transformation. For each f : X ⟶ Y the square

[figure: ω(Y) ⊗ ω(Y)^{∗}; ω(Y) ⊗ ω(X)^{∗}.; 1⊗ω(f)^{∗}; I; ω(X) ⊗ ω(X)^{∗}; db_{X}; db_{Y}; ω(f)⊗1]

commutes by Corollary 3.3.9. Thus the following diagram commutes

Now let M ∈ C be an object and φ : ω ⟶ ω ⊗ M a natural transformation. Observe that commutes by Corollary 3.3.10. Thus also the diagram

[figure: ω(X); ω(X) ⊗ ω(X)^{∗} ⊗ ω(X); db ⊗1; ω(X) ⊗ coend(ω); 1⊗ι(X,X); ω(Y) ⊗ ω(Y)^{∗} ⊗ ω(X); ω(Y) ⊗ ω(X)^{∗} ⊗ ω(X); 1⊗ω(f)^{∗}⊗1; ω(Y); ω(Y) ⊗ ω(Y)^{∗} ⊗ ω(Y); db ⊗1; ω(Y) ⊗ coend(ω).; 1⊗ι(Y,Y); ω(f); ω(f)⊗1; db ⊗1; 1⊗1⊗ω(f); ω(f)⊗1⊗1; 1⊗ι(X,X)]

[figure: ω(Y)^{∗} ⊗ ω(Y); I; ev; ω(Y)^{∗} ⊗ ω(X); ω(X)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; 1⊗ω(f); ev]

[figure: ω(X)^{∗} ⊗ ω(X); ω(X)^{∗} ⊗ ω(X) ⊗ M; 1⊗φ(X); ω(f)^{∗}⊗1; ω(f)^{∗}⊗1⊗1; M; ev ⊗1; ω(Y)^{∗} ⊗ ω(X); ω(Y)^{∗} ⊗ ω(X) ⊗ M; 1⊗φ(X); 1⊗ω(f); 1⊗ω(f)⊗1; ev ⊗1; ω(Y)^{∗} ⊗ ω(Y); ω(Y)^{∗} ⊗ ω(Y) ⊗ M; 1⊗φ(Y)]

commutes. We define φ̃ : coend(ω) ⟶ M from the colimit property as universal factorization

[figure: ω(X)^{∗} ⊗ ω(X); ω(Y)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; ι(X,X); (ev ⊗1)(1⊗φ(X)); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y); coend(ω); M; φ̃; (ev ⊗1)(1⊗φ(Y))]

Hence the diagram

[figure: ω(X); ω(X) ⊗ coend(ω); δ(X); ω(X) ⊗ M; ω(X) ⊗ M; 1⊗1; φ(X); 1⊗φ̃; ω(X) ⊗ ω(X)^{∗} ⊗ ω(X); ω(X) ⊗ ω(X)^{∗} ⊗ ω(X) ⊗ M; 1⊗1⊗φ(X); db ⊗1; 1⊗ev ⊗1; 1⊗ι(X,X); db ⊗1⊗1]

commutes. The exterior portion of this diagram yields

[figure: ω(X); ω(X) ⊗ coend(ω); δ(X); ω(X) ⊗ M.; 1⊗φ̃; φ(X)]

It remains to show that φ̃ : coend(ω) ⟶ M is uniquely determined. Let φ̃_{0} : coend(ω) ⟶ M be another morphism with φ(X) = (1 ⊗ φ̃_{0})δ(X) for all X ∈ D.

Then the following diagram commutes

[figure: ω(X)^{∗} ⊗ ω(X); coend(ω); ι(X,X); ω(X)^{∗} ⊗ ω(X) ⊗ coend(ω); 1⊗δ(X); 1⊗φ(X); ev ⊗1; 1⊗1⊗φ̃_{0}; φ̃_{0}; ω(X)^{∗} ⊗ ω(X) ⊗ M; M,; ev ⊗1]

hence we have φ̃_{0} = φ̃.

Corollary 3.4.4. The functor Nat(ω, ω ⊗ M) is a representable functor in M represented by coend(ω).

Proof. The universal problem implies the isomorphism

Nat(ω, ω ⊗ M) ≅ Mor_{C}(coend(ω), M)

and the universal natural transformation δ : ω ⟶ ω ⊗ coend(ω) is mapped to the identity under this isomorphism.

It is also possible to construct an isomorphism

Nat(ω, ω′ ⊗ M) ≅ Mor_{C}(cohom(ω′, ω), M)

for different functors ω, ω′ : D ⟶ C and thus define cohomomorphism objects. Observe that only ω′ has to take values in C_{0} since then we can build objects ω′(X)^{∗} ⊗ ω(X).