CHAPTER 2 Hopf Algebras, Algebraic, Formal, and Quantum Groups

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2. Monoids and Groups in a Category

Before we use Hopf algebras to describe quantum groups and some of the better known groups, such as affine algebraic groups and formal groups, we introduce the concept of a general group (and of a monoid) in an arbitrary category. Usually this concept is defined with respect to a categorical product in the given category. But in some categories there are in general no products. Still, one can define the concept of a group in a very simple fashion. We will start with that definition and then show that it coincides with the usual notion of a group in a category in case that category has finite products.

Definition 2.2.1. Let C be an arbitrary category. Let G ∈ C be an object. We use the notation G(X) := Mor_{C}(X, G) for all X ∈ C, G(f) := Mor_{C}(f, G) for all morphisms f : X ⟶ Y in C, and f(X) := Mor_{C}(X, f) for all morphisms f : G ⟶ G′.

G together with a natural transformation m : G(-) × G(-) ⟶ G(-) is called a group (monoid) in the category C, if the sets G(X) together with the multiplication m(X) : G(X) × G(X) ⟶ G(X) are groups (monoids) for all X ∈ C.

Let (G, m) and (G′, m′) be groups in C. A morphism f : G ⟶ G′ in C is called a homomorphism of groups in C, if the diagrams

[figure: G(X) × G(X); G(X); m(X); G′(X) × G′(X); G′(X); m′(X); f(X)×f(X); f(X)]

commute for all X ∈ C.

Let (G, m) and (G′, m′) be monoids in C. A morphism f : G ⟶ G′ in C is called a homomorphism of monoids in C, if the diagrams

[figure: G(X) × G(X); G(X); m(X); G′(X) × G′(X); G′(X); m′(X); f(X)×f(X); f(X)]

and

[figure: {∗}; u; u′; G(X); G′(X); f(X)]

commute for all X ∈ C.

Problem 2.2.1. 1) If a set Z together with a multiplication m : Z × Z ⟶ Z is a monoid, then the unit element e ∈ Z is uniquely determined. If it is a group then also the inverse i : Z ⟶ Z is uniquely determined. Unit element and inverses of groups are preserved by maps that are compatible with the multiplication.

2) Find an example of monoids Y and Z and a map f : Y ⟶ Z with f(y_{1}y_{2}) = f(y_{1})f(y_{2}) for all y_{1}, y_{2} ∈ Y, but f(e_{Y}) ≠ e_{Z}.

3) If (G, m) is a group in C and i_{X} : G(X) ⟶ G(X) is the inverse, then i is a natural transformation. The Yoneda Lemma provides a morphism S : G ⟶ G such that i_{X} = Mor_{C}(X, S) = S(X) for all X ∈ C.

Proposition 2.2.2. Let C be a category with finite (categorical) products. An object G in C carries the structure m : G(-) × G(-) ⟶ G(-) of a group in C if and only if there are morphisms m : G × G ⟶ G, u : E ⟶ G, and S : G ⟶ G such that the diagrams

[figure: G × G × G; G × G; m×1; G × G; G; m; 1×m; m; E × G ≅ G ≅ G × E; G × G; 1×u; G × G; G; m; u×1; m; 1]

[figure: G; E; G; u; G × G; G × G; 1×S; S×1; Δ; m]

commute where Δ is the morphism defined in A.2. The multiplications are related by m_{X} = Mor_{C}(X, m) = m(X).

An analogous statement holds for monoids.

Proof. The Yoneda Lemma defines a bijection between the set of morphisms f : X ⟶ Y and the set of natural transformations f(-) : X(-) ⟶ Y (-) by f(Z) = Mor_{C}(Z, f). In particular we have m_{X} = Mor_{C}(X, m) = m(X). The diagram

[figure: G(-) × G(-) × G(-); G(-) × G(-); m-×1; G(-) × G(-); G(-); m-; 1×m-; m-]

commutes if and only if Mor_{C}(-, m(m × 1)) = Mor_{C}(-, m)(Mor_{C}(-, m) × 1) = m-(m- × 1) = m-(1 × m-) = Mor_{C}(-, m)(1 × Mor_{C}(-, m)) = Mor_{C}(-, m(1 × m)) if and only if m(m × 1) = m(1 × m) if and only if the diagram

[figure: G × G × G; G × G; m×1; G × G; G; m; 1×m; m]

commutes. In a similar way one shows the equivalence of the other diagram(s).

Problem 2.2.2. Let C be a category with finite products. Show that a morphism f : G ⟶ G′ in C is a homomorphism of groups if and only if

[figure: G × G; G; m; G′ × G′; G′; m′; f×f; f]

commutes.

Definition 2.2.3. A cogroup (comonoid) G in C is a group (monoid) in C^{op}, i.e. an object G ∈ Ob C = Ob C^{op} together with a natural transformation m(X) : G(X) × G(X) ⟶ G(X) where G(X) = Mor_{C^{op}}(X, G) = Mor_{C}(G, X), such that (G(X), m(X)) is a group (monoid) for each X ∈ C.

Remark 2.2.4. Let C be a category with finite (categorical) coproducts. An object G in C carries the structure m : G(-) × G(-) ⟶ G(-) of a cogroup in C if and only if there are morphisms Δ : G ⟶ G ⨿ G, ε : G ⟶ I, and S : G ⟶ G such that the diagrams

[figure: G ⨿ G; G ⨿ G ⨿ G; Δ⨿1; G; G ⨿ G; Δ; 1⨿Δ; Δ; G ⨿ G; I ⨿ G ≅ G ≅ G ⨿ I; ε⨿1; G; G ⨿ G; Δ; 1⨿ε; Δ; 1]

[figure: G; I; G; ε; G ⨿ G; G ⨿ G; 1⨿S; S⨿1; Δ; ∇]

commute where ∇ is dual to the morphism Δ defined in A.2. The multiplications are related by Δ_{X} = Mor_{C}(Δ, X) = Δ(X).

Let C be a category with finite coproducts and let G and G′ be cogroups in C. Then a homomorphism of groups f : G′ ⟶ G is a morphism f : G ⟶ G′ in C such that the diagram

[figure: G; G × G; Δ; G′; G′ × G′; Δ′; f×f; f]

commutes. An analogous result holds for comonoids.

Remark 2.2.5. Obviously similar observations and statements can be made for other algebraic structures in a category C. So one can introduce vector spaces and covector spaces, monoids and comonoids, rings and corings in a category C. The structures can be described by morphisms in C if C is a category with finite (co-) products.

Problem 2.2.3. Determine the structure of a covector space on a vector space V from the fact that Hom(V, W) is a vector space for all vector spaces W.

Proposition 2.2.6. Let G ∈ C be a group with multiplication a ∗ b, unit e, and inverse a^{−1} in G(X). Then the morphisms m : G × G ⟶ G, u : E ⟶ G, and S : G ⟶ G are given by

m = p_{1} ∗ p_{2},  u = e_{E},  S = id_{G}^{−1}.

Proof. By the Yoneda Lemma A.9.1 these morphisms can be constructed from the natural transformation as follows. Under Mor_{C}(G×G, G×G) = G×G(G×G) ≅

G(G × G) × G(G × G) ⟶^{∗} G(G × G) = Mor_{C}(G × G, G) the identity id_{G×G} = (p_{1}, p_{2})

is mapped to m = p_{1} ∗ p_{2}. Under Mor_{C}(E, E) = E(E) ⟶ G(E) = Mor_{C}(E, G) the identity of E is mapped to the neutral element u = e_{E}. Under Mor_{C}(G, G) = G(G) ⟶ G(G) = Mor_{C}(G, G) the identity is mapped to its ∗-inverse S = id_{G}^{−1}.

Corollary 2.2.7. Let G ∈ C be a cogroup with multiplication a ∗ b, unit e, and inverse a^{−1} in G(X). Then the morphisms Δ : G ⟶ G ⨿ G, ε : G ⟶ I, and S : G ⟶ G are given by

Δ = ι_{1} ∗ ι_{2},  ε = e_{I},  S = id_{G}^{−1}.