CHAPTER 8 Toolbox

26

6. Coalgebras

Definition 8.6.1. A 𝕂-coalgebra is a 𝕂-module C together with a comultiplication or diagonal Δ : C ⟶ C ⊗ C that is coassociative:

[figure: C ⊗ C; C ⊗ C ⊗ C; id ⊗Δ; C; C ⊗ C; Δ; Δ; Δ⊗id]

and a counit or augmentation ϵ : C ⟶ 𝕂:

[figure: C; C ⊗ C; Δ; Δ; id ⊗ϵ; C ⊗ C; 𝕂 ⊗ C ≅ C ≅ C ⊗ 𝕂.; ϵ⊗id; id]

A 𝕂-coalgebra C is cocommutative if the following diagram commutes

[figure: C ⊗ C; C ⊗ C; τ; C; Δ; Δ]

Let C and D be 𝕂-coalgebras. A homomorphism of coalgebras f : C ⟶ D is a 𝕂-linear map such that the following diagrams commute:

[figure: C ⊗ C; D ⊗ D; f⊗f; C; D; f; Δ_{C}; Δ_{D}]

and

[figure: 𝕂; ϵ_{C}; ϵ_{D}; C; D.; f]

Remark 8.6.2. Obviously the composition of two homomorphisms of coalgebras is again a homomorphism of coalgebras. Furthermore the identity map is a homomorphism of coalgebras. Hence the 𝕂-coalgebras form a category 𝕂-Coalg. The category of cocommutative 𝕂-coalgebras will be denoted by 𝕂-cCoalg.

Problem 8.6.1. 1. Show that V ⊗ V^{∗} is a coalgebra for every finite dimensional vector space V over a field 𝕂 if the comultiplication is defined by Δ(v ⊗ v^{∗}) := ∑_{i=1}^{n} v ⊗ v_{i}^{∗} ⊗ v_{i} ⊗ v^{∗} where {v_{i}} and {v_{i}^{∗}} are dual bases of V resp. V^{∗}.

2. Show that the free 𝕂-modules 𝕂X with the basis X and the comultiplication Δ(x) = x ⊗ x is a coalgebra. What is the counit? Is the counit unique?

3. Show that 𝕂 ⊕ V with Δ(1) = 1 ⊗ 1, Δ(v) = v ⊗ 1 + 1 ⊗ v defines a coalgebra.

4. Let C and D be coalgebras. Then C⊗D is a coalgebra with the comultiplication Δ_{C⊗D} := (1_{C} ⊗ τ ⊗ 1_{D})(Δ_{C} ⊗ Δ_{D}) : C ⊗ D ⊗ C ⊗ D ⟶ C ⊗ D and counit ε = ε_{C⊗D} : C ⊗ D ⟶ 𝕂 ⊗ K ⟶ 𝕂. (The proof is analogous to the proof of Lemma 8.5.3.)

To describe the comultiplication of a 𝕂-coalgebra in terms of elements we introduce a notation first introduced by Sweedler similar to the notation ∇(a ⊗ b) = ab used for algebras. Instead of Δ(c) = ∑ c_{i} ⊗ c′_{i} we write

Δ(c) = ∑ c_{(1)} ⊗ c_{(2)}.

Observe that only the complete expression on the right hand side makes sense, not the components c_{(1)} or c_{(2)} which are not considered as families of elements of C. This notation alone does not help much in the calculations we have to perform later on. So we introduce a more general notation.

Definition 8.6.3. (Sweedler Notation) Let M be an arbitrary 𝕂-module and C be a 𝕂-coalgebra. Then there is a bijection between all multilinear maps

f : C × … × C ⟶ M

and all linear maps

f′ : C ⊗ … ⊗ C ⟶ M.

These maps are associated to each other by the formula

f(c_{1}, …, c_{n}) = f′(c_{1} ⊗ … ⊗ c_{n}).

For c ∈ C we define

∑ f(c_{(1)}, …, c_{(n)}) := f′(Δ^{n−1}(c)),

where Δ^{n−1} denotes the n − 1-fold application of Δ, for example Δ^{n−1} = (Δ ⊗ 1 ⊗ … ⊗ 1) ∘ (Δ ⊗ 1) ∘ Δ.

In particular we obtain for the bilinear map ⊗ : C × C ∋ (c, d) ↦ c ⊗ d ∈ C ⊗ C

∑ c_{(1)} ⊗ c_{(2)} = Δ(c),

and for the multilinear map ⊗^{2} : C × C × C ⟶ C ⊗ C ⊗ C

∑ c_{(1)} ⊗ c_{(2)} ⊗ c_{(3)} = (Δ ⊗ 1)Δ(c) = (1 ⊗ Δ)Δ(c).

With this notation one verifies easily

∑ c_{(1)} ⊗ … ⊗ Δ(c_{(i)}) ⊗ … ⊗ c_{(n)} = ∑ c_{(1)} ⊗ … ⊗ c_{(n+1)}

and

\begin{aligned}∑ c_{(1)} ⊗ … ⊗ ϵ(c_{(i)}) ⊗ … ⊗ c_{(n)} & = ∑ c_{(1)} ⊗ … ⊗ 1 ⊗ … ⊗ c_{(n−1)} \\ & = ∑ c_{(1)} ⊗ … ⊗ c_{(n−1)} \end{aligned}

This notation and its application to multilinear maps will also be used in more general contexts like comodules.

Proposition 8.6.4. Let C be a coalgebra and A an algebra. Then the composition f ∗ g := ∇_{A}(f ⊗ g)Δ_{C} defines a multiplication

Hom(C, A) ⊗ Hom(C, A) ∋ f ⊗ g ↦ f ∗ g ∈ Hom(C, A),

such that Hom(C, A) becomes an algebra. The unit element is given by 𝕂 ∋ α ↦ (c ↦ η(αϵ(c))) ∈ Hom(C, A).

Proof. The multiplication of Hom(C, A) obviously is a bilinear map. The multiplication is associative since (f ∗ g) ∗ h = ∇_{A}((∇_{A}(f ⊗ g)Δ_{C}) ⊗ h)Δ_{C} = ∇_{A}(∇_{A} ⊗ 1)((f ⊗g)⊗h)(Δ_{C} ⊗1)Δ_{C} = ∇_{A}(1⊗∇_{A})(f ⊗(g ⊗h))(1⊗Δ_{C})Δ_{C} = ∇_{A}(f ⊗(∇_{A}(g ⊗ h)Δ_{C}))Δ_{C} = f ∗ (g ∗ h). Furthermore it is unitary with unit 1_{Hom(C,A)} = η_{A}ϵ_{C} since η_{A}ϵ_{C} ∗ f = ∇_{A}(η_{A}ϵ_{C} ⊗ f)Δ_{C} = ∇_{A}(η_{A} ⊗ 1_{A})(1_{K} ⊗ f)(ϵ_{C} ⊗ 1_{C})Δ_{C} = f and similarly f ∗ η_{A}ϵ_{C} = f.

Definition 8.6.5. The multiplication ∗ : Hom(C, A)⊗Hom(C, A) ⟶ Hom(C, A) is called convolution.

Corollary 8.6.6. Let C be a 𝕂-coalgebra. Then C^{∗} = Hom_{K}(C, 𝕂) is an 𝕂algebra.

Proof. Use that 𝕂 itself is a 𝕂-algebra.

Remark 8.6.7. If we write the evaluation as C^{∗} ⊗ C ∋ a ⊗ c ↦ ⟨a, c⟩ ∈ 𝕂 then an element a ∈ C^{∗} is completely determined by the values of ⟨a, c⟩ for all c ∈ C. So the product of a and b in C^{∗} is uniquely determined by the formula

⟨a ∗ b, c⟩ = ⟨a ⊗ b, Δ(c)⟩ = ∑ a(c_{(1)})b(c_{(2)}).

The unit element of C^{∗} is ϵ ∈ C^{∗}.

Lemma 8.6.8. Let 𝕂 be a field and A be a finite dimensional 𝕂-algebra. Then A^{∗} = Hom_{K}(A, 𝕂) is a 𝕂-coalgebra.

Proof. Define the comultiplication on C^{∗} by

Δ : A^{∗} ⟶^{∇^{∗}} (A ⊗ A)^{∗} ⟶^{can^{−1}} A^{∗} ⊗ A^{∗}.

The canonical map can : A^{∗}⊗A^{∗} ⟶ (A⊗A)^{∗} is invertible, since A is finite dimensional. By a diagrammatic proof or by calculation with elements it is easy to show that A^{∗} becomes a 𝕂-coalgebra.

Remark 8.6.9. If 𝕂 is an arbitrary commutative ring, then A^{∗} = Hom_{K}(A, 𝕂) is a 𝕂-coalgebra if A is a finitely generated projective 𝕂-module.

Problem 8.6.2. Find sufficient conditions for an algebra A resp. a coalgebra C such that Hom(A, C) becomes a coalgebra with co-convolution as comultiplication.

Definition 8.6.10. Let C be a 𝕂-coalgebra. A left C-comodule is a 𝕂-module M together with a homomorphism δ_{M} : M ⟶ C ⊗ M, such that the diagrams

[figure: C ⊗ M; C ⊗ C ⊗ M; id ⊗δ; M; C ⊗ M; δ; δ; Δ⊗id]

and

[figure: M; δ; C ⊗ M; 𝕂 ⊗ M ≅ M.; ϵ⊗id; id]

commute.

Let ^{C}M and ^{C}N be C-comodules and let f : M ⟶ N be a 𝕂-linear map. The map f is called a homomorphism of comodules if the diagram

[figure: N; C ⊗ N; δ_{N}; M; C ⊗ M; δ_{M}; f; 1⊗f]

commutes.

The left C-comodules and their homomorphisms form the category ^{C}M of comodules.

Let N be an arbitrary 𝕂-module and M be a C-comodule. Then there is a bijection between all multilinear maps

f : C × … × M ⟶ N

and all linear maps

f′ : C ⊗ … ⊗ M ⟶ N.

These maps are associated to each other by the formula

f(c_{1}, …, c_{n}, m) = f′(c_{1} ⊗ … ⊗ c_{n} ⊗ m).

For m ∈ M we define

∑ f(m_{(1)}, …, m_{(n)}, m_{(M)}) := f′(δ^{n}(m)),

where δ^{n} denotes the n-fold application of δ, i.e. δ^{n} = (1 ⊗ … ⊗ 1 ⊗ δ) ∘ (1 ⊗ δ) ∘ δ.

In particular we obtain for the bilinear map ⊗ : C × M ⟶ C ⊗ M

∑ m_{(1)} ⊗ m_{(M)} = δ(m),

and for the multilinear map ⊗^{2} : C × C × M ⟶ C ⊗ C ⊗ M

∑ m_{(1)} ⊗ m_{(2)} ⊗ m_{(M)} = (1 ⊗ δ)δ(c) = (Δ ⊗ 1)δ(m).

Problem 8.6.3. Show that a finite dimensional vector space V is a comodule over the coalgebra V ⊗ V^{∗} as defined in problem 8.11.1 with the coaction δ(v) := ∑ v ⊗ v_{i}^{∗} ⊗ v_{i} ∈ (V ⊗ V^{∗}) ⊗ V where ∑ v_{i}^{∗} ⊗ v_{i} is the dual basis of V in V^{∗} ⊗ V.

Theorem 8.6.11. (Fundamental Theorem for Comodules) Let 𝕂 be a field. Let M be a left C-comodule and let m ∈ M be given. Then there exists a finite dimensional subcoalgebra C′ ⊆ C and a finite dimensional C′-comodule M′ with m ∈ M′ ⊆ M where M′ ⊆ M is a 𝕂-submodule, such that the diagram

[figure: M; C ⊗ M; δ; M′; C′ ⊗ M′; δ′]

commutes.

Corollary 8.6.12. 1. Each element c ∈ C of a coalgebra is contained in a finite dimensional subcoalgebra of C.

2. Each element m ∈ M of a comodule is contained in a finite dimensional subcomodule of M.

Corollary 8.6.13. 1. Each finite dimensional subspace V of a coalgebra C is contained in a finite dimensional subcoalgebra C′ of C.

2. Each finite dimensional subspace V of a comodule M is contained in a finite dimensional subcomodule M′ of M.

Corollary 8.6.14. 1. Each coalgebra is a union of finite dimensional subcoalgebras.

2. Each comodule is a union of finite dimensional subcomodules.

Proof. (of the Theorem) We can assume that m ≠ 0 for else we can use M′ = 0 and C′ = 0.

Under the representations of δ(m) ∈ C ⊗M as finite sums of decomposable tensors pick one

δ(m) = ∑_{i=1}^{s} c_{i} ⊗ m_{i}

of shortest length s. Then the families (c_{i}|i = 1, …, s) and (m_{i}|i = 1, …, s) are linearly independent. Choose coefficients c_{ij} ∈ C such that

Δ(c_{j}) = ∑_{i=1}^{t} c_{i} ⊗ c_{ij}, ∀j = 1, …, s,

by suitably extending the linearly independent family (c_{i}|i = 1, …, s) to a linearly independent family (c_{i}|i = 1, …, t) and t ≥ s.

We first show that we can choose t = s. By coassociativity we have ∑_{i=1}^{s} c_{i} ⊗ δ(m_{i}) = ∑_{j=1}^{s} Δ(c_{j}) ⊗ m_{j} = ∑_{j=1}^{s} ∑_{i=1}^{t} c_{i} ⊗ c_{ij} ⊗ m_{j}. Since the c_{i} and the m_{j} are linearly independent we can compare coefficients and get

δ(m_{i}) = ∑_{j=1}^{s} c_{ij} ⊗ m_{j}, ∀i = 1, …, s (1)

and 0 = ∑_{j=1}^{s} c_{ij} ⊗ m_{j} for i > s. The last statement implies

c_{ij} = 0, ∀i > s, j = 1, …, s.

Hence we get t = s and

Δ(c_{j}) = ∑_{i=1}^{s} c_{i} ⊗ c_{ij}, ∀j = 1, …, s.

Define finite dimensional subspaces C′ = ⟨c_{ij}|i, j = 1, …, s⟩ ⊆ C and M′ = ⟨m_{i}|i = 1, …, s⟩ ⊆ M. Then by (1) we get δ : M′ ⟶ C′ ⊗ M′. We show that m ∈ M′ and that the restriction of Δ to C′ gives a linear map Δ : C′ ⟶ C′ ⊗ C′ so that the required properties of the theorem are satisfied. First observe that m = ∑ ε(c_{i})m_{i} ∈ M′ and c_{j} = ∑ ε(c_{i})c_{ij} ∈ C′. Using coassociativity we get

\begin{aligned}∑_{i,j=1}^{n} c_{i} ⊗ Δ(c_{ij}) ⊗ m_{j} & = ∑_{k,j=1}^{s} Δ(c_{k}) ⊗ c_{kj} ⊗ m_{j} \\ & = ∑_{i,j,k=1}^{s} c_{i} ⊗ c_{ik} ⊗ c_{kj} ⊗ m_{j} \end{aligned}

hence

Δ(c_{ij}) = ∑_{k=1}^{s} c_{ik} ⊗ c_{kj}. (2)

Remark 8.6.15. We give a sketch of a second proof which is somewhat more technical. Since C is a 𝕂-coalgebra, the dual C^{∗} is an algebra. The comodule structure δ : M ⟶ C ⊗ M leads to a module structure by ρ = (ev ⊗1)(1 ⊗ δ) : C^{∗} ⊗ M ⟶ C^{∗}⊗C⊗M ⟶ M. Consider the submodule N := C^{∗}m. Then N is finite dimensional, since c^{∗}m = ∑_{i=1}^{n}⟨c^{∗}, c_{i}⟩m_{i} for all c^{∗} ∈ C^{∗} where ∑_{i=1}^{n} c_{i} ⊗ m_{i} = δ(m). Observe that C^{∗}m is a subspace of the space generated by the m_{i}. But it does not depend on the choice of the m_{i}. Furthermore if we take δ(m) = ∑ c_{i} ⊗ m_{i} with a shortest representation then the m_{i} are in C^{∗}m since c^{∗}m = ∑⟨c^{∗}, c_{i}⟩m_{i} = m_{i} for c^{∗} an element of a dual basis of the c_{i}.

N is a C-comodule since δ(c^{∗}m) = ∑⟨c^{∗}, c_{i}⟩δ(m_{i}) = ∑⟨c^{∗}, c_{i(1)}⟩c_{i(2)} ⊗ m_{i} ∈ C ⊗ C^{∗}m.

Now we construct a subcoalgebra D of C such that N is a D-comodule with the induced coaction. Let D := N ⊗ N^{∗}. By 8.13 N is a comodule over the coalgebra N ⊗ N^{∗}. Construct a linear map ϕ : D ⟶ C by n ⊗ n^{∗} ↦ ∑ n_{(1)}⟨n^{∗}, n_{(N)}⟩. By definition of the dual basis we have n = ∑ n_{i}⟨n_{i}^{∗}, n⟩. Thus we get

(ϕ ⊗ ϕ)Δ_{D}(n ⊗ n^{∗}) = (ϕ ⊗ ϕ)(∑ n ⊗ n_{i}^{∗} ⊗ n_{i} ⊗ n^{∗})
= ∑ n_{(1)}⟨n_{i}^{∗}, n_{(N)}⟩ ⊗ n_{i(1)}⟨n^{∗}, n_{i(N)}⟩
= ∑ n_{(1)} ⊗ n_{i(1)}⟨n^{∗}, n_{i(N)}⟩⟨n_{i}^{∗}, n_{(N)}⟩
= ∑ n_{(1)} ⊗ n_{(2)}⟨n^{∗}, n_{(N)}⟩ = ∑ Δ_{C}(n_{(1)})⟨n^{∗}, n_{(N)}⟩
= Δ_{C}ϕ(n ⊗ n^{∗}).

Furthermore ε_{C}ϕ(n ⊗ n^{∗}) = ε(∑ n_{(1)}⟨n^{∗}, n_{(N)}⟩ = ⟨n^{∗}, ∑ ε(n_{(1)})n_{(N)}⟩ = ⟨n^{∗}, n⟩ = ε(n⊗n^{∗}). Hence ϕ : D ⟶ C is a homomorphism of coalgebras, D is finite dimensional and the image C′ := ϕ(D) is a finite dimensional subcoalgebra of C. Clearly N is also a C′-comodule, since it is a D-comodule.

Finally we show that the D-comodule structure on N if lifted to the C-comodule structure coincides with the one defined on M. We have

δ_{C}(c^{∗}m) = δ_{C}(∑⟨c^{∗}, m_{(1)}⟩m_{(M)}) = ∑⟨c^{∗}, m_{(1)}⟩m_{(2)} ⊗ m_{(M)}

= ∑⟨c^{∗}, m_{(1)}⟩m_{(2)} ⊗ m_{i}⟨m_{i}^{∗}, m_{(M)}⟩ = ∑⟨c^{∗}, m_{(1)}⟩m_{(2)}⟨m_{i}^{∗}, m_{(M)}⟩ ⊗ m_{i} = (ϕ ⊗ 1)(∑⟨c^{∗}, m_{(1)}⟩m_{(M)} ⊗ m_{i}^{∗} ⊗ m_{i}) = (ϕ ⊗ 1)(∑ c^{∗}m ⊗ m_{i}^{∗} ⊗ m_{i})

= (ϕ ⊗ 1)δ_{D}(c^{∗}m).