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

105

8. Reconstruction and C-categories

Now we show that an arbitrary coalgebra C can be reconstructed by the methods introduced above from its (co-)representations or more precisely from the underlying functor ω : Comod-C ⟶ Vec. In this case one can not use the usual construction of coend(ω) that is restricted to finite dimensional comodules.

The following Theorem is an example that shows that the restriction to finite dimensional comodules in general is too strong for Tannaka reconstruction. There may be universal coendomorphism bialgebras for more general diagrams. On the other hand the following Theorem also holds if one only considers finite dimensional corepresentations of C. However the proof then becomes somewhat more complicated.

Definition 3.8.1. Let C be a monoidal category. A category D together with a bifunctor ⊗ : C ×D ⟶ D and natural isomorphisms β : (A⊗B)⊗M ⟶ A⊗(B ⊗M), η : I ⊗ M ⟶ M is called a C-category if the following diagrams commute

[figure: ((A ⊗ B) ⊗ C) ⊗ M; (A ⊗ (B ⊗ C)) ⊗ M; α(A,B,C)⊗1; A ⊗ ((B ⊗ C) ⊗ M); β(A,B⊗C,M); β(A⊗B,C,M); 1⊗β(B,C,M); (A ⊗ B) ⊗ (C ⊗ M); A ⊗ (B ⊗ (C ⊗ M)); β(A,B,C⊗M)]

[figure: (A ⊗ I) ⊗ M; A ⊗ (I ⊗ M); β(A,I,M); A ⊗ M; ρ(A)⊗1; 1⊗η(M)]

A C-category is called strict if the morphisms β, η are the identities.

Let (D, ⊗) and (D′, ⊗) be C-categories. A functor F : D ⟶ D′ together with a natural transformation ζ(A, M) : A ⊗ F(M) ⟶ F(A ⊗ M) is called a weak C-functor if the following diagrams commute

[figure: (A ⊗ B) ⊗ F(M); F((A ⊗ B) ⊗ M); ζ; β; F(β); A ⊗ (B ⊗ F(M)); A ⊗ F(B ⊗ M); 1⊗ζ; F(A ⊗ (B ⊗ M)); ζ]

[figure: I ⊗ F(M); F(I ⊗ M); ζ; F(M); η; F(η)]

If, in addition, ζ is an isomorphism then we call F a C-functor. The functor is called a strict C-functor if ζ is the identity morphism.

8. RECONSTRUCTION AND C-CATEGORIES 107

A natural transformation φ : F ⟶ F′ between (weak) C-functors is called a C-transformation if

[figure: A ⊗ F′(M); F′(A ⊗ M); ζ′; A ⊗ F(M); F(A ⊗ M); ζ; 1_{A}⊗φ(M); φ(A⊗M)]

commutes.

Example 3.8.2. Let C be a coalgebra and C := Vec. Then the category Comod-C of right C-comodules is a C-category since N ∈ Comod-C and V ∈ C = Vec implies that V ⊗ N is a comodules with the comodule structure of N.

The underlying functor ω : Comod-C ⟶ Vec is a strict C-functor since we have V ⊗ ω(N) = ω(V ⊗ N). Similarly ω ⊗ M : Comod-C ⟶ Vec is a C-functor since V ⊗ (ω(N) ⊗ M) ≅ ω(V ⊗ N) ⊗ M.

Lemma 3.8.3. Let C be a coalgebra. Let ω : Comod-C ⟶ Vec be the underlying functor. Let φ : ω ⟶ ω ⊗ M be a natural transformation. Then φ is a C-transformation with C = Vec.

Proof. It suffices to show 1_{V} ⊗ φ(N) = φ(V ⊗ N) for an arbitrary comodule N. We show that the diagram

[figure: V ⊗ N; V ⊗ N ⊗ M; 1_{V} ⊗φ(N); V ⊗ N; V ⊗ N ⊗ M; φ(V ⊗N); 1; 1]

commutes. Let (v_{i}) be a basis of V. For an arbitrary vector space W let p_{i} : V ⊗ W ⟶ W be the projections defined by p_{i}(t) = p_{i}(∑_{j} v_{j} ⊗ w_{j}) = w_{i} where ∑_{j} v_{j} ⊗ w_{j} is the unique representation of an arbitrary tensor in V ⊗ W. So we get

t = ∑_{i} v_{i} ⊗ p_{i}(t)

for all t ∈ V ⊗W. Now we consider V ⊗N as a comodule by the comodule structure of N. Then the p_{i} : V ⊗ N ⟶ N are homomorphisms of comodules. Hence all diagrams of the form commute. Expressed in formulas this means φ(N)p_{i}(t) = p_{i}φ(V ⊗ N)(t) for all t ∈ V ⊗ N. Hence we have

[figure: N; N ⊗ M.; φ(N); V ⊗ N; V ⊗ N ⊗ M; φ(V ⊗N); p_{i}; p_{i}⊗M]

\begin{matrix}(1_{V} ⊗ φ(N))(t) = (1_{V} ⊗ φ(N))(∑ v_{i} ⊗ p_{i}(t)) = ∑ v_{i} ⊗ φ(N)p_{i}(t) \\ = ∑_{i} v_{i} ⊗ p_{i}φ(V ⊗ N)(t) = φ(V ⊗ N)(t) \end{matrix}

So we have 1_{V} ⊗ φ(N) = φ(V ⊗ N) as claimed.

We prove the following Theorem only for the category C = Vec of vector spaces. The Theorem holds in general and says that in an arbitrary symmetric monoidal category C the coalgebra C represents the functor C- Nat(ω, ω ⊗ M) ≅ Mor_{C}(C, M) of natural C-transformations.

Theorem 3.8.4. (Reconstruction of coalgebras) Let C be a coalgebra. Let ω : Comod-C ⟶ Vec be the underlying functor. Then C ≅ coend(ω).

Proof. Let M in Vec and let φ : ω ⟶ ω ⊗ M be a natural transformation. We define the homomorphism φ̃: C ⟶ M by φ̃ = (ϵ ⊗ 1)φ(C) using the fact that C is a comodule.

Let N be a C-comodule. Then N is a subcomodule of N ⊗ C by δ : N ⟶ N ⊗ C since the diagram

[figure: N ⊗ C; N ⊗ C ⊗ C; δ⊗1; N; N ⊗ C; δ; δ; 1⊗Δ]

commutes. Thus the following diagram commutes

In particular we have shown that the diagram

commutes.

[figure: N; N ⊗ C; δ; N ⊗ M; N ⊗ C ⊗ M; δ⊗1; φ(N); φ(N⊗C)=1_{N} ⊗φ(C); N ⊗ M; 1; 1⊗ϵ⊗1; 1⊗φ̃]

[figure: ω; ω ⊗ C; δ; φ; ω ⊗ M; 1⊗φ̃]

To show the uniqueness of φ̃ let g : C ⟶ M be another homomorphism with (1 ⊗ g)δ = φ. For c ∈ C we have g(c) = g(ϵ ⊗ 1)Δ(c) = (ϵ ⊗ 1)(1 ⊗ g)Δ(c) = (ϵ ⊗ 1)φ(C)(c) = φ̃(c).

The coalgebra structure from Corollary 3.5.1 is the original coalgebra structure of C. This can be seen as follows. The comultiplication δ : ω ⟶ ω ⊗ C is a natural transformation hence (δ ⊗ 1_{C})δ : ω ⟶ ω ⊗ C ⊗ C is also a natural transformation. As in Corollary 3.5.1 this induced a unique homomorphism Δ : C ⟶ C ⊗ C so that the diagram

[figure: ω ⊗ coend(ω); ω ⊗ coend(ω) ⊗ coend(ω); δ⊗1; ω; ω ⊗ coend(ω); δ; δ; 1⊗Δ]

commutes. In a similar way the natural isomorphism ω ≅ ω ⊗ 𝕂 induces a unique homomorphism ϵ : C ⟶ 𝕂 so that the diagram

[figure: ω; ω ⊗ coend(ω); δ; id_{ω}; ω ⊗ I; 1⊗ϵ]

commutes. Because of the uniqueness these must be the structure homomorphisms of C.

We need a more general version of this Theorem in the next chapter. So let C be a coalgebra. Let ω : Comod-C ⟶ Vec be the underlying functor and δ : ω ⟶ ω ⊗C the universal natural transformation for C ≅ coend(ω).

We use the permutation map τ on the tensor product that gives the natural isomorphism

τ : N_{1} ⊗ T_{1} ⊗ N_{2} ⊗ T_{2} ⊗ … ⊗ N_{n} ⊗ T_{n} ≅ N_{1} ⊗ N_{2} ⊗ … ⊗ N_{n} ⊗ T_{1} ⊗ T_{2} … ⊗ ⊗T_{n}

which is uniquely determined by the coherence theorems and is constructed by suitable applications of the flip τ : N ⊗ T ≅ T ⊗ N.

Let ω^{n} : Comod-C × Comod-C × … × Comod-C ⟶ Vec be the functor ω^{n}(N_{1}, N_{2}, …, N_{n}) = ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}). For notational convenience we abbreviate {N}^{n} := N_{1} ⊗ N_{2} ⊗ … ⊗ N_{n}, similarly {C}^{n} = C ⊗ C ⊗ … ⊗ C and {f}^{n} := f_{1} ⊗ f_{2} ⊗ … ⊗ f_{n}. So we get τ : {N ⊗ T }^{n} ≅ {N}^{n} ⊗ {T }^{n}.

Lemma 3.8.5. Let φ : ω^{n} ⟶ ω^{n} ⊗ M be a natural transformation. Then φ is a C-transformation in the sense that the diagrams

[figure: {V }^{n} ⊗ {N}^{n}; {V }^{n} ⊗ {N}^{n} ⊗ M; {V }^{n}⊗φ(N_{1},... ,N_{n}); {V ⊗ N}^{n}; {V ⊗ N}^{n} ⊗ M; φ(V_{1}⊗N_{1},... ,V_{n}⊗N_{n}); τ; τ⊗M]

commute for all vector spaces V_{i} and C-comodules N_{i}.

Proof. Choose bases {v_{ij}} of the vector spaces V_{i} with corresponding projections p_{ij} : V_{i} ⊗ N_{i} ⟶ N_{i}. Then we have τ(t_{1} ⊗ … ⊗ t_{n}) = ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ p_{1i_{1}}(t_{1}) ⊗ … ⊗ p_{ni_{n}}(t_{n}) so τ = ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ {p}^{n}.

The p_{ij_{i}} : V_{i} ⊗N_{i} ⟶ N_{i} are homomorphisms of C-comodules. Hence the diagrams

[figure: {N}^{n}; {N}^{n} ⊗ M; φ(N_{1},... ,N_{n}); {V ⊗ N}^{n}; {V ⊗ N}^{n} ⊗ M; φ(V_{1}⊗N_{1},... ,V_{n}⊗N_{n}); {p}^{n}; {p}^{n}⊗M]

commute for all choices of {p}^{n} = p_{1i_{1}} ⊗ … ⊗ p_{ni_{n}}.

So we get for all t_{i} ∈ V_{i} ⊗ N_{i}

({V }^{n} ⊗ φ(N_{1}, …, N_{n}))τ(t_{1} ⊗ … ⊗ t_{n}) =

= ({V }^{n} ⊗ φ(N_{1}, …, N_{n}))(∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ p_{1i_{1}}(t_{1}) ⊗ … ⊗ p_{ni_{n}}(t_{n}))

= ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ φ(N_{1}, …, N_{n}){p}^{n}(t_{1} ⊗ … ⊗ t_{n})

= ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ ({p}^{n} ⊗ M)φ(V_{1} ⊗ N_{1}, …, V_{n} ⊗ N_{n})(t_{1} ⊗ … ⊗ t_{n})

= (τ ⊗ M)φ(V_{1} ⊗ N_{1}, …, V_{n} ⊗ N_{n})(t_{1} ⊗ … ⊗ t_{n}).

Theorem 3.8.6. With the notation given above we have

coend(ω^{n}) ≅ C ⊗ C ⊗ … ⊗ C

with the universal natural transformation

δ^{(n)}(N_{1}, N_{2}, …, N_{n}) := τ(δ(N_{1}) ⊗ δ(N_{2}) ⊗ … ⊗ δ(N_{n})) :

ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}) ⟶ ω(N_{1}) ⊗ C ⊗ ω(N_{2}) ⊗ C ⊗ … ⊗ ω(N_{n}) ⊗ C ≅ ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}) ⊗ C ⊗ C ⊗ … ⊗ C.

Proof. We proceed as in the proof of the previous Theorem.

Let M in Vec and let φ : ω^{n} ⟶ ω^{n} ⊗ M be a natural transformation. We define the homomorphism φ̃: C^{n} = ω(C) ⊗ ω(C) ⊗ … ⊗ ω(C) = C ⊗ C ⊗ … ⊗ C ⟶ M by φ̃ = (ε^{n} ⊗ 1_{M})φ(C, …, C) using the fact that C is a comodule.

As in the preceding proof we get that δ : N_{i} ⟶ N_{i} ⊗ C are homomorphisms of C-comodules. Thus the following diagram commutes

[figure: N_{1} ⊗ … ⊗ N_{n}; N_{1} ⊗ C ⊗ … ⊗ N_{n} ⊗ C; δ⊗...⊗δ; N_{1} ⊗ … ⊗ N_{n}⊗; τ; C ⊗ … ⊗ C; N_{1} ⊗ … ⊗ N_{n} ⊗ M; N_{1} ⊗ C ⊗ … ⊗ N_{n} ⊗ C ⊗ M; δ⊗...⊗δ⊗M; N_{1} ⊗ … ⊗ N_{n}⊗; τ⊗M; C ⊗ … ⊗ C ⊗ M; φ(N_{1}⊗...⊗N_{n}); φ(N_{1}⊗C,... ,N_{n}⊗C); N_{1}⊗...⊗N_{n}⊗φ(C,... ,C); 1; {1⊗ε}^{n}⊗1; N_{1} ⊗ … ⊗ N_{n} ⊗ M; 1⊗{ϵ}^{n}⊗1]

Hence we get the commutative diagram

[figure: ω^{n}; ω^{n} ⊗ {C}^{n}; δ^{(n)}; φ; ω^{n} ⊗ M; 1⊗φ̃]

To show the uniqueness of φ̃ let g : C^{n} ⟶ M be another homomorphism with (1_{ω^{n}} ⊗ g)δ^{(n)} = φ. We have g = g(ε^{n} ⊗ 1_{C^{n}})τΔ^{n} = g(ε^{n} ⊗ 1_{C^{n}})δ^{(n)}(C, …, C) = (ε^{n} ⊗ 1_{M})(1_{C^{n}} ⊗ g)δ^{(n)}(C, …, C) = (ε^{n} ⊗ 1_{M})φ(C, …, C) = φ̃.

Now we prove the finite dimensional case of reconstruction of coalgebras.

Proposition 3.8.7. (Reconstruction) Let C be a coalgebra. Let Comod_{0}-C be the category of finite dimensional C-comodules and ω : Comod_{0}-C ⟶ Vec be the underlying functor. Then we have C ≅ coend(ω).

Proof. Let M be in Vec and let φ : ω ⟶ ω ⊗ M be a natural transformation. We define the homomorphism φ̃ : C ⟶ M as follows. Let c ∈ C. Let N be a finite dimensional C-subcomodule of C containing c. Then we define g(c) := (ϵ|_{N} ⊗ 1)φ(N)(c). If N′ is another finite dimensional subcomodule of C with c ∈ N′ and with N ⊆ N′ then the following commutes

[figure: N; N ⊗ M; φ(N); N′; N′ ⊗ M; φ(N′); C ⊗ M; M; ϵ⊗1]

Thus the definition of φ̃(c) is independent of the choice of N. Furthermore φ̃ : N ⟶ M is obviously a linear map. For any two elements c, c′ ∈ C there is a finite dimensional subcomodule N ⊆ C with c, c′ ∈ N e. g. the sum of the finite dimensional subcomodules containing c and c′ separately. Thus φ̃ can be extended to all of C.

The rest of the proof is essentially the same as the proof of the first reconstruction theorem.

The representations allow to reconstruct further structure of the coalgebra. We prove a reconstruction theorem about bialgebras. Recall that the category of Bcomodules over a bialgebra B is a monoidal category, furthermore that the underlying functor ω : Comod-B ⟶ Vec is a monoidal functor. From this information we can reconstruct the full bialgebra structure of B. We have

Theorem 3.8.8. Let B be a coalgebra. Let Comod-B be a monoidal category such that the underlying functor ω : Comod-B ⟶ Vec is a monoidal functor. Then there is a unique bialgebra structure on B that induces the given monoidal structure on the corepresentations.

Proof. First we prove the uniqueness of the multiplication ∇ : B ⊗ B ⟶ B and of the unit η : 𝕂 ⟶ B. The natural transformation δ : ω ⟶ ω ⊗ B becomes a monoidal natural transformation with ∇ : B ⊗ B ⟶ B and η : 𝕂 ⟶ B We show that ∇ and η are uniquely determined by ω and δ.

Let ∇′ : B ⊗ B ⟶ B and η′ : B ⟶ 𝕂 be morphisms that make δ a monoidal natural transformation. The diagrams

and

commute. In particular the following diagrams commute and

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ B ⊗ B; δ(X)⊗δ(Y); ρ; ρ⊗∇′; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ B; δ(X⊗Y)]

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; 1 ⊗ η′; ω(𝕂); ω(𝕂) ⊗ B; δ(𝕂)] [figure: ω(B) ⊗ ω(B); ω(B) ⊗ ω(B) ⊗ B ⊗ B; δ(B)⊗δ(B); ρ; ρ⊗∇′; ω(B ⊗ B); ω(B ⊗ B) ⊗ B; δ(B⊗B)]

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; 1 ⊗ η′; ω(𝕂); ω(𝕂) ⊗ B; δ(𝕂)]

Hence we get ∑ b_{(1)}⊗c_{(1)}⊗b_{(2)}c_{(2)} = ∑ b_{(1)}⊗c_{(1)}⊗∇′(b_{(2)}⊗c_{(2)}) and 1⊗1 = 1⊗η′(1). This implies bc = ∑ ϵ(b_{(1)})ϵ(c_{(1)})b_{(2)}c_{(2)} = ∑ ϵ(b_{(1)})ϵ(c_{(1)})∇′(b_{(2)} ⊗ c_{(2)}) = ∇′(b ⊗ c) and 1 = η′(1).

Now we show the existence of a bialgebra structure. Let B be a coalgebra only and let ω : Comod-B ⟶ Vec be a monoidal functor with ξ : ω(M)⊗ω(N) ⟶ ω(M ⊗N) and ξ_{0} : 𝕂 ⟶ ω(𝕂). First we observe that the new tensor product between the comodules M and N coincides with the tensor product of the underlying vector spaces (up to an isomorphism ξ). Because of the coherence theorems for monoidal categories (that also hold in our situation) we may identify along the maps ξ and ξ_{0}.

We define η := (𝕂 ⟶^{δ(𝕂)} 𝕂 ⊗ B ≅ B) and ∇ := (B ⊗ B ⟶^{δ(B⊗B)} B ⊗ B ⊗ B ⟶^{ϵ⊗ϵ⊗1_{B}}

𝕂 ⊗ 𝕂 ⊗ B ≅ B).

Since the structural morphism for the comodule δ : M ⟶ M ⊗ B is a homomorphism of of B comodules where the comodule structure on M ⊗ B is only given by the diagonal of B that is the C-structure on ω : Comod-B ⟶ Vec we get that also δ(M) ⊗ δ(N) : M ⊗ N ⟶ M ⊗ N ⊗ B is a comodule homomorphism. Hence the first square in the following diagram commutes

[figure: M ⊗ N; M ⊗ B ⊗ N ⊗ B; δ(M)⊗δ(N); M ⊗ N ⊗ B; M ⊗ B ⊗ N ⊗ B ⊗ B; δ(M)⊗δ(N)⊗1_{B}; M ⊗ N ⊗ B ⊗ B; 1⊗τ⊗1; M ⊗ N ⊗ B ⊗ B ⊗ B; 1⊗τ⊗1⊗1; δ(M⊗N); δ(M⊗B⊗N⊗B); 1⊗1⊗δ(B⊗B)]

The second square commutes by a similar reasoning since the comodule structure on M ⊗ B resp. N ⊗ B is given by the diagonal on B hence M ⊗ N can be factored out of the natural (C-)transformation. Now we attach

1_{M} ⊗ 1_{N} ⊗ ϵ ⊗ ϵ ⊗ 1_{B} : M ⊗ N ⊗ B ⊗ B ⊗ B ⟶ M ⊗ N ⊗ B

to the commutative rectangle and obtain δ(M ⊗N) = (1_{M} ⊗1_{N} ⊗∇)(1⊗τ ⊗1)(δ(M)⊗ δ(N). Thus the comodule structure on M ⊗ N is induced by the multiplication ∇ : B ⊗ B ⟶ B defined above.

So the following diagrams commute

[figure: B ⊗ B; B ⊗ B ⊗ B ⊗ B; Δ⊗Δ; B ⊗ B ⊗ B ⊗ B; 1⊗τ⊗1; B ⊗ B; B ⊗ B ⊗ B; δ(B⊗B); B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B; 1⊗1⊗Δ; B; B ⊗ B; Δ; 1_{B}⊗1_{B}; 1⊗∇; δ(B⊗B); δ(B⊗B)⊗1; ∇; ∇⊗1; ϵ⊗ϵ⊗1; ϵ⊗ϵ⊗1⊗1]

[figure: B ⊗ B; B; ∇; B ⊗ B; 𝕂; ϵ⊗ϵ; B ⊗ B ⊗ B; δ(B⊗B); ϵ⊗ϵ⊗1; 1; 1⊗1⊗ϵ; ϵ]

[figure: 𝕂; B; η; 𝕂 ⊗ B; 𝕂 ⊗ B ⊗ B; δ(𝕂)⊗1; B ⊗ B; ≅; B; B ⊗ B; η⊗1; 𝕂 ⊗ B; δ(𝕂); δ(𝕂); ≅; ≅; η⊗1; 1_{B}; η; 1⊗Δ; ≅; Δ]

and

[figure: 𝕂; 𝕂; 1; 𝕂 ⊗ B; δ(𝕂); 1⊗ϵ; η; ϵ; B; ≅]

Hence η and ∇ are coalgebra homomorphisms.

To show the associativity of ∇ we identify along the maps α : (M ⊗ N) ⊗ P ≅ M ⊗(N ⊗P) and furthermore simplify the relevant diagram by fixing that σ represents a suitable permutation of the tensor factors. Then the following commute

[figure: B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B ⊗ B ⊗ B; σ(δ(B)⊗δ(B)⊗δ(B); B ⊗ B ⊗ B; ϵ⊗ϵ⊗ϵ⊗1; B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B ⊗ B ⊗ B; σ(δ(B⊗B)⊗δ(B)); σ(δ(B)⊗δ(B⊗B)); B ⊗ B; ϵ⊗ϵ⊗ϵ⊗1; B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B; δ(B⊗B⊗B); B; ϵ⊗ϵ⊗ϵ⊗1; 1; 1; 1⊗(∇⊗1); 1⊗(1⊗∇); ∇⊗1; 1⊗∇; 1⊗∇; ∇]

The upper row is the identity hence we get the associative law.

For the proof that η has the properties of a unit we must explicitly consider the coherence morphisms λ and ρ By reasons of symmetry we will only show one half of of the unit axiom. This axiom follows from the commutativity of the following diagram

[figure: B; B ⊗ B; δ(B); B ⊗ B ⊗ 𝕂; ρ^{−1}; B ⊗ B; ρ; B; ϵ⊗1; B ⊗ 𝕂; B ⊗ B ⊗ 𝕂 ⊗ B; δ(B)⊗δ(𝕂); B ⊗ 𝕂 ⊗ B ⊗ B; 1⊗τ⊗1; B ⊗ B ⊗ B; ρ⊗1⊗1; B ⊗ B; ϵ⊗1⊗1; B ⊗ 𝕂; B ⊗ 𝕂 ⊗ B; δ(B⊗𝕂); B; B ⊗ B; δ(B); B; ϵ⊗1; ρ^{−1}; 1⊗1⊗δ(𝕂); 1⊗1⊗η; 1⊗η; =; 1⊗1⊗∇; 1⊗∇; ∇; ρ; δ(B)⊗1; ρ⊗1]