7. Bialgebras

Definition 1.7.1. 1. A bialgebra (B, ∇, η, Δ, ϵ) consists of an algebra (B, ∇, η) and a coalgebra (B, Δ, ϵ) such that the diagrams

[figure: B ⊗ B; B ⊗ B ⊗ B ⊗ B; Δ⊗Δ; 1⊗τ⊗1; ∇; B; B ⊗ B; Δ; B ⊗ B ⊗ B ⊗ B; ∇⊗∇]

and

[figure: 𝕂; B; η; η⊗η; B; B ⊗ B; Δ; B ⊗ B; B; ∇; 𝕂; ϵ⊗ϵ; ϵ; 𝕂; 𝕂; id; B; η; ϵ]

commute, i.e. Δ and ϵ are homomorphisms of algebras resp. ∇ and η are homomorphisms of coalgebras.

2. Given bialgebras A and B. A map f : A ⟶ B is called a homomorphism of bialgebras if it is a homomorphism of algebras and a homomorphism of coalgebras.

3. The category of bialgebras is denoted by 𝕂-Bialg.

Problem 1.1. 1. Let (B, ∇, η) be an algebra and (B, Δ, ε) be a coalgebra. The following are equivalent:

a) (B, ∇, η, Δ, ε) is a bialgebra.

b) Δ : B ⟶ B ⊗ B and ε : B ⟶ 𝕂 are homomorphisms of 𝕂-algebras.

c) ∇ : B ⊗ B ⟶ B and η : 𝕂 ⟶ B are homomorphisms of 𝕂-coalgebras.

2. Let B be a finite dimensional bialgebra over field 𝕂. Show that the dual space B^{∗} is a bialgebra.

One of the most important properties of bialgebras B is that the tensor product over 𝕂 of two B-modules or two B-comodules is again a B-module.

Proposition 1.7.2. 1. Let B be a bialgebra. Let M and N be left B-modules. Then M ⊗_{𝕂} N is a B-module by the map

B ⊗ M ⊗ N ⟶^{Δ⊗1} B ⊗ B ⊗ M ⊗ N ⟶^{1⊗τ⊗1} B ⊗ M ⊗ B ⊗ N ⟶^{μ⊗μ} M ⊗ N.

2. Let B be a bialgebra. Let M and N be left B-comodules. Then M ⊗_{𝕂} N is a B-comodule by the map

M ⊗ N ⟶^{δ⊗δ} B ⊗ M ⊗ B ⊗ N ⟶^{1⊗τ⊗1} B ⊗ B ⊗ M ⊗ N ⟶^{∇⊗1} B ⊗ M ⊗ N.

3. 𝕂 is a B-module by the map B ⊗ 𝕂 ≅ B ⟶^{ε} 𝕂.

4. 𝕂 is a B-comodule by the map 𝕂 ⟶^{η} B ≅ B ⊗ 𝕂.

Proof. We give a diagrammatic proof for 1. The associativity law is given by

[figure: B ⊗ B ⊗ M ⊗ N; B ⊗ B ⊗ B ⊗ M ⊗ N; 1⊗Δ⊗1⊗1; B ⊗ B ⊗ M ⊗ B ⊗ N; 1⊗1⊗τ⊗1; B ⊗ M ⊗ N; 1⊗μ⊗μ; B ⊗ B ⊗ B ⊗ B ⊗ M ⊗ N; B ⊗ B ⊗ B ⊗ M ⊗ B ⊗ N; 1⊗1⊗1⊗τ⊗1; B ⊗ B ⊗ M ⊗ N; 1⊗1⊗μ⊗μ; B ⊗ B ⊗ B ⊗ B ⊗ M ⊗ N; B ⊗ B ⊗ M ⊗ B ⊗ B ⊗ N; 1⊗1⊗τ(B⊗B,M)⊗1; B ⊗ M ⊗ B ⊗ N; 1⊗μ⊗1⊗μ; B ⊗ M ⊗ N; B ⊗ B ⊗ M ⊗ N; Δ⊗1⊗1; B ⊗ M ⊗ B ⊗ N; 1⊗τ⊗1; M ⊗ N; μ⊗μ; ∇⊗1⊗1; Δ⊗1⊗1⊗1⊗1; Δ⊗1⊗1⊗1⊗1; Δ⊗1⊗1; 1⊗τ⊗1⊗1⊗1; 1⊗τ(B,B⊗M)⊗1⊗1; 1⊗τ⊗1; ∇⊗∇⊗1⊗1; ∇⊗1⊗∇⊗1; μ⊗μ]

The unit law is the commutativity of

[figure: M ⊗ N ≅ 𝕂 ⊗ M ⊗ N; B ⊗ M ⊗ N; η⊗1⊗1; 𝕂 ⊗ 𝕂 ⊗ M ⊗ N; B ⊗ B ⊗ M ⊗ N; η⊗η⊗1⊗1; M ⊗ N ≅ 𝕂 ⊗ M ⊗ 𝕂 ⊗ N; B ⊗ M ⊗ B ⊗ N; η⊗1⊗η⊗1; ≅; 1⊗τ⊗1; Δ⊗1⊗1; 1⊗τ⊗1; μ⊗μ; =; M ⊗ N; 1]

The corresponding properties for comodules follows from the dualized diagrams. The module and comodule properties of 𝕂 are easily checked.

Definition 1.7.3. 1. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let A be a left B-module with structure map μ : B ⊗ A ⟶ A. Let furthermore (A, ∇_{A}, η_{A}) be an algebra such that ∇_{A} and η_{A} are homomorphisms of B-modules. Then (A, ∇_{A}, η_{A}, μ) is called a B-module algebra.

2. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let C be a left B-module with structure map μ : B ⊗C ⟶ C. Let furthermore (C, Δ_{C}, ε_{C}) be a coalgebra such that Δ_{C} and ε_{C} are homomorphisms of B- modules. Then (C, Δ_{C}, ε_{C}, μ) is called a B-module coalgebra.

3. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let A be a left B-comodule with structure map δ : A ⟶ B ⊗ A. Let furthermore (A, ∇_{A}, η_{A}) be an algebra such that ∇_{A} and η_{A} are homomorphisms of B-comodules. Then (A, ∇_{A}, η_{A}, δ) is called a B-comodule algebra.

4. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let C be a left B-comodule with structure map δ : C ⟶ B ⊗ C. Let furthermore (C, Δ_{C}, ε_{C}) be a coalgebra such that Δ_{C} and ε_{C} are homomorphisms of B-comodules. Then (C, Δ_{C}, ε_{C}, δ) is called a B-comodule coalgebra.

Remark 1.7.4. If (C, Δ_{C}, ε_{C}) is a 𝕂-coalgebra and (C, μ) is a B-module, then (C, Δ_{C}, ε_{C}, μ) is a B-module coalgebra iff μ is a homomorphism of 𝕂-coalgebras.

If (A, ∇_{A}, η_{A}) is a 𝕂-algebra and (A, δ) is a B-comodule, then (A, ∇_{A}, η_{A}, δ) is a B-comodule algebra iff δ is a homomorphism of 𝕂-algebras.

Similar statement for module algebras or comodule coalgebras do not hold.