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

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7. The quantum monoid of a quantum space

Problem 3.7.1. If A is a finite dimensional algebra and δ : A ⟶ M(A) ⊗ A the universal cooperation of the Tambara bialgebra on A from the left then τδ : A ⟶ A ⊗ M(A) (with the same multiplication on M(A)) is a universal cooperation of M(A) on A from the right. The comultiplication defined by this cooperation is τΔ : M(A) ⟶ M(A) ⊗ M(A). Thus we have to distinguish between the left and the right Tambara bialgebra on A and we have M_{r}(A) = M_{l}(A)^{cop}.

Now consider the special monoidal diagram scheme D := D[X; m, u]. To make things simpler we assume that Vec is strict monoidal. The category D[X; m, u] has the objects X ⊗ … ⊗ X = X^{⊗n} for all n ∈ ℕ (and I := X^{⊗0}) and the morphisms m : X ⊗ X ⟶ X, u : I ⟶ X and all morphisms formally constructed from m, u, id by taking tensor products and composition of morphisms.

Let A be an algebra with multiplication m_{A} : A ⊗ A ⟶ A and unit u_{A} : 𝕂 ⟶ A. Then ω_{A} : D ⟶ C defined by ω(X) = A, ω(X^{⊗n}) = A^{⊗n}, ω(m) = m_{A} and ω(u) = u_{A} is a strict monoidal functor. If A is finite dimensional then the diagram is finite. We get

Theorem 3.7.1. Let A be a finite dimensional algebra. Then the algebra M(A) coacting universally from the right on A (the right Tambara bialgebra) M(A) and coend(ω_{A}) are isomorphic as bialgebras.

Proof. We have studied the Tambara bialgebra for left coaction f : A ⟶ M(A)⊗ A but here we need the analogue for universal right coaction f : A ⟶ A ⊗ M(A) (see Problem 3.9).

Let B be an algebra and f : A ⟶ A ⊗ B be a homomorphism of algebras. For ω = ω_{A} we define

φ(X^{⊗n}) : ω(X^{⊗n}) = A^{⊗n} ⟶^{f^{⊗n}} A^{⊗n} ⊗ B^{⊗n} ⟶^{1⊗m_{B}^{n}} A^{⊗n} ⊗ B = ω(X^{⊗n}) ⊗ B,

where m_{B}^{n} : B^{⊗n} ⟶ B is the n-fold multiplication on B. The map φ is a natural transformation since the diagrams

[figure: A; A ⊗ B; φ(X); 𝕂; 𝕂 ⊗ B; φ(I); u; 1⊗u]

and

[figure: A ⊗ A; A ⊗ A ⊗ B; φ(X⊗X); A ⊗ A ⊗ B ⊗ B; f⊗f; 1⊗m; m; m⊗m; m⊗1; A ⊗ B; f; 1⊗1; A; A ⊗ B; φ(X)]

commute. Furthermore the following commute

[figure: A^{⊗r} ⊗ A^{⊗s}; A^{⊗r} ⊗ A^{⊗s} ⊗ B ⊗ B; φ(X^{⊗r})⊗φ(X^{⊗s}); A^{⊗r} ⊗ A^{⊗s} ⊗ B^{⊗r} ⊗ B^{⊗s}; A^{⊗(r+s)} ⊗ B^{⊗(r+s)}; A^{⊗(r+s)}; A^{⊗(r+s)} ⊗ B; φ(X^{⊗(r+s)})]

so that φ : ω_{A} ⟶ ω_{A} ⊗ B is a monoidal natural transformation.

Conversely let φ : ω_{A} ⟶ ω_{A} ⊗ B be a natural transformation. Let f := φ(X) : A ⟶ A ⊗ B. Then the following commute

[figure: A ⊗ A; A ⊗ A ⊗ B ⊗ B; f⊗f; =; 1⊗m; A ⊗ A; A ⊗ A ⊗ B; φ(X⊗X); m; m⊗1; A; A ⊗ B; f]

and

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; =; 𝕂; 𝕂 ⊗ B; u; u⊗1; A; A ⊗ B.; f]

Hence f : A ⟶ A ⊗ B is a homomorphism of algebras.

Thus we have defined an isomorphism

𝕂Alg(A, A ⊗ B) ≅ Nat^{⊗}(ω_{A}, ω_{A} ⊗ B)

that is natural in B. If A is finite dimensional then the left hand side is represented by the Tambara bialgebra M_{r}(A) and the right hand side by the bialgebra coend(ω_{A}). Thus both bialgebras must be isomorphic.

Corollary 3.7.2. There is a unique isomorphism of bialgebras M_{r}(A) ≅ coend(ω_{A}) such that the diagram

[figure: A; A ⊗ M_{r}(A); A ⊗ coend(ω_{A})]

commutes

Proof. This is a direct consequence of the universal property.

Thus the Tambara bialgebra that represents the universal quantum monoid acting on a finite quantum space may be reconstructed by the Tannaka-Krein reconstruction from representation theory. Similar reconstructions can be given for more complicated quantum spaces such as so called quadratic quantum spaces.