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

58

6. Quantum Automorphism Groups

Lemma 2.6.1. The category 𝕂-Alg of 𝕂-algebras has arbitrary coproducts.

Proof. This is a well known fact from universal algebra. In fact all equationally defined algebraic categories are complete and cocomplete. We indicate the construction of the coproduct of a family (A_{i}|i ∈ I) of 𝕂-algebras.

Define ∐_{i∈I} A_{i} := T (⨁_{i∈I} A_{i})/L where T denotes the tensor algebra and where L is the two sided ideal in T (⨁_{i∈I} A_{i}) generated by the set

J := {ιj_{k}(x_{k}y_{k}) − ι(j_{k}(x_{k}))ι(j_{k}(y_{k})), 1_{T (⨁ A_{i})} − ιj_{k}(1_{A_{k}})|x_{k}, y_{k} ∈ A_{k}, k ∈ I}.

Then one checks easily for a family of algebra homomorphisms (f_{k} : A_{k} ⟶ B|k ∈ I) that the following diagram gives the required universal property

[figure: A_{k}; ⨁ A_{i}; j_{k}; T (⨁ A_{i}); ι; T (⨁ A_{i})/L; ν; B; f_{k}; f; f′; f̄]

Corollary 2.6.2. The category of bialgebras has finite coproducts.

Proof. The coproduct ∐ B_{i} of bialgebras (B_{i}|i ∈ I) in 𝕂-Alg is an algebra. For the diagonal and the counit we obtain the following commutative diagrams

[figure: B_{k} ⊗ B_{k}; ∐ B_{i} ⊗ ∐ B_{i}; j_{k}⊗j_{k}; B_{k}; ∐ B_{i}; j_{k}; Δ_{k}; ∃_{1}Δ]

[figure: B_{k}; ∐ B_{i}; j_{k}; ε_{k}; 𝕂; ∃_{1}ε]

since in both cases ∐ B_{i} is a coproduct in 𝕂-Alg. Then it is easy to show that these homomorphisms define a bialgebra structure on ∐ B_{i} and that ∐ B_{i} satisfies the universal property for bialgebras.

Theorem 2.6.3. Let B be a bialgebra. Then there exists a Hopf algebra H(B) and a homomorphism of bialgebras ι : B ⟶ H(B) such that for every Hopf algebra H and for every homomorphism of bialgebras f : B ⟶ H there is a unique homomorphism of Hopf algebras g : H(B) ⟶ H such that the diagram

[figure: B; H(B); ι; f; H; g]

commutes.

Proof. Define a sequence of bialgebras (B_{i}|i ∈ ℕ) by

\begin{matrix}B_{0} := B, \\ B_{i+1} := B_{i}^{opcop}, i ∈ ℕ. \end{matrix}

Let B′ be the coproduct of the family (B_{i}|i ∈ ℕ) with injections ι_{i} : B_{i} ⟶ B′. Because B′ is a coproduct of bialgebras there is a unique homomorphism of bialgebras S′ : B′ ⟶ B′^{opcop} such that the diagrams

[figure: B_{i+1}^{opcop}; B′^{opcop}; ι_{i+1}; B_{i}; B′; ι_{i}; id; S′]

commute.

Now let I be the two sided ideal in B′ generated by

{(S′ ∗ 1 − uε)(x_{i}), (1 ∗ S′ − uε)(x_{i})|x_{i} ∈ ι_{i}(B_{i}), i ∈ ℕ}.

I is a coideal, i.e. ε_{B′}(I) = 0 and Δ_{B′}(I) ⊆ I ⊗ B′ + B′ ⊗ I.

Since ε_{B′} and Δ_{B′} are homomorphisms of algebras it suffices to check this for the generating elements of I. Let x ∈ B_{i} be given. Then we have ε((1 ∗ S′)ι_{i}(x)) = ε(∇(1⊗S′)Δι_{i}(x)) = ∇_{𝕂}(ε⊗εS′)(ι_{i}⊗ι_{i})Δ_{i}(x) = (ει_{i}⊗ει_{i})Δ_{i}(x) = ε_{i}(x) = ε(uει_{i}(x)). Symmetrically we have ε((S′ ∗ 1)ι_{i}(x)) = ε(uει_{i}(x)). Furthermore we have

Δ((1 ∗ S′)ι_{i}(x))
= Δ∇(1 ⊗ S′)Δι_{i}(x)
= (∇ ⊗ ∇)(1 ⊗ τ ⊗ 1)(Δ ⊗ Δ)(1 ⊗ S′)(ι_{i} ⊗ ι_{i})Δ_{i}(x)
= (∇ ⊗ ∇)(1 ⊗ τ ⊗ 1)(Δ ⊗ τ(S′ ⊗ S′)Δ)(ι_{i} ⊗ ι_{i})Δ_{i}(x)
= ∑(∇ ⊗ ∇)(1 ⊗ τ ⊗ 1)(ι_{i}(x_{(1)}) ⊗ ι_{i}(x_{(2)}) ⊗ S′ι_{i}(x_{(4)}) ⊗ S′ι_{i}(x_{(3)}))
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(4)}) ⊗ ι_{i}(x_{(2)})S′ι_{i}(x_{(3)})
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′)ι_{i}(x_{(2)}).

Hence we have

Δ((1 ∗ S′−uε)ι_{i}(x))
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′)ι_{i}(x_{(2)}) − Δuει_{i}(x)
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ ((1 ∗ S′) − uε)ι_{i}(x_{(2)})
+ ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ uει_{i}(x_{(2)}) − Δuει_{i}(x)
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′ − uε)ι_{i}(x_{(2)})
+ ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(2)}) ⊗ 1_{B′} − uει_{i}(x) ⊗ 1_{B′}
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′ − uε)ι_{i}(x_{(2)})
+(1 ∗ S′ − uε)ι_{i}(x) ⊗ 1_{B′}
∈ B′ ⊗ I + I ⊗ B′.

Thus I is a coideal and a biideal of B′.

Now let H(B) := B′/I and let ν : B′ ⟶ H(B) be the residue class homomorphism. We show that H(B) is a bialgebra and ν is a homomorphism of bialgebras. H(B) is an algebra and ν is a homomorphism of algebras since I is a two sided ideal. Since I ⊆ Ker(ε) there is a unique factorization

[figure: B′; B′/I; ν; ε′; 𝕂; ε]

where ε : B′/I ⟶ 𝕂 is a homomorphism of algebras. Since Δ(I) ⊆ B′ ⊗ I + I ⊗ B′ ⊆ Ker(ν ⊗ ν : B′ ⊗ B′ ⟶ B′/I ⊗ B′/I) and thus I ⊆ Ker(Δ(ν ⊗ ν)) we have a unique factorization

[figure: B′ ⊗ B′; B′/I ⊗ B′/I; ν⊗ν; B′; B′/I; ν; Δ_{B′}; Δ]

by an algebra homomorphism Δ : B′/I ⟶ B′/I ⊗ B′/I. Now it is easy to verify that B′/I becomes a bialgebra and ν a bialgebra homomorphism.

We show that the map νS′ : B′ ⟶ B′/I can be factorized through B′/I in the commutative diagram

[figure: B′; B′/I; ν; νS′; B′/I; S]

This holds if I ⊆ Ker(νS′). Since Ker(ν) = I it suffices to show S′(I) ⊆ I. We have

S′((S′ ∗ 1)ι_{i}(x)) =
= ∇τ(S′^{2}ι_{i} ⊗ S′ι_{i})Δ_{i}(x)
= ∇τ(S′ ⊗ 1)(ι_{i+1} ⊗ ι_{i+1})Δ_{i}(x)
= ∇(1 ⊗ S′)(ι_{i+1} ⊗ ι_{i+1})τΔ_{i}(x)
= ∇(1 ⊗ S′)(ι_{i+1} ⊗ ι_{i+1})Δ_{i+1}(x)
= (1 ∗ S′)ι_{i+1}(x)

and

S′(uει_{i}(x)) = S′(1)ε_{i}(x) = S′(1)ε_{i+1}(x) = S′(uει_{i+1}(x))

hence we get

S′((S′ ∗ 1 − uε)ι_{i}(x)) = (1 ∗ S′ − uε)ι_{i+1}(x) ∈ I.

This shows S′(I) ⊆ I. So there is a unique homomorphism of bialgebras S : H(B) ⟶ H(B)^{opcop} such that the diagram

[figure: B′^{opcop}; H(B)^{opcop}; ν; B′; H(B); ν; S′; S]

commutes.

Now we show that H(B) is a Hopf algebra with antipode S. By Proposition 2.1.3 it suffices to test on generators of H(B) hence on images νι_{i}(x) of elements x ∈ B_{i}. We have

\begin{matrix}(1 ∗ S)νι_{i}(x) = ∇(ν ⊗ Sν)Δι_{i}(x) = ∇(ν ⊗ ν)(1 ⊗ S′)Δι_{i}(x) = \\ = ν(1 ∗ S′)ι_{i}(x) = νuει_{i}(x) = uενι_{i}(x). \end{matrix}

By Proposition 2.1.3 S is an antipode for H(B).

We prove now that H(B) together with ι := νι_{0} : B ⟶ H(B) is a free Hopf algebra over B. Let H be a Hopf algebra and let f : B ⟶ H be a homomorphism of bialgebras. We will show that there is a unique homomorphism f̄: H(B) ⟶ H such that

[figure: B; H(B); ι; f; H; f̄]

commutes.

We define a family of homomorphisms of bialgebras f_{i} : B_{i} ⟶ H by

\begin{matrix}f_{0} := f, \\ f_{i+1} := S_{H}f_{i}, i ∈ ℕ. \end{matrix}

We have in particular f_{i} = S_{H}^{i}f for all i ∈ ℕ. Thus there is a unique homomorphism of bialgebras f′ : B′ = ∐ B_{i} ⟶ H such that f′ι_{i} = f_{i} for all i ∈ ℕ.

We show that f′(I) = 0. Let x ∈ B_{i}. Then

f′((1 ∗ S′)ι_{i}(x))= f′(∇(1 ⊗ S′)(ι_{i} ⊗ ι_{i})Δ_{i}(x))
= ∑ f′ι_{i}(x_{(1)})f′S′ι_{i}(x_{(2)})
= ∑ f′ι_{i}(x_{(1)})f′ι_{i+1}(x_{(2)})
= ∑ f_{i}(x_{(1)})f_{i+1}(x_{(2)})
= ∑ f_{i}(x_{(1)})Sf_{i}(x_{(2)})
= (1 ∗ S)f_{i}(x) = uεf_{i}(x) = uε_{i}(x)
= f′(uει_{i}(x)).

This together with the symmetric statement gives f′(I) = 0. Hence there is a unique factorization through a homomorphism of algebras f̄: H(B) ⟶ H such that f′ = f̄ν.

The homomorphism f̄: H(B) ⟶ H is a homomorphism of bialgebras since the diagram

[figure: B′/I; ν; f′; f̄; f′⊗f′; B′/I ⊗ B′/I; ν⊗ν; f̄⊗f̄; H; H ⊗ H; Δ_{H}; Δ′; B′; B′ ⊗ B′; Δ]

commutes with the possible exception of the right hand square Δf̄ and (f̄⊗f̄)Δ′. But ν is surjective so also the last square commutes. Similarly we get ε_{H} f̄ = ε_{H(B)}. Thus f̄ is a homomorphism of bialgebras and hence a homomorphism of Hopf algebras.

Remark 2.6.4. In chapter 1 we have constructed universal bialgebras M(A) with coaction δ : A ⟶ M(A) ⊗ A for certain algebras A (see 1.3.12). This induces a homomorphism of algebras

δ′ : A ⟶ H(M(A)) ⊗ A

such that A is a comodule-algebra over the Hopf algebra H(M(A)). If H is a Hopf algebra and A is an H-comodule algebra by ∂ : A ⟶ H ⊗ A then there is a unique homomorphism of bialgebras f : M(A) ⟶ H such that

[figure: A; M(A) ⊗ A; δ; ∂; H ⊗ A; f⊗1]

commutes. Since the f : M(A) ⟶ H factorizes uniquely through f̄: H(M(A)) ⟶ H we get a commutative diagram

[figure: A; H(M(A)) ⊗ A; δ′; ∂; H ⊗ A; f̄⊗1]

with a unique homomorphism of Hopf algebras f̄: H(M(A)) ⟶ H.

This proof depends only on the existence of a universal algebra M(A) for the algebra A. Hence we have

Corollary 2.6.5. Let X be a quantum space with universal quantum space (and quantum monoid) M(X). Then there is a unique (up to isomorphism) quantum group H(M(X)) acting universally on X.

This quantum group H(M(X)) can be considered as the “quantum subgroup of invertible elements” of M(X) or the quantum group of “quantum automorphisms” of X.