2a. Hopf algebras

[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

In this chapter we introduce the compact quantum groups. Let us start with the finite case, which is elementary, and easy to explain. The idea will be that of calling “finite quantum groups” the compact quantum spaces [math]G[/math] appearing via a formula of type [math]A=C(G)[/math], with the algebra [math]A[/math] being finite dimensional, and having some suitable extra structure. In order to simplify the presentation, we use the following terminology:

Definition

Given a finite dimensional [math]C^*[/math]-algebra [math]A[/math], any morphisms of type

[[math]] \Delta:A\to A\otimes A [[/math]]

[[math]] \varepsilon:A\to\mathbb C [[/math]]

[[math]] S:A\to A^{opp} [[/math]]

will be called comultiplication, counit and antipode.

The terminology comes from the fact that in the commutative case, [math]A=C(X)[/math], the morphisms [math]\Delta,\varepsilon,S[/math] are transpose to group-type operations, as follows:

[[math]] m:X\times X\to X [[/math]]

[[math]] u:\{.\}\to X [[/math]]

[[math]] i:X\to X [[/math]]

The reasons for using the opposite algebra [math]A^{opp}[/math] instead of [math]A[/math] will become clear in a moment. Now with these conventions in hand, we can formulate:

Definition

A finite dimensional Hopf algebra is a finite dimensional [math]C^*[/math]-algebra [math]A[/math], with a comultiplication, counit and antipode, satisfying

[[math]] (\Delta\otimes id)\Delta=(id\otimes \Delta)\Delta [[/math]]

[[math]] (\varepsilon\otimes id)\Delta=id [[/math]]

[[math]] (id\otimes\varepsilon)\Delta=id [[/math]]

[[math]] m(S\otimes id)\Delta=\varepsilon(.)1 [[/math]]

[[math]] m(id\otimes S)\Delta=\varepsilon(.)1 [[/math]]

along with the condition [math]S^2=id[/math]. Given such an algebra we write [math]A=C(G)=C^*(H)[/math], and call [math]G,H[/math] finite quantum groups, dual to each other.

In this definition everything is standard, except for our choice to use [math]C^*[/math]-algebras in all that we are doing, and also for the last axiom, [math]S^2=id[/math]. This axiom corresponds to the fact that, in the corresponding quantum group, we have:

[[math]] (g^{-1})^{-1}=g [[/math]]

It is possible to prove that this condition is automatic, in the present [math]C^*[/math]-algebra setting. However, this is something non-trivial, and since all this is just a preliminary discussion, not needed later, we have opted for including [math]S^2=id[/math] in our axioms.

For reasons that will become clear in a moment, we say that a Hopf algebra [math]A[/math] as above is cocommutative if, with [math]\Sigma(a\otimes b)=b\otimes a[/math] being the flip, we have:

[[math]] \Sigma\Delta=\Delta [[/math]]

With this convention made, we have the following result, which summarizes the basic theory of finite quantum groups, and justifies the terminology and axioms:

Theorem

The following happen:

• If [math]G[/math] is a finite group then [math]C(G)[/math] is a commutative Hopf algebra, with
  [[math]] \Delta(\varphi)=(g,h)\to \varphi(gh) [[/math]]
  [[math]] \varepsilon(\varphi)=\varphi(1) [[/math]]
  [[math]] S(\varphi)=g\to\varphi(g^{-1}) [[/math]]
  as structural maps. Any commutative Hopf algebra is of this form.
• If [math]H[/math] is a finite group then [math]C^*(H)[/math] is a cocommutative Hopf algebra, with
  [[math]] \Delta(g)=g\otimes g [[/math]]
  [[math]] \varepsilon(g)=1 [[/math]]
  [[math]] S(g)=g^{-1} [[/math]]
  as structural maps. Any cocommutative Hopf algebra is of this form.
• If [math]G,H[/math] are finite abelian groups, dual to each other via Pontrjagin duality, then we have an identification of Hopf algebras [math]C(G)=C^*(H)[/math].

Show Proof

These results are all elementary, the idea being as follows:

(1) The fact that [math]\Delta,\varepsilon,S[/math] satisfy the axioms is clear from definitions, and the converse follows from the Gelfand theorem, by working out the details, regarding [math]\Delta,\varepsilon,S[/math].

(2) Once again, the fact that [math]\Delta,\varepsilon,S[/math] satisfy the axioms is clear from definitions, with the remark that the use of the opposite multiplication [math](a,b)\to a\cdot b[/math] in really needed here, in order for the antipode [math]S[/math] to be an algebra morphism, as shown by:

[[math]] S(gh) =(gh)^{-1} =h^{-1}g^{-1} =g^{-1}\cdot h^{-1} =S(g)\cdot S(h) [[/math]]

For the converse, we use a trick. Let [math]A[/math] be an arbitrary Hopf algebra, as in Definition 2.2, and consider its comultiplication, counit, multiplication, unit and antipode maps. The transposes of these maps are then linear maps as follows:

[[math]] \Delta^t:A^*\otimes A^*\to A^* [[/math]]

[[math]] \varepsilon^t:\mathbb C\to A^* [[/math]]

[[math]] m^t:A^*\to A^*\otimes A^* [[/math]]

[[math]] u^t:A^*\to\mathbb C [[/math]]

[[math]] S^t:A^*\to A^* [[/math]]

It is routine to check that these maps make [math]A^*[/math] into a Hopf algebra. Now assuming that [math]A[/math] is cocommutative, it follows that [math]A^*[/math] is commutative, so by (1) we obtain [math]A^*=C(G)[/math] for a certain finite group [math]G[/math], which in turn gives [math]A=C^*(G)[/math], as desired.

(3) This follows from the discussion in the proof of (2) above.

■

This was for the basics of finite quantum groups, under the strongest possible axioms. It is possible to further build on this, but we will discuss this directly in the compact setting. For more on Hopf algebras, over [math]\mathbb C[/math] as above, or over [math]\mathbb C[/math] with weaker axioms, or over other fields [math]k[/math], we refer to Abe , Chari-Pressley and Majid .

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].