Exercises

BBot

Apr 20'25

Work out the details for the Peter-Weyl decomposition formula

[[math]] \mathcal C(G)=\bigoplus_{v\in Irr(G)}M_{\dim(v)}(\mathbb C) [[/math]]

in an explicit way, for some finite non-abelian groups, of your choice.

BBot

Apr 20'25

Given a finite group [math]G[/math], setting [math]A=C(G)[/math], prove that the maps

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

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

[[math]] S:A\to A [[/math]]

which are transpose to the multiplication [math]m:G\times G\to G[/math], unit [math]u:\{.\}\to G[/math] and inverse map [math]i:G\to G[/math], are subject to the following conditions

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

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

in usual tensor product notation, along with the extra condition [math]S^2=id[/math].

BBot

Apr 20'25

Given a finite group [math]H[/math], setting [math]A=C^*(H)[/math], prove that the maps

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

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

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

given by the formulae [math]\Delta(g)=g\otimes g[/math], [math]\varepsilon(g)=1[/math], [math]S(g)=g^{-1}[/math] and linearity, are subject to the same conditions as above, namely

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

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

in usual tensor product notation, along with the extra condition [math]S^2=id[/math].

BBot

Apr 20'25

Let us call finite Hopf algebra a finite dimensional [math]C^*[/math]-algebra, with maps as follows, called comultiplication, counit and antipode,

[[math]] \Delta:A\to A\otimes A\quad,\quad \varepsilon:A\to\mathbb C\quad,\quad S:A\to A^{opp} [[/math]]

satisfying the following conditions, which are those found above,

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

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

along with [math]S^2=id[/math]. Prove that if [math]G,H[/math] are finite abelian groups, dual to each other via Pontrjagin duality, then we have an identification of Hopf algebras as follows,

[[math]] C(G)=C^*(H) [[/math]]

and based on this, go ahead and formally write any finite Hopf algebra as

[[math]] A=C(G)=C^*(H) [[/math]]

and call [math]G,H[/math] finite quantum groups, dual to each other.

BBot

Apr 20'25

Given a finite dimensional Hopf algebra [math]A[/math], prove that its dual [math]A^*[/math] is a Hopf algebra too, with structural 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]]

Also, check that [math]A[/math] is commutative if and only if [math]A^*[/math] is cocommutative, and also discuss what happens in the cases [math]A=C(G)[/math] and [math]A=C^*(H)[/math], with [math]G,H[/math] being finite groups.

BBot

Apr 20'25

Develop a theory of compact and discrete quantum groups, generalizing at the same time the theory of usual compact and discrete groups, and the Pontrjagin duality for them, in the abelian case, and the theory of finite quantum groups, and the abstract duality for them, developed in the above series of exercises.