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Quantum subgroups

Let

be a closed or algebraic subgroup of

$\begin{displaymath} I_H=\{f\in F[G] : f\vert _H =0\} \end{displaymath}$

is a Hopf ideal and

$\begin{displaymath} F[G]/I_H\cong F[H] \end{displaymath}$

as Hopf algebras. To put it another way

$\begin{displaymath} H\text{ subgroup of }G \quad\Longleftrightarrow\quad F[G]\to F[H] \quad \text{ Hopf algebra epimorphism} \end{displaymath}$

Alternatively thinking at the infinitesimal level

$\begin{displaymath} \gh \text{ subalgebra of }\gerg \quad\Longleftrightarrow\quad U(\gh)\to U(\gerg)\quad\text{ Hopf algebra monomorphism} \end{displaymath}$

It is therefore natural to say
$\begin{defn} A \textbf{quantum subgroup} of a (global, local, special) quantize... ...bra epimorphism \begin{displaymath} F_q[G]\to F_q[H] \end{displaymath}\end{defn}$
Therefore quantum subgroups correspond to Hopf ideals in

Pawel Witkowski 2006-06-26