  Suppose that we have a semisimple, connected, simply connected algebraic
group $G$ with corresponding Lie algebra $\mathfrak{g}$. There is a Hopf
pairing between the universal enveloping algebra $U(\mathfrak{g})$ and the
coordinate ring $O(G)$. By introducing a parameter $q$, we can consider quantum
deformations $U_q(\mathfrak{g})$ and $O_q(G)$ respectively, between which there
again exists a Hopf pairing. We show that the category of crystals associated
with $U_q(\mathfrak{g})$ is a monoidal category. We define subgroups of
$U_q(\mathfrak{g})$ to be right coideal subalgebras, and subgroups of $O_q(G)$
to be quotient left $O_q(G)$-module coalgebras. Furthermore, we discuss a
categorical approach to subgroups of quantum groups which we hope will provide
us with a link to crystal basis theory.
