q-deformation of the permutation group? The only definition of a quantum group I know of involves q-deforming the relation $EF-FE=H$ or for SL(2):
\[ \left[ \left( \begin{array}{cc} 0 & 1 \\\\ 0 & 0  \end{array} \right), \left( \begin{array}{cc} 0 & 0 \\\\ 1 & 0  \end{array} \right) \right] =
\left( \begin{array}{cr} 1 & 0 \\\\ 0 & -1  \end{array} \right) \]
All the axioms I have seen are very confusing and don't help me with much.  I also get the sense, these should be called 'quantum lie algebras' rather than quantum groups.  And I never understood the point of co-commutativity.

For now, what does a q-deformation of the permutation group look like? Or the dihedral group?
 A: The algebras you are looking for are called Iwahori-Hecke algebras.  In the case of the symmetric groups the Iwahori-Hecke algebras are generated by `transpositions' $T_i$ which satisfy the braid relations but don't square to zero; instead there is a relation which looks like
$$T_i^2 = qT_i + (1-q)$$
I'd recommend you read up on the monoidal category of modules for a Hopf algebra.  The various properties of a Hopf algebra determine properties of its module category.  For instance if the coproduct is cocommutative then the category of modules is symmetric monoidal.  Many of these q-deformations aren't cocommutative but their module categories still have structure, they become braided monoidal.
Once I understood this everything became much clearer, good luck!
A: In 


*

*Wang, Shuzhou. Quantum symmetry groups of finite spaces. Comm. Math. Phys. 195 (1998), no. 1, 195--211. MR1637425 (99h:58014), link
a quantum version of the symmetric group $\mathbb{S}_n$ is defined. 
Let me sketch Wang's construction. 
Let $u_{ij}$ be the characteristic function of the set of $\sigma\in\mathbb{S}_n$ such that $\sigma(j)=i$. 
Assume that all entries $u_{ij}$ are projections, and
on each row and column of $u=(u_{ij})$ these projections are orthogonal, and sum up to $1$.
Then the commutative $C^*$-algebra generated by these $u$ is $C(\mathbb{S}_n)$.
Now drop the commutativity condition and let $A_s(n)$ be the $C^*$-algebra generated by all the $u_{ij}$. Then we have a quantum analogue of $\mathbb{S}_n$. 
It turns out that $A_s(n)$ is a finitely generated Hopf algebra.
The group $\mathbb{S}_n$ acts on an set $X=[1,2,...,n]$ with $|X|=n$. The corresponding action map $(i,\sigma)\mapsto \sigma(i)$ gives by transposition a certain morphism $\alpha$ ($\alpha$ is called coaction). This coaction can be expressed as 
$\alpha(\delta_i)=\sum\delta_j\otimes u_{ji}$. Furthermore, $\alpha$ is a sort of universal coaction. 
It is possible to prove that the following diagram is commutative
$$
\begin{array}{ccc}
C(X) & \to & C(X)\otimes A_s(n)\\\\
\downarrow & & \downarrow\\\\
C(X) & \to & C(X)\otimes C(\mathbb{S}_n)
\end{array}
$$
Furthermore, $C(\mathbb{S}_n)=A_s(n)$ if $n=1,2,3$. For $n\geq4$, $A_s(n)$ is not commutative and infinite dimensional.
For a nice survey about quantum permutation groups and some applications see the following paper:


*

*Banica, Teodor; Bichon, Julien; Collins, Benoît. Quantum permutation groups: a survey. Noncommutative harmonic analysis with applications to probability, 13--34, Banach Center Publ., 78, Polish Acad. Sci. Inst. Math., Warsaw, 2007. MR2402345 (2009f:46094), link
For a quantum version of the automorphism group of finite graphs (and a quantum version of the dihedral group $\mathbb{D}_4$):


*

*Bichon, Julien. Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc. 131 (2003), no. 3, 665--673 (electronic). MR1937403 (2003j:16049), link
A complete classification of quantum permutation groups acting on 4 points was given in:


*

*Banica, Teodor; Bichon, Julien. Quantum groups acting on 4 points. J. Reine Angew. Math. 626 (2009), 75--114. MR2492990 (2010c:46153), link
A: Consider the braid group $B_n$ on $n$ strands. This is generated by $s_1,\cdots, s_{n-1}$ with relations $s_is_j=s_js_i$ if $\mid i-j\mid \geq 2$ and $s_is_js_i=s_js_is_j$ otherwise. The group $B_n$  has a representation called the Burau representation with parameter $q$ given on the generators $s_i$ as follows. If $e_1\cdots ,e_{n-1}$ is the standard basis of ${\mathbb Z}^{n-1}$ then $s_i(e_i)=(-q)e_i$, $s_i(e_{i+1})= e_{i+1}+e_i$ and $s_i(e_{i-1})=e_{i-1}+qe_i$. When you specialise $q=1$, the image is the symmetric group $S_n$. So you may think of the Burau representation at the parameter $q$ as a defomration of the permutation group $S_n$. 
