I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of Section 2 [here](http://www.maths.ed.ac.uk/~tom/belfastXXX.pdf)). As a theoretical physicist, I'm struggling to understand in what sense these objects are quantum!


Has anyone got any references or knowledge which might help answer


1. Where do these odd looking relations come from?
2. In what sense are the matrices quantum (can the non-commutativity in the coordinate ring be understood as emerging from some quantisation procedure...)?
3. Should I think of quantum groups as a controlled mechanism for introducing non-commutativity into coordinate rings in general?


Many thanks in advance for your expertise!



Why are quantum groups so called?
Typically in math "quantum X" means a deformation of "X" which is in some sense "less commutative." So quantum groups should be deformations of groups which are "less commutative." Interpreting this is slightly tricky since groups are already non-commutative, but nonetheless they do have some "commutativity" built in which you can see either by noting:


1. The ring of functions on the group is a commutative ring.
2. The tensor product of representations of the group has a symmetric isomorphism $V \otimes W \rightarrow W \otimes V$.


You can use either of these to motivate versions of quantum groups. A quantum group is:


1. A Hopf algebra which deforms the Hopf algebra of functions on a group, but where the ring structure is non-commutative.
2. Something whose category of representations deforms the category of representations of a group, but where the tensor product structure is not symmetric.


