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). 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
Where do these odd looking relations come from?
In what sense are the matrices quantum?
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!