Quantum groups and $q$-deformed $SU(2)$ I am looking for a self-contained introduction to quantum groups and $SU(2)_q$ in particular, which a person with background in relativity and particle physics would understand. It has to contain the basic definitions and some representation theory. (I know that there are a finite amount of irreps, but I don't know how to classify them.) 
 A: For a physicist, who would rather learn from shared examples, you might look at a mini-review of mine, paradigms of quantum algebras, (C K Zachos, Elementary Paradigms of Quantum Algebras,    Contemporary Mathematics 134 (1992) 351-377, (Deformation Theory and Quantum Groups with Applications to Mathematical Physics,  J Stasheﬀ & M Gerstenhaber (eds), AMS)... contains the missing figure of p 8 of the above weblink provided.)
The basic theme of this type of presentation (which you might as well always keep in mind when reading turgid abstract math texts!) is to connect the deformed algebras to the undeformed ones through simple, intuitive maps, and 
investigate the (sometimes singular) limits of the deformation parameters going to, e.g., a root of unity, as the multiplets break up. In the above link, this is done on pp 7-8, but the "picture worth a thousand words" is missing--here it is:


My advice, however, bearing repeating!, is that much of the culture shock you encounter is mere language: always test terms like coproduct, antipode, etc... with, first q=1, and then a deforming map (might start from deformed oscillators), before delving into recondite structures. They very often are trivialities, dressed up in heavy artillery garb---I've had mathematicians writing entire papers riffing on all-but-trivial deforming maps I and my collaborators have invented!
A: One good book that I've found is the one by Christian Kassel simply titled, Quantum Groups. 
They're basically used as a rigorous algebraic construction of TQFTs, which is where their physical interest lies as the original one was constructed by Witten by path-integral at the physics level of rigour. 
