By this I mean the specialisation of the quantum group U_q(g) with q a root of unity, and the 'correct' meaning of 'correct' (enclosed in quotations since there isn't necessarily a correct answer) is likely to mean that its category of representations is the 'correct' one.

My suspicion is that I want to take an integral form of U_q(g) defined over ℤ[q^±1/2] and base change to the appropriate ring of integers in a cyclotomic field. Having heard of the 'small quantum group' and Lusztigs algebra U dot (notation in his quantum groups book), I suspect the existence of multiple approaches, which diverge at least when an integral form is desired, and hence turn to mathoverflow for clarification and enlightenment.

Afficionados of this type of question can consider it as a continuation in a sequence initiated by this question on the universal enveloping algebra in positive characteristic.

By this I mean the specialisation of the quantum group U_q(g) with q a root of unity, and the 'correct' meaning of 'correct' (enclosed in quotations since there isn't necessarily a correct answer) is likely to mean that its category of representations is the 'correct' one.

My suspicion is that I want to take an integral form of U_q(g) defined over ℤ[q^±1/2] and base change to the appropriate ring of integers in a cyclotomic field. Having heard of the 'small quantum group' and Lusztigs algebra U dot (notation in his quantum groups book), I suspect the existence of multiple approaches, which diverge at least when an integral form is desired, and hence turn to mathoverflow for clarification and enlightenment.

Afficionados of this type of question can consider it as a continuation in a sequence initiated by this question on the universal enveloping algebra in positive characteristic.