So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of idea of what it's all about, but piecing together what I've read, I understand that the idea is to "deform" a group (Hopf) algebra to one that's not quite as nice but is still very workable.
To a certain extent, I get what's implied by "deformation"; the idea is to take some relations defining our Hopf algebra and introduce a new parameter, which specializes to the classical case at a certain point. What I don't understand is:
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How and when we can do this and have it still make sense;
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Why this should "obviously" be a construction worth looking at, and why it should be useful and meaningful.
The problem is when I look for stuff (in the library catalogue, on the Internet) on deformation theory, everything that turns up is really technical and assumes some familiarity with the basic definitions and intuitions about the subject. Does anyone know of a more basic introduction that can be understood by the "general mathematical audience" and answers (1) and (2)?
Best Answer
Quoting from the first line of this paper by Barry Mazur (PDF file):