Introduction to deformation theory (of algebras)?

Question

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:

1. How and when we can do this and have it still make sense;

2. 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)?

Answer 1

[Gerstenhaber, Murray; Schack, Samuel D. Algebraic cohomology and deformation theory. Deformation theory of algebras and structures and applications (Il Ciocco, 1986), 11--264, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 247, Kluwer Acad. Publ., Dordrecht, 1988. MR0981619 (90c:16016)] is a rather good introduction to the subject. Gerstenhaber papers (the series called On the deformation of rings and algebras) is extremely readable.

As to why should one expect deformation to yield something interesting... I once asked this to Jacques Alev, and he observed that the interest of really interesting things should survive small deformations.

Answer 2

Quoting from the first line of this paper by Barry Mazur (PDF file):

One can learn a lot about a mathematical object by studying how it behaves under small perturbations.

Answer 3

In answer to your last paragraph, a good starting point for deformation theory, not specifically of quantum groups, is the first order deformation theory of associative algebras. Good references for this have been mentioned by Mariano (Gerstenhaber's papers) and Kevin Lin (Kontsevich's notes), but I wanted to add that before even opening them, there are some pleasingly simple exercises you can do to get a feel of the subject. Try extending an associative $k$-linear product on $A$ to a $t$-linear one on $A\otimes (k[t]/t^2)$ and see that what you need is a Hochschild 2-cocycle (definition in homological algebra books, e.g Weibel's); and that the extensions that become trivial after a change of variable are the coboundaries. If you insist on unital products, you'll get cocycles for the reduced Hochschild complex; if you impose commutativity you'll see Harrison cocycles.

One more reference: a paper of Goldman-Millson (link requires MR subscription) which readably explains the DGLA philosphy of char zero deformation theory.

Answer 4

You might try The unbearable lightness of deformation theory by Balázs Szendrői.

Answer 5

I like "Why Deformations are Cohomological" by M Anel

Answer 6

I know almost nothing about quantum groups, but nevertheless I think the first thing to realize is that "deformation" can really be taken as just a synonym for "family". If you are interested in moduli problems, then you are interested in families, and thus deformations. As for why one would be interested in moduli problems... there are many reasons, so maybe you can ask about that in a different question :-)

1. There is a draft of a book on deformation theory by Kontsevich-Soibelman. It's available here. There are also these notes from an old course on deformation theory that Kontsevich taught back when he was at Berkeley in 1994. This is all very much in a similar vein to the references that Mariano has cited. This material is a bit more "modern" and is not quite the same as deformation theory in algebraic geometry, though they do share many characteristics. For deformation theory in algebraic geometry, try taking a look at "Moduli of Curves" by Harris-Morrison, "Deformations of Algebraic Schemes" by Sernesi, or these notes of Hartshorne.

2. One motivation to look at deformations comes from physics, see for example Kontsevich's famous paper on deformation quantization of Poisson manifolds. Another motivation, as I already mentioned, is moduli theory. Even if you are just interested in blahs and not deformations/families/moduli of blahs, it can still be useful to study deformations/families/moduli of blahs. For example, suppose we are interested in some object X. If X has a moduli space of deformations, then we can study how X changes, or how some property of X changes, as we move around in the moduli space. This can then be taken as an invariant of X itself. If X is a smooth projective variety and the property we are looking at is the Hodge structure of X, then this leads to a beautiful theory of variation of Hodge structure, which was developed by Griffiths and others. Finally, deformations themselves can be interesting in their own right: they sometimes have very rich and complex mathematical structures (leading to, for example, applications to knot theory in the case of quantum groups) that we would not see if we just looked at non-deformed objects. This is probably not "obvious" at all, but that's probably largely why it's so awesome.

Answer 7

My answer is more concerned with your second question ``Why this should "obviously" be a construction worth looking at, and why it should be useful and meaningful."

I would quote Arnold who, by saying that Mathematics and physics are the two opposite sides of a same medal, conveys the platonistic idea that the coherence and unity of mathematics comes in fact from the coherence and unity of nature, since it is the natural language to describe it.

If one believes in this, one realizes that most of our mathematics is classical, in the sense that most of mathematical objects come from the study of concepts originated in classical mechanics (geometry, Lie groups, ...). But it is known since the early 20th century that classical mechanics is the shadow of quantum mechanics.

Therefore, there should exist a whole brave new world of quantum mathematics, of which classical mathematics should be the ``semiclassical limit".

The paper of Bayern, Flato, Lichnerowicz and Sternheimer gives a paradigm to explore it : they show that quantum mechanics can be interpreted in terms of deformations of associative algebras of classical commutative algebras of observables on Poisson manifolds.

Therefore, an approach to quantize a mathematical concept is to encode its structure in terms of properties of an algebra of functions, and deform this algebra to a non commutative algebra with similar properties. If you apply it to the algebra of functions on a Lie group, you arrive to the concept of quantum group.

Answer 8

If you haven't already, you might find it worthwhile to read the paper by Drinfeld:

Drinfeld, V.G.: Quantum Groups, Proceedings of ICM (Berkeley 1986) Providence RI American Math. Soc. 1987, 198–820.

I think it is appropriate to a general audience, although not all statements are explained completely.

Answer 9

I first saw an introduction to the deformations of associative algebras in the very nice paper of Braverman and Gaitsgory. I think it's a very good place to start.

Answer 10

Our page at nlab is still very unfinished but at the end of the paget we created a long list of mostly carefully chosen references (majority not that elementary though).

Answer 11

In addition the excellent references listed in the other answers, see these 2007 lecture notes on deformation theory by Doubek, Markl, and Zima.