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Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like any situation where you take some interesting geometric construction, characterize it in terms of some structure on the corresponding ring of functions, and then ask what happens when you no longer require such algebras to be commutative.

However, most of the people I know who study Hopf algebras study a very particular class of them often called 'quantum groups'. This term very often applies to much more general class of Hopf algebras, so for clarity I will call them Drinfel'd-Jimbo quantum groups, as wikipedia does. I am referring to a very specific q-deformation of the universal enveloping algebra of a semi-simple Lie algebra $g$ (the definition is long, so I will defer to wikipedia).

My question is, what is so special about these Hopf algebras in particular? Why not some other deformation of the Hopf algebra structure on $\mathcal{U}g$? I have seen some cool things you can do with these, such as characterizing crystal bases of modules; but are these the reasons people started studying these in the first place? Is there a natural reason why these are the 'best' deformations of $\mathcal{U}g$?

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6 Answers 6

These Hopf algebras are distinguished by the property that their representation categories admit a braided tensor structure. The space of deformations of the braided tensor structure on these particular categories is one dimensional (I believe this is a result of Drinfeld), so we get a universal family of braided deformations of U(g)-mod by varying q in U_q(g). The symmetric structure in U(g)-mod makes it a special point in this space, and one could argue that U(g)-mod is a "degeneration" of the generic braided behavior. There is unpublished work of Lurie on algebraic groups over the sphere spectrum that lends homotopy-theoretic support to this idea, since the symmetric structure doesn't manifest over the sphere.

Braided structures are important when studying topological (and conformal) field theories, since they describe the local behavior of embedded codimension 2 objects, such as points in a surface or links in a three-manifold. If you like homotopy theory, a braided tensor category is one that admits an action of the E[2] operad, whose spaces are (homotopy equivalent to) configuration spaces of points in the plane. Physically, these are the points where one inserts fields.

In principle, any statement about semisimple groups that can be phrased in braided-commutative (rather than fully commutative) language should be reconfigurable to a statement about these quantum groups. For example, there is a quantum local Langlands program (see the introduction of Gaitsgory's twisted Whittaker paper). Also, the representation theory of U_q(g) is interesting because of its connections to the representation theory of affine algebras and mod p representations (I think Kazhdan, Lusztig, and Bezrukavnikov are among the key names here).

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This tends not to be a satisfying answer to questions of the type of yours... but one can prove that the enveloping algebra of a semisimple Lie algebra has a unique non-trivial deformation as a quasi bialgebra over the ring of formal series up to change of formal parameter and twisting, namely, the Drinfel'd-Jimbo one. This is proven in various forms by various people---for example, the paper by Schnider in "Deformation theory and quantum groups with applications to mathematical physics", edited by Gerstenhaber and Stasheff---or in Ch. Kassel's book.

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4  
Let's agree that if there is only one deformation available, then it's important, even if you didn't otherwise care about deformations. –  Greg Kuperberg Nov 14 '09 at 18:18
    
Surely anything of which there is such a small supply should at least attract some attention! Yet I think the interest in D-J's deformations precedes its uniqueness, which came (I may well be mixing up the chronology...) later as a nice extra confimation that focusing on them was not at all misguided. –  Mariano Suárez-Alvarez Nov 14 '09 at 20:47
    
Yes, I think the DJ definition is from the late 80s, and most of the existence and uniqueness results are from the late 90s and early 00s. –  Theo Johnson-Freyd Nov 14 '09 at 22:10
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Of course, SL(2), for instance, supports more than one nonisomorphic Lie bialgebra structure (see Chari and Pressley, if memory serves), and it is a result originally of Etingof and Khazdan, and improved by Enriquez, that any Lie bialgebra admits a quantization. Actually, Enriquez classifies all quantization functors from \{Lie bialgebras\} to \{Hopf algebras\} that play well with duals and doubles: there is only one up to the multiplicative group of formal power series 1 + O(h). [Well, that's what I remember from the MathSciNet reviews of the papers. I've been meaning to read them.] –  Theo Johnson-Freyd Nov 14 '09 at 22:11
    
The point is that there is an interesting canonical choice of bialgebra structure on semisimple Lie algebras (perhaps up to a normalization, which is probably fixed by picking a Killing form?). –  Theo Johnson-Freyd Nov 14 '09 at 22:13

I think one good answer is Chern-Simons theory; this is a topological quantum field theory which has a path integral definition just using a compact Lie group, but when you extend it to 2- and 1- manifolds, the category of representations of a quantum group show up.

As a general rule, in many cases where one finds some natural construction for the Lie algebra of its universal enveloping algebra, a slight tweak gives the quantum group. Perhaps the best example of this is the theory of Hall algebras, which give the universal enveloping algebra if you just use Euler characteristics, but if you keep motivic informations (for example, by counting points over finite fields) you get a quantum group. Closely related include Lusztig's categorification of the upper half, and Rouquier-Khovanov-Lauda's categorification of the whole universal enveloping algebra. A similar, but slightly tweaked, story is Nakajima's quiver variety construction of the affine quantum group as equivariant K-theory of a quiver variety.

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Could you tell me a reference for the Nakajima construction? –  Dinakar Muthiah Feb 12 '10 at 8:35

I have seen some cool things you can do with these, such as characterizing crystal bases of modules; but are these the reasons people started studying these in the first place?

No. The original motivations came from quantum inverse scattering method, as they are the natural place where the quantum R-matrices live in abstract form (universal R-element); in particular godo representations of quantum groups supply examples of concrete R-matrices. Thus at least one restricts to quasitriangular or coquasitriangular Hopf algebras (cf. nlab entry).

Later it appeared that apart of some deformed integrable models (spin chains and alike), most physics applications belong to quantum groups at root of unity. For example, the Chern-Simons theory which somebody mentions has a Hamiltonian reduction to WZNW model which has roughly speaking a quantum group symmetry. But the representation theory at the root of unity is related to the representation theory of affine Lie algebras. There are many aspects of this, including the hints from deep work of Kazhdan and Lusztig mentioned in other answers, but also hypergeometric pairing

A.Varchenko, "Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups," Advanced Series in Mathematical Physics, Vol. 21, World Scientific (1995)

which can be interpreted as evaluating cohomology cocycles on certain homology cycles; this picture is later extended in many directions. Big step is a deep work of then extremely young Bezrukavnikov, summarized in the joint book Factorizable sheaves and quantum groups with Finkelberg and Schechtman. More recently this is further understood by joint work of Lurie and Gaitsgory, I know of the published part by

Dennis Gaitsgory, Twisted Whittaker model and factorizable sheaves, Selecta Math. (N.S.) 13 (2008), no. 4, 617--659.

the newer part with even more exciting idea mentioned by Scott I did not see in any detailed form yet.

As far as constructing part of the quantum group (say just the Borel part) from Cartan data withouit explicit formulas but somehow a priori has with various degrees of success being attempted by various people. Luzstig, Nakajima and others take an appropriate variant of configuration spaces and look at certain perverse sheaves there (this kind of ideas is not unrelated to the business of Bezrukavnikov and others involving another category of (factorizable) sheaves to study quantum groups). Lyubashenko, Majid, Schauenburg and have tried starting with some simple braided tensor category and few other data to construct the quantum group; some categorical meaning of relations, though still complicated and with much input by hand is in Aguiar's thesis, allowing for some neat generalizations. Rosenberg (here) starts with only a braided tensor category and a finite family of distinguished objects to create a quantum group-like structure and claims without proof that the category of vector spaces with certain easy choice of braiding (imvolving formula q to the power involving Cartan matrix elements) gives Drinfel'd-Jimbo case; all the relations, including Serre relations are for free by general nonsense.

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I forgot to mention that there is a yet newer correspondence between the representations of LOGARITHMIC vertex algebras and of root of unity quantum groups in an improved variant which involves also the quantum dimension zero representations. Thus this correspondence is even better in logaritmic CFT than the usual CFT where the spurious quantum dimension zero representations drop out. See recent papers by Adamović and Milas where some examples are worked out (a general case is just a conjecture). –  Zoran Skoda Feb 25 '10 at 19:25

So, here's some of my own investigation into this. Define the n-th partial flag variety $Fl_n(\mathbb{C}^m)$ to be the set of all n-step partial flags

$$F_0 \subseteq F_1 \subseteq ... \subseteq F_n \subseteq \mathbb{C}^m$$

where we do NOT require the inclusions be proper (this is a variety in a natural way).

There is a natural way to get a representation of the undeformed group $GL(n,k)$ out of this, specifically by taking the Borel-Moore homology $H_*(Fl_n(\mathbb{C}^m),k)$ with coefficients in some field $k$. Define $Z_i$ to be the set of all 'pairs of n-step partial flags $(F,F')$ which are the same except at the ith-step, where $dim(F'_i)=dim(F_i)+1$'.

There are two natural maps $\pi_1, \pi_2:Z_i\rightarrow Fl_n$, each forgetting one of the flags. Then pulling back along $\pi_1$ and pushing forward along $\pi_2$ gives a map $E_i$ on Borel-Moore homology. Similarly, pulling back along $\pi_2$ and pushing forward along $\pi_1$ gives a map $F_i$ on Borel-Moore homology.

Claim: The maps $\{E_i,F_i\}$ generate an action of $GL(n,k)$ on $H_*(Fl_n(\mathbb{C}^m)$. I believe this can be found in "Representation Theory and Complex Geometry" by Chriss and Ginzburg, and this is sketched in Joel Kamnitzer's notes from a summer school here.


So what does this have to do with quantum groups? Well, redo the above construction, except replace the field $\mathbb{C}$ with $\mathbb{F}_q$. Then the partial flag varieties only have a finite set of points, so instead of taking Borel-Moore homology, take $k$-linear combinations of these points $k[Fl_n(\mathbb{F}_q)]$.

You still have push-forward and pullback operators, where pushing-forward is now just summing up values in the fiber. Therefore, you can still define operators $E_i$ and $F_i$, but its not clear what the generate.

My conjecture: $E_i$ and $F_i$ generate the action of the quantum group $GL_q(n,k)$, where q is the value of the deformation parameter.

I strongly suspect this is true, but the only thing I have to base it on is a passing remark of Joel Kamnitzer's. If it is, it gives a pretty natural motivation for looking at these quantum groups, at least in narrow case of q a prime power.

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This is closely related to the Hall algebra stuff I mentioned earlier. I don't know anywhere this is properly written down, but it should be easy to just do Ginzburg's proof a little more carefully. I'll note, this really the decategorification of the Kac-Moody 2-category action on perverse sheaves on those flag varieties, so its implicit in that. –  Ben Webster Nov 16 '09 at 4:45
    
There is a famous paper by Beilinson, Lusztig, and MacPherson where they develop this idea of constructing representations of quantum groups using $\mathbb F_q $ points of partial flag varieties. –  Joel Kamnitzer Jun 15 '11 at 1:37

Conc. the definition of them: If I remember correctly (my excuses if I don't!), Manin mentioned once in an article that quantum groups are so far not really intrinsically defined, but only given in several ways as modifications of other structures (like groups before the invention of abstract algebra were given by examples). Is that still the case?

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