Intuition behind the definition of quantum groups

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Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to improve the situation.

The definition of the quantum group I saw is that it is a Hopf algebra given by some explicit generators and relations. Though I have heard that at least the case of the quantum $sl_2$ was motivated by physics, this was not explicit enough. If someone defined the classical (i.e. non-quantum) Lie algebra $gl_n$ or the symmetric group $S_n$ using generators and relations rather than operators acting on a vector space or on a finite set respectively, such a definition would be equally unclear to me. The abstract approach to quantum groups as deformations of a universal enveloping algebras in the class of Hopf algebras is very useful, but still not very intuitive and explicit.

While any clarifying remarks would be appreciated, I can ask the following more specific questions. (1) In simple examples of quantum groups, such as quantum $sl_2, sl_n$, are there "natural" examples of their representations, like the standard representation of the classical $sl_n$, its dual representation and their tensor powers? (2) Are there mathematical problems which are not about quantum groups, but whose solution does require this notion?

This post imported from StackExchange MathOverflow at 2015-04-28 14:42 (UTC), posted by SE-user MKO
retagged Apr 28, 2015
See for instance the book "A Guide to Quantum Groups" by Chari and Pressley. This is related to deformation quantization that "gives" the sheaf a structure of Poisson-Lie group.

This post imported from StackExchange MathOverflow at 2015-04-28 14:42 (UTC), posted by SE-user user40276
A quantum group is a new tensor product on the category of representations. I think that something along the lines of the KZ equations give a coordinate-free construction. At first glance, they just give the braiding, but that's a good start.

This post imported from StackExchange MathOverflow at 2015-04-28 14:42 (UTC), posted by SE-user Ben Wieland
@BenWieland The KZ equations give a new "tensor structure" in the sense that they give the same monoidal functor, but a new braiding AND a new associator. It's really the associator that makes the new tensor structure new in a meaningful way. For example, it changes which are the "associative $G$-algebras".

This post imported from StackExchange MathOverflow at 2015-04-28 14:42 (UTC), posted by SE-user Theo Johnson-Freyd
+1 for the second question.

This post imported from StackExchange MathOverflow at 2015-04-28 14:42 (UTC), posted by SE-user José Figueroa-O'Farrill

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Here is an answer to question (1). I recommend that you split of question (2) as a separate question.

Define the quantum plane to be the "spectrum" of the noncommutative ring $\mathbb K\langle x,y\rangle / (xy = qyx)$, where $\mathbb K$ is some ground commutative ring in which $q$ is invertible (e.g. $\mathbb K = \mathbb C(q)$). So a "point" in this "plane" is determined by the "values" of the two "coordinates" $x$ and $y$. Note that these "values" don't commute: they are "valued" in some space of "noncommutative numbers".

Now, recall that $2\times 2$ matrices act on the usual plane. Are there "$2 \times 2$ matrices" that act on the quantum plane?

Well, in the usual case, a matrix $\bigl( \begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\bigr)$ takes the vector $\bigl( \begin{smallmatrix} x \\ y \end{smallmatrix}\bigr)$ to: $$\begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}.$$ Let's try to keep the same rule. For this to work, the coordinates of the new point need to satisfy the same "$q$-mmutation" (as opposed to "commutation") law as $\{x,y\}$, which is to say:

$$(ax + by)(cx+ dy) = q(cx+dy)(ax+by).$$

If we suppose that variables $a,b,c,d$ commute with variables $x,y$, then we get:

$$ac = qca \quad bd = qdb \quad ad + q^{-1}bc = qcb + da. \quad\quad (\star)$$

What about, say, how $a$ and $b$ commute? Well, I should have mentioned that the quantum plane has a "dual quantum plane". Recall that in classical linear algebra, the dual to the space of column vectors is the space of row vectors. Let's, then, take the dual quantum plane to consist of row vectors $(v,w)$ whose coordinates satisfy $vw = qwv$. Then by letting quantum matrices act from the right on quantum row vectors, you can compute:

$$ab = qba \quad cd = qdc \quad ad + q^{-1}cb = qbc + da. \quad\quad (\star\star)$$

Note that the last lines in each are slightly different. Together, the rules $(\star)$ and $(\star\star)$ define the space of "quantum $2\times 2$ matrices". Let me simplify the last two rules:

$$bc = cb \quad ad - da = (q-q^{-1})bc$$

Now here's an exercise: suppose that $\bigl( \begin{smallmatrix} a_1 & b_1 \\ c_1 & d_1\end{smallmatrix}\bigr)$ and $\bigl( \begin{smallmatrix} a_2 & b_2 \\ c_2 & d_2\end{smallmatrix}\bigr)$ are each quantum $2\times 2$ matrices, by which I mean that their coordinates each independently satisfy the rules $(\star,\star\star)$, and suppose that the $r_1$s commute with the $r_2$s. Then their product $$\begin{pmatrix} a_1a_2 + b_1 c_2 & a_1 b_2 + b_1 d_2 \\ c_1 a_2 + d_1 c_2 & c_1 b_2 + d_1 d_2 \end{pmatrix}$$ is again a quantum $2\times 2$ matrix.

Thus our "quantum space" of quantum $2\times 2$ matrix (defined by $(\star,\star\star)$, which just determines if at some space in which each point is determined by the values of 4 noncommuting coordinates) is in face a "ring", or at least a "monoid". Oh, also you should check that (exercise) $\bigl( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix} \bigr)$ is a quantum matrix, and is the unit. So now always do the coordinates have to evaluate to noncommuting numbers.

Now, an important property of matrices is the "determinant". Exercise: the number $\Delta = ad - qbc$ commutes with all coordinates.

Exericse: If $X = \bigl( \begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\bigr)$ is a quantum $2\times 2$ matrix in which $\Delta^{-1}$ exists, then $X$ is invertible, with inverse $\Delta^{-1}\bigl( \begin{smallmatrix} d & -b \\ -qc & a\end{smallmatrix}\bigr)$.

Anyway, by definition, the group quantum $SL(2)$ is the subgroup of the quantum $2\times 2$ matrices consisting of those matrices for which $\Delta = 1$. By definition, the group quantum $GL(2)$ is the subgroup for which $\Delta^{-1}$ exists (thus you could say that a point in quantum $GL(2)$ has five coordinates, $a,b,c,d,\Delta^{-1}$, satisfying the equations $(\star,\star\star)$ and $\Delta^{-1}(ad - qbc) = 1$).

Similar constructions give other quantum groups as well, by starting with higher-dimensional quantum spaces and/or quantum versions of symmetric or skew-symmetric pairings.

Finally, you might have seen "groups" like $U_q \mathfrak{sl}(2)$, which really are the universal enveloping algebras of "lie algebras" of these groups.

This post imported from StackExchange MathOverflow at 2015-04-28 14:42 (UTC), posted by SE-user Theo Johnson-Freyd
answered Apr 28, 2015 by (275 points)
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(1) Concerning natural actions:

The quantum groups U_q(G) of a semisimple Lie group is a deformation of the corresponding classical Lie group and acts naturally on the universal enveloping algebra of the corresponding Lie algebra. (As the answer of Theo Johnson-Freyd shows, a very explicit description is already messy in the simplest cases; thus I won't give an example.)

(2) Concerning applications:

(i) Quantum groups are useful in understanding the structure of certain infinite-dimensional Lie algebras, namely Kac-Moody algebras. See, e.g., http://arxiv.org/abs/math/0605460

(ii) Quantum groups and their variants describe generalized symmetries of many conformal field theories. See, e.g., http://arxiv.org/abs/hep-th/9109023

(iii) Quantum groups and their variants describe generalized symmetries  of many massive integrable quantum field theories in 2 dimensions. See, e.g., http://arxiv.org/abs/hep-th/9108007, and topological field theories in 3 dimensions.

(iv) Applications to knot invariants in 3 dimensional manifolds are given in Chapter 15 of the book "A Guide to Quantum Groups" by Chari and Pressley. (This very readable book was also mentioned by ​user40276.) The connection is given by the fact that quantum groups provide representations of the braid group. A more sophisticated connection is given through the relations with 3-dimensional topological field theories.

(v) Applications to the structure theory of monoidal categories and fusion categories are discussed here.

(vi) Applications to the representation theory of Lie algebras over fields of characteristic $p$ are given in http://home.math.au.dk/jantzen/mntrl.pdf.

(vii) Quantum group representations are closely related to the analysis and combinatorics of q-special functions; see Section 13.5 of the book by Chari and Pressley mentioned above.

(viii) The smallest quantum group $SL_q(2)$ appears in algebraic combinatorics where it encodes the representation theory of certain distance-regular graphs (P- and Q-polynomial association schemes) and the associated tridiagonal pairs. See, e.g., http://arxiv.org/abs/1307.7968

Many more references could be given - the subject is vast.

answered Apr 28, 2015 by (13,229 points)
edited Apr 29, 2015

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