Have the Quantum Group Theorists taught the Group Theorists Anything?

Consider for a moment the two (very) soft definitions.

• An abstraction of an object $$X$$ is a category $$\mathcal{C}_0$$ such that $$X$$ is an object in the class $$\operatorname{ob}(\mathcal{C}_0)$$.

• A generalisation of an abstraction $$\mathcal{C}_0$$ is a category $$\mathcal{C}$$ such that $$\mathcal{C}_0$$ is a proper subcategory of $$\mathcal{C}$$ (so that in this soft definition regime, a generalisation is also an abstraction).

It is a storied theme of mathematics that by abstracting an object $$X$$ to $$\mathcal{C}_0$$, that we can prove theorems for a whole class of objects, rather than just for the single object $$X$$.

Moreover, often when interested just in the object $$X$$, it can be easier to work in the abstraction $$\mathcal{C}_0$$, as this sometimes allows us to disregard the irrelevant idiosyncrasies of $$X$$.

Mathematical history --- with all its humanity --- is littered with examples of theorems proved in an abstraction $$\mathcal{C}_0$$ before they were known or considered in the specific context of $$X$$. This is of a subjectively different flavour to just putting together a slicker proof or proving a general result.

Of course, when you move from $$X$$ to $$\mathcal{C}_0$$ some theorems are no longer true.

The same is true when we look at a generalisation $$\mathcal{C}$$ of $$\mathcal{C}_0$$. However, of course, theorems true in $$\mathcal{C}$$ will be true for $$X$$ but moreover $$\mathcal{C}_0$$.

Moving towards the specific, the Peter-Weyl Theorem in the category of compact groups is also true (with suitable definitions) in the generalisation to compact matrix quantum groups.

There are many definitions/categories of quantum groups. In those categories which are (in the sense above) generalisations of categories of classical groups (classical in the sense of "has a set of points $$G$$" --- I believe all such definitions of quantum groups include at the very least the category of finite groups), have the quantum group theorists ever 'discovered' something that group theorists either were interested in, or would plausibly be interested in?

When a generalisation $$\mathcal{C}$$ of an abstraction $$\mathcal{C}_0$$ is developed to help study objects in $$\mathcal{C}_0$$, you can imagine that this happens, but as quantum groups are, arguably, studied for their non-commutative aspects, rather than as an attempt to understand classical groups better, this may not have happened.

To bookend; my question:

Have quantum group theorists discovered something new about groups that is interesting to group theorists?

• It seems to me that the many, many paragraphs of motivation add only empty generalisations, not examples; I think just the paragraph on Peter–Weyl and the question itself would make a good post. – LSpice Apr 12 at 18:13
• In representation theory, there are examples where specilializing the parameter $q$ to a suitable root of unity turns difficult characterisiic $p$ problems into characteristic zero problems. – Geoff Robinson Apr 12 at 18:28
• There is e.g. the famous example of Lusztig's conjecture about representations of quantum groups at root of unity and representation of Lie algebras in positive characteristic (see the lots of discussion at mathoverflow.net/questions/138310/…). But I think one problem with the question as phrased is that Lie theorists might not consider themselves group theorists anyways. – Sam Hopkins Apr 12 at 18:49
• Canonical bases? (e.g. Chari-Pressley 1994, Chap. 14.) – Francois Ziegler Apr 12 at 19:11
• @LSpice : There is some discussion of this in arxiv.org/abs/math/0308281 , for example. – Geoff Robinson Apr 12 at 19:27

The term "Quantum groups" itself, implies that the development of the hopf algebra theory generalizes -in some categorical sense- usual group theory. There are various points that might support this view (although i am not sure if this is what you are really looking for):

• If $$H$$ is a cocommutative, finite dimensional hopf algebra over an algebraically closed field $$k$$, of characteristic zero then $$H\cong kG$$, for some finite group $$G$$. Similarly, the commutative hopf algebras over $$k$$ are isomorphic to the duals of group hopf algebras of finite groups.
• The above results can be viewed categorically: there is an equivalence (but not necessarily an isomorphism) of categories, between the category $$\mathcal{H}$$, of commutative, cocommutative, f.d. hopf algebras over $$k$$ and the category $$\mathcal{Ab}_f$$ of finite abelian groups. This implies that in finite dimensions and under the constraints imposed by commutativity and cocommutativity, the hopf algebra theory is "essentially" the theory of finite abelian groups. If we drop commutativity and keep only cocommutativity we have the finite group theory.

I do not know if these are new discoveries, in the sense that they are classical results of the hopf algebra theory; cocommutativity after all is an obvious property in the "tensoring" of group representations. (and of the lie algebra representations as well).

However, -as mentioned in the OP- it is the noncommutative (and the non-cocommutative i would add) aspects of quantum group theory or hopf algebra theory that are really interesting. The notions of quasitriangularity (QT) and coquasitriangularity (CQT), generalize cocommutativity and commutativity respectively. However they still keep close touch to group theory: CQT group hopf algebras are abelian and equipped with a bicharacter $$\langle . | . \rangle:G\times G\rightarrow k$$. The set of bicharacters on $$G$$ is in bijection with the set of the homomorphisms of $$G$$ to its character group $$\hat{G}$$.
In the f.d. case and for $$k=\mathbb{C}$$ the complex numbers, the bicharacters of the finite, abelian group $$G$$ are in bijection with the QT and the CQT structures of the group hopf algebra $$\mathbb{C}G$$ (that is, its universal $$R$$-matrices) and in bijection with the braidings of the monoidal category $${}_{\mathbb{C}G}\mathcal{M}$$ of the group algebra representations.

In this sense, the non-trivial (co)quasitriangular structures of the group hopf algebra (if the group is finite, abelian these are non-trivial $$R$$-matrices), correspond to non-trivial bicharacters of the group or to non-trivial braidings of its category of representations.

These notions contribute to the expansion of the definition of quantum groups. Braided groups, are hopf algebras in the braided monoidal categories of representations of (co)quasitriangular group hopf algebras. (i.e. group hopf algebras equipped with non-trivial $$R$$-matrices or non-trivial bicharacters of the corresponding group).

Edit: Since the OP cites generalizations of group theoretic results to the quantum groups/hopf algebra setting (like the Peter-Weyl theorem), maybe it would be interesting to mention results on the generalizations of Frobenius-Schur indicator for compact groups: In arXiv:math/0004097 [math.RT], the Frobenius-Schur theorem for finite groups, is generalized for semisimple hopf algebras over algebraically closed fields of zero char and to semisimple/cosemisimple hopf algebras if the characteristic is greater than zero. Some more recent results are presented in FSZ groups and Frobenius-Schur indicators for quantum doubles. There, the authors study the problem of

when higher indicators of the reps of the quantum double of a finite group are all integers?

They characterize this as an

interesting group-theoretic question

and proceed in finding groups which have this property and counterexamples as well.

Concluding, i am not claiming that quantum group theory has answered unsolved problems of group theory but it may have contributed some ideas, or at least some descriptions, or even has posed some questions, of interest to a group theorist.

• Konstantinos, will it be correct to say that the theory of quantum groups is exactly the theory of Hopf algebras? It seems to me one might argue that the first one is something narrower... – Sergei Akbarov Apr 13 at 4:13
• Sergei, to tell the truth, your remark has also concerned me before posting. To my understanding, there is no globally accepted definition of quantum groups, which distinguishes them in some clear way from hopf algebras. I think that initially the term was used as a synonym for quasitriangular hopf algebras. (If i am not mistaken this reflected Drinfeld's initial use of the terminology). Later, various authors have used the term as a synonym for all hopf algebras. Other authors have used the term quantum groups as synonym for non-commutative, non-cocommutative hopf algebras. – Konstantinos Kanakoglou Apr 13 at 18:28
• In my post, i essentially use the term quantum groups as a synonym to all hopf algebras. I think this is also what the OP is doing. If however there is some objection, i could edit as long as some clear distinction has been made. – Konstantinos Kanakoglou Apr 13 at 18:28
• I am thinking primarily of those who came after Woronowicz (compact and compact matrix quantum groups) including the locally compact quantum groups of Kustermans and Vaes, which while have a lot of Hopf algebra in the background, and in the motivation, are not strictly Hopf algebras. – JP McCarthy Apr 15 at 8:12
• Your answer is still a positive contribution of course. I am interested in any definition/category of quantum groups that includes classical/commutative groups – JP McCarthy Apr 15 at 18:17