12c. Beyond easiness

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Let us discuss now the general, non-easy case. In order to do so, we must find extensions of the notions of uniformity, twistability and orientability. Regarding the notion of uniformity, the definition here is straightforward, with only some minor changes with respect to the easy quantum group case, as follows:

Definition

A series [math]G=(G_N)[/math] of closed subgroups [math]G_N\subset U_N^+[/math] is called:

• Weakly uniform, if for any [math]N\in\mathbb N[/math] we have [math]G_{N-1}=G_N\cap U_{N-1}^+[/math], with respect to the embedding [math]U_{N-1}^+\subset U_N^+[/math] given by [math]u\to diag(u,1)[/math].
• Uniform, if for any [math]N\in\mathbb N[/math] we have [math]G_{N-1}=G_N\cap U_{N-1}^+[/math], with respect to the [math]N[/math] possible embeddings [math]U_{N-1}^+\subset U_N^+[/math], of type [math]u\to diag(u,1)[/math].

In the easy quantum group case these two notions coincide, due to the presence of the symmetric group [math]S_N\subset G_N[/math], which acts on everything, and allows one to pass from one embedding [math]U_{N-1}^+\subset U_N^+[/math] to another. In general, these two notions do not coincide.

Regarding the examples, in the classical case we have substantially more examples than in the easy case, obtained by using the determinant, and its powers:

Proposition

The following compact groups are uniform,

• The complex reflection groups
  [[math]] H_N^{sd}=\left\{g\in\mathbb Z_s\wr S_N\Big|(\det g)^d=1\right\} [[/math]]
  for any values of the parameters [math]s\in\{1,2,\ldots,\infty\}[/math] and [math]d\in\mathbb N[/math], [math]d|s[/math],
• The orthogonal group [math]O_N[/math], the special orthogonal group [math]SO_N[/math], and the series
  [[math]] U_N^d=\left\{g\in U_N\Big|(\det g)^d=1\right\} [[/math]]
  of modified unitary groups, with [math]s\in\{1,2,\ldots,\infty\}[/math],

and so are the bistochastic versions of these groups.

Show Proof

Both these assertions are clear from definitions, the idea being as follows:

(1) These groups are well-known objects in finite group theory, and more precisely form the series of complex reflection groups, and generalize the groups [math]H_N^s[/math] from chapter 10 above, which appear at [math]d=s[/math]. See Shephard-Todd .

(2) These groups are well-known as well, in compact Lie group theory, with [math]U_N^1[/math] being equal to [math]SU_N[/math], and with [math]U_N^\infty[/math] being by definition [math]U_N[/math] itself.

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In the free case now, corresponding to the condition [math]S_N^+\subset G_N\subset U_N^+[/math], it is widely believed that the only examples are the easy ones. A precise conjecture in this sense, which is a bit more general, valid for any [math]G_N\subset U_N^+[/math], states that we should have:

[[math]] \lt G_N,S_N^+ \gt =\{G_N',S_N^+\} [[/math]]

Here [math]G_N'[/math] denotes as usual the easy envelope of [math]G_N[/math], and [math]\{\,,\}[/math] is an easy generation operation. This conjecture is probably something quite difficult.

Now back to our questions, we have definitely no new examples in the free case. So, the basic examples will be those that we previously met, namely:

Proposition

The following free quantum groups are uniform,

• Liberations [math]H_N^{s+}=\mathbb Z_s\wr_*S_N^+[/math] of the complex reflection groups [math]H_N^s=\mathbb Z_s\wr S_N[/math],
• Liberations [math]O_N^+,U_N^+[/math] of the continuous groups [math]O_N,U_N[/math],

and so are the bistochastic versions of these quantum groups.

Show Proof

This is something that we basically know, with the uniformity check for [math]H_N^{s+}[/math] being the same as for [math]S_N^+,H_N^+,K_N^+[/math], which appear at [math]s=1,2,\infty[/math].

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We would need now a second axiom, such as the twistability condition [math]T_N\subset G_N[/math] used in chapter 11. However, if we carefully look at Proposition 12.13, and we want to have as examples the groups there, a condition of type [math]A_N\subset G_N[/math] would be more appropriate.

In order to comment on this dillema, let us recall from chapter 11 that, in view of the considerations there, “taking the bistochastic version” is a bad direction, geometrically speaking. But the operations “taking the diagonal torus” and “taking the special version”, that we are currently discussing, are bad too. Thus, we have 3 bad directions, and so we end up with a cube formed by these bad 3 directions, as follows:

Proposition

We have the following diagram of finite groups,

[[math]] \xymatrix@R=18pt@C=18pt{ &S_N\ar[rr]&&H_N\\ A_N\ar[rr]\ar[ur]&&SH_N\ar[ur]\\ &\{1\}\ar[rr]\ar[uu]&&T_N\ar[uu]\\ \{1\}\ar[uu]\ar[ur]\ar[rr]&&ST_N\ar[uu]\ar[ur] } [[/math]]

obtained from [math]H_N[/math] by taking bistochastic, special and diagonal versions.

Show Proof

This is clear from definitions, with the operations of taking bistochastic versions, special versions and diagonal subgroups corresponding respectively to going left, backwards, and downwards, with respect to the coordinates in the statement.

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Now back to our classification questions, the vertices of the above cube are all interesting groups, and assuming that the quantum groups [math]G_N\subset U_N^+[/math] that we want to classify contain any of them is something quite natural.

Let us just select here three such conditions, as follows:

Definition

A closed subgroup [math]G_N\subset U_N^+[/math] is called:

• Twistable, if [math]T_N\subset G_N[/math].
• Homogeneous, if [math]S_N\subset G_N[/math].
• Half-homogeneous, if [math]A_N\subset G_N[/math].

As before with the notion of uniformity, things simplify in the easy case. To be more precise, any easy quantum group is automatically homogeneous, and half-homogeneous as well. As for the notion of twistability, this coincides with the old one.

Let us go ahead now, and formulate our third and last definition, regarding the orientability axiom. Things are quite tricky here, and we must start as follows:

Definition

Associated to any closed subgroup [math]G_N\subset U_N^+[/math] are its classical, discrete and real versions, given by

[[math]] G_N^c=G_N\cap U_N [[/math]]

[[math]] G_N^d=G_N\cap K_N^+ [[/math]]

[[math]] G_N^r=G_N\cap O_N^+ [[/math]]

as well as its free, smooth and unitary versions, given by

[[math]] G_N^f= \lt G_N,H_N^+ \gt [[/math]]

[[math]] G_N^s= \lt G_N,O_N \gt [[/math]]

[[math]] G_N^u= \lt G_N,K_N \gt [[/math]]

where [math] \lt \,, \gt [/math] is the usual, non-easy topological generation operation.

Observe the difference, and notational clash, with some of the notions used in chapter 11. To be more precise, as explained in chapter 7, it is believed that we should have [math]\{\,,\}= \lt \,, \gt [/math], but this is not clear at all, and the problem comes from this.

A second issue comes when composing the above operations, and more specifically those involving the generation operation, once again due to the conjectural status of the formula [math]\{\,,\}= \lt \,, \gt [/math]. Due to this fact, instead of formulating a result here, we have to formulate a second definition, complementary to Definition 12.7, as follows:

Definition

Associated to any closed subgroup [math]G_N\subset U_N^+[/math] are the mixes of its classical, discrete and real versions, given by

[[math]] G_N^{cd}=G_N\cap K_N [[/math]]

[[math]] G_N^{cr}=G_N\cap O_N^+ [[/math]]

[[math]] G_N^{dr}=G_N\cap H_N^+ [[/math]]

as well as the mixes of its free, smooth and unitary versions, given by

[[math]] G_N^{fs}= \lt G_N,O_N^+ \gt [[/math]]

[[math]] G_N^{fu}= \lt G_N,K_N^+ \gt [[/math]]

[[math]] G_N^{us}= \lt G_N,U_N \gt [[/math]]

where [math] \lt \,, \gt [/math] is the usual, non-easy topological generation operation.

Now back to our orientation questions, the slicing and bi-orientability conditions lead us again into [math]\{\,,\}[/math] vs. [math] \lt \,, \gt [/math] troubles, and are therefore rather to be ignored. The orientability conditions from Definition 12.11, however, have the following analogue:

Definition

A closed subgroup [math]G_N\subset U_N^+[/math] is called “oriented” if

[[math]] G_N= \lt G_N^{cd},G_N^{cr},G_N^{dr} \gt [[/math]]

[[math]] G_N=G_N^{fs}\cap G_N^{fu}\cap G_N^{su} [[/math]]

and “weakly oriented” if the following conditions hold,

[[math]] G_N= \lt G_N^c,G_N^d,G_N^r \gt [[/math]]

[[math]] G_N=G_N^f\cap G_N^s\cap G_N^u [[/math]]

where the various versions are those in Definition 12.17 and Definition 12.18.

With these notions, our claim is that some classification results are possible:

(1) In the classical case, we believe that the uniform, half-homogeneous, oriented groups are those in Proposition 12.13, with some bistochastic versions excluded. This is of course something quite heavy, well beyond easiness, with the potential tools available for proving such things coming from advanced finite group theory and Lie algebra theory. Our uniformity axiom could play a key role here, when combined with , in order to exclude all the exceptional objects which might appear on the way.

(2) In the free case, under similar assumptions, we believe that the solutions should be those in Proposition 12.14, once again with some bistochastic versions excluded. This is something heavy, too, related to the above-mentioned well-known conjecture [math] \lt G_N,S_N^+ \gt =\{G_N',S_N^+\}[/math]. Indeed, assuming that we would have such a formula, and perhaps some more formulae of the same type as well, we can in principle work out our way inside the cube, from the edge and face projections to [math]G_N[/math] itself, and in this process [math]G_N[/math] would become easy. This would be the straightforward strategy here.

(3) In the group dual case, the orientability axiom simplifies, because the group duals are discrete in our sense. We believe that the uniform, twistable, oriented group duals should appear as combinations of certain abelian groups, which appear in the classical case, with duals of varieties of real reflection groups, which appear in the real case. This is probably the easiest question in the present series, and the most reasonable one, to start with. However, there are no concrete results so far, in this direction.

We refer to and related papers for further comments, on all the above.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].