12b. Edge results

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Another interesting result, dealing this time with the unitary edge of the standard cube, is the one obtained by Mang-Weber in . To be more precise, the problem here is that of classifying the intermediate easy quantum groups as follows:

[[math]] U_N\subset G\subset U_N^+ [[/math]]

A first construction of such quantum groups is as follows:

Proposition

Associated to any [math]r\in\mathbb N[/math] is the easy quantum group

[[math]] U_N\subset U_N^{(r)}\subset U_N^+ [[/math]]

coming from the category [math]\mathcal P_2^{(r)}[/math] of matching pairings having the property that

[[math]] \#\circ=\#\bullet(r) [[/math]]

holds between the legs of each string. These quantum groups have the following properties:

• At [math]r=1[/math] we obtain the usual unitary group, [math]U_N^{(1)}=U_N[/math].
• At [math]r=2[/math] we obtain the half-classical unitary group, [math]U_N^{(2)}=U_N^*[/math].
• For any [math]r|s[/math] we have an embedding [math]U_N^{(r)}\subset U_N^{(s)}[/math].
• In general, we have an embedding [math]U_N^{(r)}\subset U_N^r\rtimes\mathbb Z_r[/math].
• We have as well a cyclic matrix model [math]C(U_N^{(r)})\subset M_r(C(U_N^r))[/math].
• In this latter model, [math]\int_{U_N^{(r)}}[/math] appears as the restriction of [math]tr_r\otimes\int_{U_N^r}[/math].

Show Proof

This is something quite compact, summarizing various findings from , and from . Here are a few brief explanations on all this:

(1) This is clear from [math]\mathcal P_2^{(1)}=\mathcal P_2[/math], and from the Brauer theorem .

(2) This is because [math]\mathcal P_2^{(2)}[/math] is generated by the partitions with implement the relations [math]abc=cba[/math] between the variables [math]\{u_{ij},u_{ij}^*\}[/math], used in for constructing [math]U_N^*[/math].

(3) This simply follows from [math]\mathcal P_2^{(s)}\subset\mathcal P_2^{(r)}[/math], by functoriality.

(4) This is the original definition of [math]U_N^{(r)}[/math], from . We refer to for the formula of the embedding, and to for the compatibility with the Tannakian definition.

(5) This is also from , more specifically it is an alternative definition for [math]U_N^{(r)}[/math].

(6) Once again, this is something from , and we will be back to it.

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Let us discuss now the second known construction of unitary quantum groups, from . This construction uses an additive semigroup [math]D\subset\mathbb N[/math], but as pointed out there, using instead the complementary set [math]C=\mathbb N-D[/math] leads to several simplifications. So, let us call “cosemigroup” any subset [math]C\subset\mathbb N[/math] which is complementary to an additive semigroup, [math]x,y\notin C\implies x+y\notin C[/math]. The construction from is then:

Proposition

Associated to any cosemigroup [math]C\subset\mathbb N[/math] is the easy quantum group

[[math]] U_N\subset U_N^C\subset U_N^+ [[/math]]

coming from the category [math]\mathcal P_2^C\subset P_2^{(\infty)}[/math] of pairings having the property

[[math]] \#\circ-\#\bullet\in C [[/math]]

between each two legs colored [math]\circ,\bullet[/math] of two strings which cross. We have:

• For [math]C=\emptyset[/math] we obtain the quantum group [math]U_N^+[/math].
• For [math]C=\{0\}[/math] we obtain the quantum group [math]U_N^\times[/math].
• For [math]C=\{0,1\}[/math] we obtain the quantum group [math]U_N^{**}[/math].
• For [math]C=\mathbb N[/math] we obtain the quantum group [math]U_N^{(\infty)}[/math].
• For [math]C\subset C'[/math] we have an inclusion [math]U_N^{C'}\subset U_N^C[/math].
• Each quantum group [math]U_N^C[/math] contains each quantum group [math]U_N^{(r)}[/math].

Show Proof

Once again this is something very compact, coming from recent work in , with our convention that the semigroup [math]D\subset\mathbb N[/math] which is used there is replaced here by its complement [math]C=\mathbb N-D[/math]. Here are a few explanations on all this:

(1) The assumption [math]C=\emptyset[/math] means that the condition [math]\#\circ-\#\bullet\in C[/math] can never be applied. Thus, the strings cannot cross, we have [math]\mathcal P_2^\emptyset=\mathcal{NC}_2[/math], and so [math]U_N^\emptyset=U_N^+[/math].

(2) As explained in , here we obtain indeed the quantum group [math]U_N^\times[/math], constructed by using the relations [math]ab^*c=cb^*a[/math], with [math]a,b,c\in\{u_{ij}\}[/math].

(3) This is also explained in , with [math]U_N^{**}[/math] being the quantum group from , which is the biggest whose full projective version, in the sense there, is classical.

(4) Here the assumption [math]C=\mathbb N[/math] simply tells us that the condition [math]\#\circ-\#\bullet\in C[/math] in the statement is irrelevant. Thus, we have [math]\mathcal P_2^\mathbb N=\mathcal P_2^{(\infty)}[/math], and so [math]U_N^\mathbb N=U_N^{(\infty)}[/math].

(5) This is clear by functoriality, because [math]C\subset C'[/math] implies [math]\mathcal P_2^{C}\subset\mathcal P_2^{C'}[/math].

(6) This is clear from definitions, and from Proposition 12.7 above.

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We have the following key result, from Mang-Weber :

Theorem

The easy quantum groups [math]U_N\subset G\subset U_N^+[/math] are as follows,

[[math]] U_N\subset\{U_N^{(r)}\}\subset\{U_N^C\}\subset U_N^+ [[/math]]

with the series covering [math]U_N[/math], and the family covering [math]U_N^+[/math].

Show Proof

This is something non-trivial, and we refer here to . The general idea is that [math]U_N^{(\infty)}[/math] produces a dichotomy for the quantum groups in the statement, and this leads, via combinatorial computations, to the series and the family. See .

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Observe that there is an obvious similarity here with the dichotomy for the liberations of [math]H_N[/math], coming from the work of Raum-Weber , explained in the above. To be more precise, the above-mentioned classification results for the liberations of [math]H_N[/math] and the liberations of [math]U_N[/math] have some obvious similarity between them. We have indeed a family followed by a series, and a series followed by a family.

All this suggests the existence of a general “contravariant duality” between these quantum groups, as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ U_N\ar[r]\ar@.[d]&U_N^{(r)}\ar[r]\ar@.[d]&U_N^C\ar[r]\ar@.[d]&U_N^+\ar@.[d]\\ H_N^+\ar@.[u]&H_N^{[r]}\ar[l]\ar@.[u]&H_N^\Gamma\ar[l]\ar@.[u]&H_N\ar[l]\ar@.[u] } [[/math]]

At the first glance, this might sound a bit strange. Indeed, we have some natural and well-established correspondences [math]H_N\leftrightarrow U_N[/math] and [math]H_N^+\leftrightarrow U_N^+[/math], obtained in one sense by taking the real reflection subgroup, [math]H=U\cap H_N^+[/math], and in the other sense by setting [math]U= \lt H,U_N \gt [/math]. Thus, our proposal of duality seems to go the wrong way.

On the other hand, obvious as well is the fact that these correspondences [math]H_N\leftrightarrow U_N[/math] and [math]H_N^+\leftrightarrow U_N^+[/math] cannot be extended as to map the series to the series, and the family to the family, because the series/families would have to be “inverted”, in order to do so.

Thus, we are led to the above contravariant duality conjecture. In practice, the idea would be that of constructing the duality by a clever use of the interesection and generation operations [math]\cap[/math] and [math] \lt \,, \gt [/math], but it is not clear so far on how to do this.

Following , let us discuss now what happens inside the standard cube, first in the easy case, and then in general. The idea here will be that of carefully looking at the Ground Zero theorem from chapter 11 above, and removing the easiness axiom there.

This is something quite technical, and in order to do so, let us start with a study of the easy case, with the goal of improving the Ground Zero theorem, by relaxing a bit the orientability axiom there. Let us start with the following definition:

Definition

A twistable easy quantum group [math]H_N\subset G_N\subset U_N^+[/math] is called “bi-oriented” if the diagram

[[math]] \xymatrix@R=17pt@C=17pt{ &G_N^d\ar[rr]&&G_N\\ G_N^{dr}\ar[rr]\ar[ur]&&G_N^r\ar[ur]\\ &G_N^{cd}\ar[rr]\ar[uu]&&G_N^c\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&G_N^{cr}\ar[uu]\ar[ur]} [[/math]]

as well as the diagram

[[math]] \xymatrix@R=17pt@C=17pt{ &G_N^{fu}\ar[rr]&&U_N^+\\ G_N^f\ar[rr]\ar[ur]&&G_N^{sf}\ar[ur]\\ &G_N^u\ar[rr]\ar[uu]&&G_N^{su}\ar[uu]\\ G_N\ar[uu]\ar[ur]\ar[rr]&&G_N^s\ar[uu]\ar[ur] } [[/math]]

are intersection and easy generation diagrams.

Observe that the first diagram is automatically an intersection diagram, and that the second diagram is automatically an easy generation diagram.

The question of replacing the slicing axiom with the bi-orientability condition makes sense. In fact, we can even talk about weaker axioms, as follows:

Definition

An easy quantum group [math]H_N\subset G_N\subset U_N^+[/math] is called “oriented” if

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

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

and “weakly oriented” if the following weaker conditions hold,

[[math]] G_N=\{G_N^c,G_N^d,G_N^r\} [[/math]]

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

where the various versions are those in chapter 11 above.

In order to prove now the uniqueness of the main 8 easy quantum groups, in the bi-orientable case, we can still proceed as in the proof of the Ground Zero theorem, but we are no longer allowed to use the coordinate system there, based at [math]O_N[/math].

To be more precise, we must use the 2 coordinate systems highlighted below, both taken in some weak sense, weaker than the slicing:

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar@=[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar@=[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar@=[uu]\\ H_N\ar@=[uu]\ar@=[ur]\ar@=[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

Skipping some details here, all this is viable, by using the known “edge results” surveyed above, along with the key fact, coming also from the above edge results, that the quantum group [math]H_N^{[\infty]}[/math] from has no orthogonal counterpart.

Thus, we obtain in principle some improvements of the Ground Zero theorem, under the bi-orientability assumption, and more generally under the orientability assumption. As for the weak orientability assumption, the situation here is more tricky, because we would need full “face results”, which are not available yet.

General references

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