guide:23a52665b8

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Let us discuss now some extensions of the above constructions. We will be mostly interested in the quantum reflection groups, but let us first discuss, with full details, the case of the quantum groups [math]S_N,S_N^+[/math]. The starting point is the semigroup [math]\widetilde{S}_N[/math] of partial permutations. This is a quite familiar object in combinatorics, defined as follows:

Definition

[math]\widetilde{S}_N[/math] is the semigroup of partial permutations of [math]\{1\,\ldots,N\}[/math],

[[math]] \widetilde{S}_N=\left\{\sigma:X\simeq Y\Big|X,Y\subset\{1,\ldots,N\}\right\} [[/math]]

with the usual composition operation, [math]\sigma'\sigma:\sigma^{-1}(X'\cap Y)\to\sigma'(X'\cap Y)[/math].

Observe that [math]\widetilde{S}_N[/math] is not simplifiable, because the null permutation [math]\emptyset\in\widetilde{S}_N[/math], having the empty set as domain/range, satisfies [math]\emptyset\sigma=\sigma\emptyset=\emptyset[/math], for any [math]\sigma\in\widetilde{S}_N[/math]. Observe also that [math]\widetilde{S}_N[/math] has a “subinverse” map, sending [math]\sigma:X\to Y[/math] to its usual inverse [math]\sigma^{-1}:Y\simeq X[/math].

A first interesting result about this semigroup [math]\widetilde{S}_N[/math], which shows that we are dealing here with some non-trivial combinatorics, is as follows:

Proposition

The number of partial permutations is given by

[[math]] |\widetilde{S}_N|=\sum_{k=0}^Nk!\binom{N}{k}^2 [[/math]]

that is, [math]1,2,7,34,209,\ldots\,[/math], and with [math]N\to\infty[/math] we have:

[[math]] |\widetilde{S}_N|\simeq N!\sqrt{\frac{\exp(4\sqrt{N}-1)}{4\pi\sqrt{N}}} [[/math]]

Show Proof

The first assertion is clear, because in order to construct a partial permutation [math]\sigma:X\to Y[/math] we must choose an integer [math]k=|X|=|Y|[/math], then we must pick two subsets [math]X,Y\subset\{1,\ldots,N\}[/math] having cardinality [math]k[/math], and there are [math]\binom{N}{k}[/math] choices for each, and finally we must construct a bijection [math]\sigma:X\to Y[/math], and there are [math]k![/math] choices here. As for the estimate, which is non-trivial, this is however something standard, and well-known.

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Another result, which is trivial, but quite fundamental, is as follows:

Proposition

We have a semigroup embedding [math]u:\widetilde{S}_N\subset M_N(0,1)[/math], defined by

[[math]] u_{ij}(\sigma)= \begin{cases} 1&{\rm if}\ \sigma(j)=i\\ 0&{\rm otherwise} \end{cases} [[/math]]

whose image are the matrices having at most one nonzero entry, on each row and column.

Show Proof

This is trivial from definitions, with [math]u:\widetilde{S}_N\subset M_N(0,1)[/math] extending the standard embedding [math]u:S_N\subset M_N(0,1)[/math], that we have been heavily using, so far.

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Let us discuss now the construction and main properties of the semigroup of quantum partial permutations [math]\widetilde{S}_N^+[/math], in analogy with the above. For this purpose, we use the above embedding [math]u:\widetilde{S}_N\subset M_N(0,1)[/math]. Due to the formula [math]u_{ij}(\sigma)=\delta_{i\sigma(j)}[/math], the matrix [math]u=(u_{ij})[/math] is “submagic”, in the sense that its entries are projections, which are pairwise orthogonal on each row and column. This suggests the following definition:

Definition

[math]C(\widetilde{S}_N^+)[/math] is the universal [math]C^*[/math]-algebra generated by the entries of a [math]N\times N[/math] submagic matrix [math]u[/math], with comultiplication and counit maps given by

[[math]] \Delta(u_{ij})=\sum_ku_{ik}\otimes u_{kj} [[/math]]

[[math]] \varepsilon(u_{ij})=\delta_{ij} [[/math]]

where submagic means formed of projections, which are pairwise orthogonal on rows and columns. We call [math]\widetilde{S}_N^+[/math] semigroup of quantum partial permutations of [math]\{1,\ldots,N\}[/math].

Here the fact that the morphisms of algebras [math]\Delta,\varepsilon[/math] as above exist indeed follows from the universality property of [math]C(\widetilde{S}_N^+)[/math], with the needed submagic checks being nearly identical to the magic checks for [math]C(S_N^+)[/math], from chapter 2. Observe also that the morphisms [math]\Delta,\varepsilon[/math] satisfy the usual axioms for a comultiplication and antipode, namely:

[[math]] (\Delta\otimes id)\Delta=(id\otimes \Delta)\Delta [[/math]]

[[math]] (\varepsilon\otimes id)\Delta=(id\otimes\varepsilon)\Delta=id [[/math]]

Thus, we have a bialgebra structure of [math]C(\widetilde{S}_N^+)[/math], which tells us that the underlying noncommutative space [math]\widetilde{S}_N^+[/math] is a compact quantum semigroup. This semigroup is of quite special type, because [math]C(\widetilde{S}_N^+)[/math] has as well a subantipode map, defined by:

[[math]] S(u_{ij})=u_{ji} [[/math]]

To be more precise here, this map exists because the transpose of a submagic matrix is submagic too. As for the subantipode axiom satisfied by it, this is as follows, where [math]m^{(3)}[/math] is the triple multiplication, and [math]\Delta^{(2)}[/math] is the double comultiplication:

[[math]] m^{(3)}(S\otimes id\otimes S)\Delta^{(2)}=S [[/math]]

Finally, observe that [math]\Delta,\varepsilon,S[/math] restrict to [math]C(\widetilde{S}_N)[/math], and correspond there, via Gelfand duality, to the usual multiplication, unit element, and subinversion map of [math]\widetilde{S}_N[/math].

As a conclusion to this discussion, the basic properties of the quantum semigroup [math]\widetilde{S}_N^+[/math] that we constructed in Definition 6.14 can be summarized as follows:

Proposition

We have maps as follows,

[[math]] \begin{matrix} C(\widetilde{S}_N^+)&\to&C(S_N^+)\\ \\ \downarrow&&\downarrow\\ \\ C(\widetilde{S}_N)&\to&C(S_N) \end{matrix} \quad \quad \quad:\quad \quad\quad \begin{matrix} \widetilde{S}_N^+&\supset&S_N^+\\ \\ \cup&&\cup\\ \\ \widetilde{S}_N&\supset&S_N \end{matrix} [[/math]]

with the bialgebras at left corresponding to the quantum semigroups at right.

Show Proof

This is clear from the above discussion, and from the well-known fact that projections which sum up to [math]1[/math] are pairwise orthogonal.

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As a first example, we have [math]\widetilde{S}_1^+=\widetilde{S}_1[/math]. At [math]N=2[/math] now, recall that the algebra generated by two free projections [math]p,q[/math] is isomorphic to the group algebra of [math]D_\infty=\mathbb Z_2*\mathbb Z_2[/math]. We denote by [math]\varepsilon:C^*(D_\infty)\to\mathbb C1[/math] the counit map, given by the following formulae:

[[math]] \varepsilon(1)=1 [[/math]]

[[math]] \varepsilon(\ldots pqpq\ldots)=0 [[/math]]

With these conventions, we have the following result:

Proposition

We have an isomorphism

[[math]] C(\widetilde{S}_2^+)\simeq\left\{(x,y)\in C^*(D_\infty)\oplus C^*(D_\infty)\Big|\varepsilon(x)=\varepsilon(y)\right\} [[/math]]

which is given by the formula

[[math]] u=\begin{pmatrix}p\oplus 0&0\oplus r\\0\oplus s&q\oplus 0\end{pmatrix} [[/math]]

where [math]p,q[/math] and [math]r,s[/math] are the standard generators of the two copies of [math]C^*(D_\infty)[/math].

Show Proof

Consider an arbitrary [math]2\times 2[/math] matrix formed by projections:

[[math]] u=\begin{pmatrix}P&R\\S&Q\end{pmatrix} [[/math]]

This matrix is submagic when the following conditions are satisfied:

[[math]] PR=PS=QR=QS=0 [[/math]]

But these conditions mean that [math]X= \lt P,Q \gt [/math] and [math]Y= \lt R,S \gt [/math] must commute, and must satisfy [math]xy=0[/math], for any [math]x\in X,y\in Y[/math]. Thus, if we denote by [math]Z[/math] the universal non-unital algebra generated by two projections, we have an isomorphism as follows:

[[math]] C(\widetilde{S}_2^+)\simeq\mathbb C1\oplus Z\oplus Z [[/math]]

Now since [math]C^*(D_\infty)=\mathbb C1\oplus Z[/math], we obtain an isomorphism as follows:

[[math]] C(\widetilde{S}_2^+)\simeq\left\{(\lambda+a,\lambda+b)\Big|\lambda\in\mathbb C, a,b\in Z\right\} [[/math]]

Thus, we are led to the conclusion in the statement.

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Summarizing, the semigroups of partial permutations [math]\widetilde{S}_N[/math] have non-trivial liberations, a bit like the permutation groups [math]S_N[/math] used to have non-trivial liberations, and this starting from [math]N=2[/math] already. In order to reach now to homogeneous spaces, in the spirit of the partial isometry spaces discussed before, we can use the following simple observation:

Proposition

Any partial permutation [math]\sigma:X\simeq Y[/math] can be factorized as

[[math]] \xymatrix@R=40pt@C=40pt {X\ar[r]^{\sigma}\ar[d]_\gamma&Y\\\{1,\ldots,k\}\ar[r]_\beta&\{1,\ldots,k\}\ar[u]_\alpha} [[/math]]

with [math]\alpha,\beta,\gamma\in S_k[/math] being certain non-unique permutations, where [math]k=\kappa(\sigma)[/math].

Show Proof

Since we have [math]|X|=|Y|=k[/math], we can pick two bijections, as follows:

[[math]] X\simeq\{1,\ldots,k\}\quad,\quad \{1,\ldots,k\}\simeq Y [[/math]]

We can complete then these bijections up to permutations [math]\gamma,\alpha\in S_N[/math]. The remaining permutation [math]\beta\in S_k[/math] is then uniquely determined by [math]\sigma=\alpha\beta\gamma[/math], as desired.

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With a bit more work, this leads to homogeneous spaces, in the spirit of the partial isometry spaces discussed before. To be more precise, we have the following notion:

Definition

Associated to any partial permutation, written [math]\sigma:I\simeq J[/math] with [math]I\subset\{1,\ldots,N\}[/math] and [math]J\subset\{1,\ldots,M\}[/math], is the real/complex partial isometry

[[math]] T_\sigma:span\left(e_i\Big|i\in I\right)\to span\left(e_j\Big|j\in J\right) [[/math]]

given on the standard basis elements by [math]T_\sigma(e_i)=e_{\sigma(i)}[/math].

We denote by [math]S_{MN}^L[/math] the set of partial permutations [math]\sigma:I\simeq J[/math] as above, with range [math]I\subset\{1,\ldots,N\}[/math] and target [math]J\subset\{1,\ldots,M\}[/math], and with:

[[math]] L=|I|=|J| [[/math]]

In analogy with the decomposition result [math]H_N^s=\mathbb Z_s\wr S_N[/math], we have:

Proposition

The space of partial permutations signed by elements of [math]\mathbb Z_s[/math],

[[math]] H_{MN}^{sL}=\left\{T(e_i)=w_ie_{\sigma(i)}\Big|\sigma\in S_{MN}^L,w_i\in\mathbb Z_s\right\} [[/math]]

is isomorphic to the following quotient space:

[[math]] (H_M^s\times H_N^s)/(H_L^s\times H_{M-L}^s\times H_{N-L}^s) [[/math]]

Show Proof

This follows by adapting the computations in the proof of Proposition 6.3. Indeed, we have an action map as follows, which is transitive:

[[math]] H_M^s\times H_N^s\to H_{MN}^{sL}\quad,\quad (A,B)U=AUB^* [[/math]]

Consider now the following point:

[[math]] U=\begin{pmatrix}1&0\\0&0\end{pmatrix} [[/math]]

The stabilizer of this point is then the following group:

[[math]] H_L^s\times H_{M-L}^s\times H_{N-L}^s [[/math]]

To be more precise, this group is embedded via:

[[math]] (x,a,b)\to\left[\begin{pmatrix}x&0\\0&a\end{pmatrix},\begin{pmatrix}x&0\\0&b\end{pmatrix}\right] [[/math]]

But this gives the result.

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In the free case now, the idea is similar, by using inspiration from the construction of the quantum group [math]H_N^{s+}=\mathbb Z_s\wr_*S_N^+[/math] in . The result here is as follows:

Proposition

The compact quantum space [math]H_{MN}^{sL+}[/math] associated to the algebra

[[math]] C(H_{MN}^{sL+})=C(U_{MN}^{L+})\Big/\left \lt u_{ij}u_{ij}^*=u_{ij}^*u_{ij}=p_{ij}={\rm projections},u_{ij}^s=p_{ij}\right \gt [[/math]]

has an action map, and is the target of a quotient map, as in Theorem 6.10.

Show Proof

We must show that if the variables [math]u_{ij}[/math] satisfy the relations in the statement, then these relations are satisfied as well for the following variables:

[[math]] U_{ij}=\sum_{kl}u_{kl}\otimes a_{ki}\otimes b_{lj}^* [[/math]]

[[math]] V_{ij}=\sum_{r\leq L}a_{ri}\otimes b_{rj}^* [[/math]]

We use the fact that the standard coordinates [math]a_{ij},b_{ij}[/math] on the quantum groups [math]H_M^{s+},H_N^{s+}[/math] satisfy the following relations, for any [math]x\neq y[/math] on the same row or column of [math]a,b[/math]:

[[math]] xy=xy^*=0 [[/math]]

We obtain, by using these relations:

[[math]] \begin{eqnarray*} U_{ij}U_{ij}^* &=&\sum_{klmn}u_{kl}u_{mn}^*\otimes a_{ki}a_{mi}^*\otimes b_{lj}^*b_{mj}\\ &=&\sum_{kl}u_{kl}u_{kl}^*\otimes a_{ki}a_{ki}^*\otimes b_{lj}^*b_{lj} \end{eqnarray*} [[/math]]

We have as well the following formula:

[[math]] \begin{eqnarray*} V_{ij}V_{ij}^* &=&\sum_{r,t\leq L}a_{ri}a_{ti}^*\otimes b_{rj}^*b_{tj}\\ &=&\sum_{r\leq L}a_{ri}a_{ri}^*\otimes b_{rj}^*b_{rj} \end{eqnarray*} [[/math]]

Consider now the following projections:

[[math]] x_{ij}=a_{ij}a_{ij}^*\quad,\quad y_{ij}=b_{ij}b_{ij}^*\quad,\quad p_{ij}=u_{ij}u_{ij}^* [[/math]]

In terms of these projections, we have:

[[math]] U_{ij}U_{ij}^*=\sum_{kl}p_{kl}\otimes x_{ki}\otimes y_{lj} [[/math]]

[[math]] V_{ij}V_{ij}^*=\sum_{r\leq L}x_{ri}\otimes y_{rj} [[/math]]

By repeating the computation, we conclude that these elements are projections. Also, a similar computation shows that [math]U_{ij}^*U_{ij},V_{ij}^*V_{ij}[/math] are given by the same formulae. Finally, once again by using the relations of type [math]xy=xy^*=0[/math], we have:

[[math]] \begin{eqnarray*} U_{ij}^s &=&\sum_{k_rl_r}u_{k_1l_1}\ldots u_{k_sl_s}\otimes a_{k_1i}\ldots a_{k_si}\otimes b_{l_1j}^*\ldots b_{l_sj}^*\\ &=&\sum_{kl}u_{kl}^s\otimes a_{ki}^s\otimes(b_{lj}^*)^s \end{eqnarray*} [[/math]]

We have as well the following formula:

[[math]] \begin{eqnarray*} V_{ij}^s &=&\sum_{r_l\leq L}a_{r_1i}\ldots a_{r_si}\otimes b_{r_1j}^*\ldots b_{r_sj}^*\\ &=&\sum_{r\leq L}a_{ri}^s\otimes(b_{rj}^*)^s \end{eqnarray*} [[/math]]

Thus the conditions of type [math]u_{ij}^s=p_{ij}[/math] are satisfied as well, and we are done.

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Let us discuss now the general case. We have the following result:

Proposition

The various spaces [math]G_{MN}^L[/math] constructed so far appear by imposing to the standard coordinates of [math]U_{MN}^{L+}[/math] the relations

[[math]] \sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}=L^{|\pi\vee\sigma|} [[/math]]

with [math]s=(e_1,\ldots,e_s)[/math] ranging over all the colored integers, and with [math]\pi,\sigma\in D(0,s)[/math].

Show Proof

According to the various constructions above, the relations defining [math]G_{MN}^L[/math] can be written as follows, with [math]\sigma[/math] ranging over a family of generators, with no upper legs, of the corresponding category of partitions [math]D[/math]:

[[math]] \sum_{j_1\ldots j_s}\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}=\delta_\sigma(i) [[/math]]

We therefore obtain the relations in the statement, as follows:

[[math]] \begin{eqnarray*} \sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s} &=&\sum_{i_1\ldots i_s}\delta_\pi(i)\sum_{j_1\ldots j_s}\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}\\ &=&\sum_{i_1\ldots i_s}\delta_\pi(i)\delta_\sigma(i)\\ &=&L^{|\pi\vee\sigma|} \end{eqnarray*} [[/math]]

As for the converse, this follows by using the relations in the statement, by keeping [math]\pi[/math] fixed, and by making [math]\sigma[/math] vary over all the partitions in the category.

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In the general case now, where [math]G=(G_N)[/math] is an arbitrary uniform easy quantum group, we can construct spaces [math]G_{MN}^L[/math] by using the above relations, and we have:

Theorem

The spaces [math]G_{MN}^L\subset U_{MN}^{L+}[/math] constructed by imposing the relations

[[math]] \sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}=L^{|\pi\vee\sigma|} [[/math]]

with [math]\pi,\sigma[/math] ranging over all the partitions in the associated category, having no upper legs, are subject to an action map/quotient map diagram, as in Theorem 6.10.

Show Proof

We proceed as in the proof of Proposition 6.9. We must prove that, if the variables [math]u_{ij}[/math] satisfy the relations in the statement, then so do the following variables:

[[math]] U_{ij}=\sum_{kl}u_{kl}\otimes a_{ki}\otimes b_{lj}^* [[/math]]

[[math]] V_{ij}=\sum_{r\leq L}a_{ri}\otimes b_{rj}^* [[/math]]

Regarding the variables [math]U_{ij}[/math], the computation here goes as follows:

[[math]] \begin{eqnarray*} &&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)U_{i_1j_1}^{e_1}\ldots U_{i_sj_s}^{e_s}\\ &=&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\sum_{k_1\ldots k_s}\sum_{l_1\ldots l_s}u_{k_1l_1}^{e_1}\ldots u_{k_sl_s}^{e_s}\otimes \delta_\pi(i)\delta_\sigma(j)a_{k_1i_1}^{e_1}\ldots a_{k_si_s}^{e_s}\otimes(b_{l_sj_s}^{e_s}\ldots b_{l_1j_1}^{e_1})^*\\ &=&\sum_{k_1\ldots k_s}\sum_{l_1\ldots l_s}\delta_\pi(k)\delta_\sigma(l)u_{k_1l_1}^{e_1}\ldots u_{k_sl_s}^{e_s}=L^{|\pi\vee\sigma|} \end{eqnarray*} [[/math]]

For the variables [math]V_{ij}[/math] the proof is similar, as follows:

[[math]] \begin{eqnarray*} &&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)V_{i_1j_1}^{e_1}\ldots V_{i_sj_s}^{e_s}\\ &=&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\sum_{l_1,\ldots,l_s\leq L}\delta_\pi(i)\delta_\sigma(j)a_{l_1i_1}^{e_1}\ldots a_{l_si_s}^{e_s}\otimes(b_{l_sj_s}^{e_s}\ldots b_{l_1j_1}^{e_1})^*\\ &=&\sum_{l_1,\ldots,l_s\leq L}\delta_\pi(l)\delta_\sigma(l)=L^{|\pi\vee\sigma|} \end{eqnarray*} [[/math]]

Thus we have constructed an action map, and a quotient map, as in Proposition 6.9 above, and the commutation of the diagram in Theorem 6.10 is then trivial.

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General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].