8c. Row spaces

[math] \newcommand{\mathds}{\mathbb}[/math]

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We discuss in what follows some constructions from , which go somehow in a direction which is opposite to what has been said in the above, namely particularization. The idea will be that of looking at the “minimal” theory of quantum homogeneous spaces generalizing at the same time the spheres [math]S[/math], and the unitary quantum groups [math]U[/math] they come form, and this time with very precise results. Such homogeneous spaces are technically covered by the general affine homogeneous space formalism from chapter 7, which is from the paper , which came some time after , but the difference of generality level being notable, there are many things that can be said, sharper than in general.

We first discuss the construction in the classical case. Given a closed subgroup [math]G\subset U_N[/math] and a number [math]k\leq N[/math], we can consider the compact group [math]H=G\cap U_k[/math], computed inside [math]U_N[/math], where the embedding [math]U_k\subset U_N[/math] that we use is given by the following formula:

[[math]] g\to \begin{pmatrix} g&0\\ 0&1_{N-k} \end{pmatrix} [[/math]]

We can form the homogeneous space [math]X=G/H[/math], and we have the following result:

Proposition

Let [math]G\subset U_N[/math] be a closed subgroup, and construct as above the closed subgroup [math]H\subset G[/math] given by the formula

[[math]] H=G\cap U_k [[/math]]

with the intersection being computed inside [math]U_N[/math]. Then the subalgebra

[[math]] C(G/H)\subset C(G) [[/math]]

that we obtain is generated by the last [math]N-k[/math] rows of coordinates on [math]G[/math].

Show Proof

Let [math]u_{ij}\in C(G)[/math] be the standard coordinates on [math]G[/math], given as usual by the formula [math]u_{ij}(g)=g_{ij}[/math], and consider the following subalgebra of [math]C(G)[/math]:

[[math]] A=\left \lt u_{ij}\Big|i \gt k,j \gt 0\right \gt [[/math]]

Since each coordinate function [math]u_{ij}[/math] with [math]i \gt k[/math] is constant on each coset [math]Hg\in G/H[/math], we have an inclusion as follows, between subalgebras of [math]C(G)[/math]:

[[math]] A\subset C(G/H) [[/math]]

In order to prove that this inclusion in a isomorphism, as to finish, we use the Stone-Weierstrass theorem. Indeed, in view of this theorem, it is enough to show that the following family of functions separates the cosets [math]\{Hg|g\in G\}[/math]:

[[math]] \left\{u_{ij}\Big|i \gt k,j \gt 0\right\} [[/math]]

But this is the same as saying that [math]Hg\neq Hh[/math] implies [math]g_{ij}\neq h_{ij}[/math], for some [math]i \gt k,j \gt 0[/math]. Equivalently, we must prove that [math]g_{ij}=h_{ij}[/math] for any [math]i \gt k,j \gt 0[/math] implies:

[[math]] Hg=Hh [[/math]]

Now since [math]Hg=Hh[/math] is equivalent to [math]gh^{-1}\in H[/math], the result follows from the usual matrix formula of [math]gh^{-1}[/math], and from the fact that [math]g,h[/math] are unitary.

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In the quantum case now, we can proceed in a similar way. Let [math]k\leq N[/math], and consider the embedding [math]U_k^+\subset U_N^+[/math] given by the same formula as before, namely:

[[math]] g\to \begin{pmatrix} g&0\\ 0&1_{N-k} \end{pmatrix} [[/math]]

That is, at the level of algebras, we use the quotient map [math]C(U_N^+)\to C(U_k^+)[/math] given by the following formula, where [math]v[/math] is the fundamental corepresentation of [math]U_k^+[/math]:

[[math]] u\to \begin{pmatrix} v&0\\ 0&1_{N-k} \end{pmatrix} [[/math]]

With this convention, we have the following definition, from :

Definition

Associated to any quantum subgroup [math]G\subset U_N^+[/math] and any [math]k\leq N[/math] are:

• The compact quantum group [math]H=G\cap U_k^+[/math].
• The algebra [math]C(G/H)\subset C(G)[/math] constructed before.
• The algebra [math]C_\times(G/H)\subset C(G/H)[/math] generated by [math]\{u_{ij}|i \gt k,j \gt 0\}[/math].

Regarding (3), let [math]u,v[/math] be the fundamental corepresentations of [math]G,H[/math], so that the quotient map [math]\pi:C(G)\to C(H)[/math] is given by [math]u\to diag(v,1_{N-k})[/math]. We have then:

[[math]] \begin{eqnarray*} (\pi\otimes id)\Delta(u_{ij}) &=&\sum_s\pi(u_{is})\otimes u_{sj}\\ &=&\begin{cases} \sum_{s\leq k}v_{is}\otimes u_{sj}&i\leq k\\ 1\otimes u_{ij}&i \gt k \end{cases} \end{eqnarray*} [[/math]]

In particular we see that the equality [math](\pi\otimes id)\Delta f=1\otimes f[/math] defining [math]C(G/H)[/math] holds on all the coefficients [math]f=u_{ij}[/math] with [math]i \gt k[/math], and this justifies the inclusion appearing in (3).

Let us first try to understand what happens in the group dual case. We will do our study here in two steps, first in the “diagonal” case, and then in the general case. We recall that given a discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], the matrix [math]D=diag(g_i)[/math] is biunitary, and produces a surjective morphism [math]C(U_N^+)\to C^*(\Gamma)[/math]. This morphism can be viewed as corresponding to a quantum embedding [math]\widehat{\Gamma}\subset U_N^+[/math], that we call “diagonal”.

We recall also that the normal closure of a subgroup [math]\Lambda\subset\Gamma[/math] is the biggest subgroup [math]\Lambda'\subset\Gamma[/math] containing [math]\Lambda[/math] as a normal subgroup. Note that [math]\Lambda'[/math] can be different from the normalizer [math]N(\Lambda)[/math]. With these conventions, we have the following result, from :

Proposition

Assume that we have a group dual [math]G=\widehat{\Gamma}[/math], with

[[math]] \Gamma= \lt g_1,\ldots,g_N \gt [[/math]]

diagonally embedded, and let [math]H=G\cap U_k^+[/math].

• [math]H=\widehat{\Theta}[/math], where [math]\Theta=\Gamma/ \lt g_{k+1}=1,\ldots,g_N=1 \gt [/math].
• [math]C_\times(G/H)=C^*(\Lambda)[/math], where [math]\Lambda= \lt g_{k+1},\ldots,g_N \gt [/math].
• [math]C(G/H)=C^*(\Lambda')[/math], where “prime” is the normal closure.
• [math]C_\times(G/H)=C(G/H)[/math] if and only if [math]\Lambda\triangleleft\Gamma[/math].

Show Proof

We use the standard fact that for any group [math]\Gamma= \lt a_i,b_j \gt [/math], the kernel of the quotient map [math]\Gamma\to\Gamma/ \lt a_i=1 \gt [/math] is the normal closure of the subgroup [math] \lt a_i \gt \subset\Gamma[/math].

(1) Since the map [math]C(U_N^+)\to C(U_k^+)[/math] is given on diagonal coordinates by [math]u_{ii}\to v_{ii}[/math] for [math]i\leq k[/math] and [math]u_{ii}\to 1[/math] for [math]i \gt k[/math], the result follows from definitions.

(2) Once again, this assertion follows from definitions.

(3) From the above and from (1) we get [math]G/H=\widehat{\Lambda'}[/math], where [math]\Lambda'=\ker(\Gamma\to\Theta)[/math]. By the above observation, this kernel is exactly the normal closure of [math]\Lambda[/math].

(4) This follows from (2) and (3).

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Let us try now to understand the general group dual case. We recall that the group dual subgroups [math]\widehat{\Gamma}\subset U_N^+[/math] appear by taking a discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] and a unitary [math]J\in U_N[/math], and constructing the morphism [math]C(U_N^+)\to C^*(\Gamma)[/math] given by [math]u\to JDJ^*[/math], where [math]D=diag(g_i)[/math]. With this in hand, Proposition 8.16 generalizes as follows:

Theorem

Assume that we have a group dual [math]G=\widehat{\Gamma}[/math], with

[[math]] \Gamma= \lt g_1,\ldots,g_N \gt [[/math]]

embedded via [math]u\to JDJ^*[/math], and let [math]H=G\cap U_k^+[/math].

• [math]H=\widehat{\Theta}[/math], where [math]\Theta=\Gamma/ \lt g_r=1|\exists\,i \gt k,J_{ir}\neq 0 \gt [/math], embedded [math]u_{ij}\to (JDJ^*)_{ij}[/math].
• [math]C_\times(G/H)=C^*(\Lambda)[/math], where [math]\Lambda= \lt g_r|\exists\,i \gt k,J_{ir}\neq 0 \gt [/math].
• [math]C(G/H)=C^*(\Lambda')[/math], where “prime” is the normal closure.
• [math]C_\times(G/H)=C(G/H)[/math] if and only if [math]\Lambda\triangleleft\Gamma[/math].

Show Proof

We basically follow the proof of Proposition 8.16:

(1) Let [math]\Lambda= \lt g_1,\ldots,g_N \gt [/math], let [math]J\in U_N[/math], and consider the embedding [math]\widehat{\Lambda}\subset U_N^+[/math] corresponding to the following morphism, where [math]D=diag(g_i)[/math]:

[[math]] C(U_N^+)\to C^*(\Lambda)\quad,\quad u\to JDJ^* [[/math]]

Let [math]G=\widehat{\Lambda}\cap U_k^+[/math]. Since we have [math]G\subset\widehat{\Lambda}[/math], the algebra [math]C(G)[/math] is cocommutative, so we have [math]G=\widehat{\Theta}[/math] for a certain discrete group [math]\Theta[/math]. Moreover, the inclusion [math]\widehat{\Theta}\subset\widehat{\Lambda}[/math] must come from a group morphism [math]\varphi:\Lambda\to\Theta[/math]. Also, since [math]\widehat{\Theta}\subset U_k^+[/math], we have a morphism as follows, where [math]V[/math] is a certain [math]k\times k[/math] biunitary matrix over the algebra [math]C^*(\Theta)[/math]:

[[math]] C(U_k^+)\to C^*(\Theta)\quad,\quad v\to V [[/math]]

With these observations in hand, let us look now at the intersection operation. We must have a group morphism [math]\varphi:\Lambda\to\Theta[/math] such that the following diagram commutes:

[[math]] \xymatrix@R=50pt@C=50pt {C(U_N^+)\ar[r]\ar[d]&C(U_k^+)\ar[d]\\ C^*(\Lambda)\ar[r]&C^*(\Theta)} [[/math]]

Thus we must have the following equality:

[[math]] (id\otimes\varphi)(JDJ^*)=diag(V,1_{N-k}) [[/math]]

With [math]f_i=\varphi(g_i)[/math], we obtain from this:

[[math]] \sum_rJ_{ir}\bar{J}_{jr}f_r =\begin{cases} V_{ij}&\mbox{if }i,j\leq k\\ \delta_{ij}&\mbox{otherwise} \end{cases} [[/math]]

Now since [math]J[/math] is unitary, the second part of the above condition is equivalent to “[math]f_r=1[/math] whenever there exists [math]i \gt k[/math] such that [math]J_{ir}\neq 0[/math]”. Indeed, this condition is easily seen to be equivalent to the “[math]=1[/math]” conditions, and implies the “[math]=0[/math]” conditions. We claim that:

[[math]] \Theta=\Lambda\Big/\left \lt g_r=1\Big|\exists\,i \gt k,J_{ir}\neq 0\right \gt [[/math]]

Indeed, the above discussion shows that [math]\Theta[/math] must be a quotient of the group on the right, say [math]\Theta_0[/math]. On the other hand, since in [math]C^*(\Theta_0)[/math] we have [math]J_{ir}g_r=J_{ir}1[/math] for any [math]i \gt k[/math], we obtain that [math](JDJ^*)_{ij}=\delta_{ij}[/math] unless [math]i,j\leq k[/math], so we have, for a certain matrix [math]V[/math]:

[[math]] JDJ^*=diag(V,1_{N-k}) [[/math]]

But the matrix [math]V[/math] must be a biunitary, so we have a morphism [math]C(U_k^+)\to C^*(\Theta_0)[/math] mapping [math]v\to V[/math], which completes the proof of our claim.

(2) Consider the standard generators of the algebra [math]C_\times(G/H)[/math] constructed in Definition 8.15 (3), which are as follows, with indices [math]i \gt k,j \gt 0[/math]:

[[math]] A_{ij}=\sum_rJ_{ir}\bar{J}_{jr}g_r [[/math]]

We have then the following formula:

[[math]] \sum_jA_{ij}J_{jm}=J_{im}g_m [[/math]]

We conclude that [math]C_\times(G/H)[/math] contains any [math]g_r[/math] such that there exists [math]i \gt k[/math] with [math]J_{ir}\neq 0[/math], i.e. contains any [math]g_r\in\Lambda[/math]. Conversely, if [math]g_r\in\Gamma-\Lambda[/math] then [math]J_{ir}g_r=0[/math] for any [math]i \gt k[/math], so [math]g_r[/math] doesn't appear in the formula of any of the generators [math]A_{ij}[/math].

(3,4) The proof here is similar to the proof of Proposition 8.16 (3,4).

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Summarizing, we have a good understanding of the row algebras for the compact quantum groups, both in the classical case, and in the group dual case.

General references

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