# Algebraic isomorphisms of quantized homogeneous spaces

Robert Yuncken

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Robert Yuncken. Algebraic isomorphisms of quantized homogeneous spaces. 2024. hal-04691053

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Distributed under a Creative Commons Attribution 4.0 International License
ALGEBRAIC ISOMORPHISMS OF QUANTIZED HOMOGENEOUS
SPACES

ROBERT YUNCKEN

ABSTRACT. We describe a proof of the following folklore theorem: If $\mathcal{X} = G/K$ is the homogeneous space of a simply connected compact semisimple Lie group with Poisson-Lie stabilizers, then the $q$-deformed algebras of regular functions $C[\mathcal{X}_q]$ with $0 < q \le 1$ are mutually non-isomorphic as $*$-algebras.

# 1. INTRODUCTION

Fix $G$ a compact simply connected semisimple Lie group and $K$ a proper closed Poisson-Lie subgroup. The homogeneous space $\mathcal{X} = G/K$ admits a family of $q$-deformations, $0 < q < \infty$, realized via their $*$-algebras of regular functions $C[\mathcal{X}_q]$.

The following theorem is folklore.

**Theorem 1.1.** $C[\mathcal{X}_p] \cong C[\mathcal{X}_q]$ if and only if $p = q^{\pm 1}$.

In this expository note, we will give a proof of this fact. The ideas are mostly not original. A proof for the quantum spheres $\mathcal{X}_q = S_q^{2n+1}$ was given by D'Andrea [DAn24]. Krähmer [Krä08] proved a similar result for the nonstandard Podleś spheres.

In the present context, the algebras $C[G_q]$ are due to Soibelman [Soi91], see also [Wor87]. The algebras $C[\mathcal{X}_q]$ were introduced over the following years, but a systematic study of their structure was undertaken by Stokman and Dijkhuizen [SD99; Sto03]. This was further developed by Neshveyev and Tuset in [NT12]. The essential points necessary for proving Theorem 1.1 are already contained in the papers just listed.

Giselsson [Gis20] proved that the enveloping $C^*$-algebras $C(\mathcal{X}_q)$ are all isomorphic for $q \in (0, \infty) \setminus \{1\}$. In other words, the quantum spaces $\mathcal{X}_q$ ($0 < q < 1$) are isomorphic as noncommutative topological spaces but not as noncommutative algebraic varieties.

<footnote>Date: September 7, 2024.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary: 20G42, Secondary: 46L67, 17B37.</footnote>

<footnote>Key words and phrases. quantum groups, quantized homogeneous spaces, quantized function algebras, Poisson-Lie groups.</footnote>

<footnote>This research was supported by ANR project OpART (ANR-23-CE40-0016), and by COST Action CaLISTA CA21109 (European Cooperation in Science and Technology), www.cost.eu. The author is grateful for the support and hospitality of the Sydney Mathematics Research Institute and the University of Wollongong, where this article was written.</footnote>
**Acknowledgements.** It is a pleasure to thank Ulrich Krähmer for discussions during a common visit to the University of Wollongong.

## 2. PRELIMINARIES

This article is not completely self-contained. We will make heavy use of the notation and results of [NT12], especially Sections 1, 2 and the beginning of Section 3.

The isomorphism $\mathbb{C}[\mathcal{X}_q] \cong \mathbb{C}[\mathcal{X}_{q-1}]$ is well-known, cf. [NT13, Lemma 2.4.2]. Also, $\mathbb{C}[\mathcal{X}_q]$ is commutative if and only if $q = 1$. We therefore work throughout with $0 < p, q < 1$.

## 3. SU_q(2) AND THE QUANTUM DISK

The algebra $\mathbb{C}[\mathrm{SU}_q(2)]$ is the universal $*$-algebra generated by two elements $\alpha$ and $\gamma$ satisfying

$$ \alpha\gamma = q\gamma\alpha, \quad \alpha\gamma^* = q\gamma^*\alpha, \quad \gamma\gamma^* = \gamma^*\gamma \\ \alpha^*\alpha + \gamma^*\gamma = 1 = \alpha\alpha^* + q^2\gamma^*\gamma. \qquad (3.1) $$

The $*$-subalgebra $\mathbb{C}[\gamma, \gamma^*]$ is a polynomial algebra in 2 commuting conjugate variables. We use the notation

$$ \alpha^{(i)} = \begin{cases} \alpha^i, & \text{if } i \ge 0, \\ (\alpha^*)^{-i}, & \text{if } i < 0. \end{cases} $$

Putting $\mathcal{A}_i = \alpha^{(i)}\mathbb{C}[\gamma, \gamma^*] = \mathbb{C}[\gamma, \gamma^*]\alpha^{(i)}$. we get an algebra grading

$$ \mathbb{C}[\mathrm{SU}_q(2)] = \bigoplus_{i \in \mathbb{Z}} \mathcal{A}_i. \qquad (3.2) $$

The standard irreducible representation $\rho_q$ of $\mathbb{C}[\mathrm{SU}_q(2)]$ on $\mathcal{B}(\ell^2(\mathbb{Z}_+))$ is given in the basis $(e_n)_{n \in \mathbb{Z}_+}$ by

$$ \rho_q(\alpha) : e_n \mapsto \sqrt{1 - q^{2n}} e_{n-1}, \quad \rho_q(\gamma) : e_n \mapsto -q^n e_n. \qquad (3.3) $$

**Lemma 3.1.** *The kernel of $\rho_q$ is the ideal generated by $\gamma - \gamma^*$.*

*Proof.* Let $a \in \ker(\rho_q)$. Write $a = \sum_i c_i \alpha^{(i)} \in \bigoplus_i \mathcal{A}_i$ where $c_i \in \mathbb{C}[\gamma, \gamma^*]$. Since $\rho_q$ maps $\mathcal{A}_i$ to weighted shifts of degree $i$, we have $\rho_q(c_i \alpha^{(i)}) = 0$ for each $i$.

Let $i \ge 0$. We have $\rho_q(c_i \alpha^{(i)} \alpha^{(i)*}) = 0$. Since $\rho_q(\alpha^{(i)} \alpha^{(i)*})$ is diagonal with no zero entries, we get $\rho_q(c_i) = 0$. The spectrum of $\rho_q(\gamma) = \rho_q(\gamma^*)$ is $-q^{\mathbb{Z}_+}$ which is Zariski dense in $\mathbb{R}$. So if we view $c_i$ as a polynomial in $\mathbb{C}[\gamma, \gamma^*] \cong \mathbb{C}[\mathbb{Z}, \bar{\mathbb{Z}}]$, it is zero on the real line and hence divisible by $\gamma - \gamma^*$. For $i < 0$, we can take the adjoint to reduce to the previous case. $\square$

We define $\mathbb{C}[\overline{\mathbb{D}}_q] = \mathbb{C}[\mathrm{SU}_q(2)]/\langle\gamma - \gamma^*\rangle \cong \rho_q(\mathbb{C}[\mathrm{SU}_q(2)])$ and let $F_q : \mathbb{C}[\mathrm{SU}_q(2)] \to \mathbb{C}[\overline{\mathbb{D}}_q]$ denote the quotient map. Then $\mathbb{C}[\overline{\mathbb{D}}_q]$ has generators

$$ y = F_q(\gamma), \quad z = F_q(\alpha)^* \qquad (3.4) $$

and relations

$$ yz = qzy, \quad y = y^*, \quad zz^* + y^2 = 1 = z^*z + q^2y^2. \qquad (3.5) $$
In Poisson language, $F_q$ is the restriction of polynomials on $SU_q(2)$ to a certain quantized symplectic leaf.

*Remark 3.2.* The $*-\text{algebra } \mathbb{C}[\overline{\mathbb{D}}_q]$ embeds into the $\mathbb{C}^*-\text{algebra}$ of the quantum disk $\mathbb{C}(\overline{\mathbb{D}}_q)$ of [NT12, §3] via the homomorphism

$$z \mapsto Z_q, \quad y \mapsto -(1 - Z_q Z_q^*)^{\frac{1}{2}}. \qquad (3.6)$$

This is not an algebraic map, which is why we add the generator $y$.

### 4. HOMOGENEOUS SPACES FOR SU_q(2)

Write $T$ for the diagonal torus in $SU(2)$. For $n \in \{1, 2, \dots, \}$, let $T_n$ denote the closed subgroup of $T$ generated by the $n$th roots of unity. We also use the convention $T_\infty = T$. The associated quantum homogeneous spaces [NT12, §2] are given by the quantized function algebras

$$\mathbb{C}[\mathrm{SU}_q(2)/T_n] = \mathrm{span}\{\alpha^{(i)}\gamma^j\gamma^{*k} | i+j-k \in n\mathbb{Z}\} \subseteq \mathbb{C}[\mathrm{SU}_q(2)], \quad (4.1)$$

where we use the convention $\infty\mathbb{Z} = \{0\}$. Explicitly $\mathrm{SU}_q(2)/T_n$ is $\mathrm{SU}_q(2)$ when $n=1$, $\mathrm{SO}_q(3)$ when $n=2$ [Pod95], the quantum lens space $L_q^3(n, 1)$ when $3 \le n < \infty$ [HS03] and the standard Podleś sphere when $n = \infty$ [Pod87].

Let us also write

$$\mathbb{C}[\overline{\mathbb{D}}_q/T_n] = F_q(\mathbb{C}[\mathrm{SU}_q(2)/T_n]) \cong \rho_q(\mathbb{C}[\mathrm{SU}_q(2)/T_n]).$$

This is the quantization of a 2-dimensional symplectic leaf of $\mathrm{SU}_q(2)/T_n$.

### 5. COMMUTATOR SPECTRUM

**Definition 5.1.** Define the commutator spectrum of an algebra $\mathcal{A}$ to be

$$\mathrm{CommSp}(\mathcal{A}) = \{\omega \in \mathbb{C}^* \mid \exists a, b \in \mathcal{A} \setminus \{0\} \text{ with } ab = \omega ba\}.$$

**Lemma 5.2.** The commutator spectrum of $\mathbb{C}[\overline{\mathbb{D}}_q/T_n]$ is

$$\mathcal{Q}(\mathbb{C}[\overline{\mathbb{D}}_q/T_n]) = \begin{cases} q^{\mathbb{Z}} & \text{if } n \text{ is odd} \\ q^{2\mathbb{Z}} & \text{if } n \text{ is even or } \infty. \end{cases}$$

*Proof.* Consider first the quantum plane algebra $\mathbb{C}_q[y, z]$ generated by $y, z$ with $yz = qzy$. We do not impose a $*-\text{structure}$ for the moment. We have

$$(y^jz^k)(y^{j'}z^{k'}) = q^{jk' - j'k}(y^{j'}z^{k'})(y^jz^k). \qquad (5.1)$$

This shows that $q^{\mathbb{Z}} \subseteq \mathrm{CommSp}[\mathbb{C}_q[y, z]]$. Moreover, by considering the leading terms for the lexicographical ordering on powers of $y$ and $z$, one can deduce that $\mathrm{CommSp}(\mathbb{C}_q[y, z]) = q^{\mathbb{Z}}$.

Now suppose $\omega \in \mathrm{CommSp}(\mathbb{C}[\overline{\mathbb{D}}_q/T_n])$, so we have $a, b \in \mathbb{C}[\overline{\mathbb{D}}_q/T_n]$ nonzero with $ab = \omega ba$. By considering the highest order terms in the grading (3.2), we can assume without loss of generality that $a \in F_q(\mathcal{A}_i)$, $b \in F_q(\mathcal{A}_{i'})$ for some $i, i'$. If both $i, i' \le 0$ then $a, b \in \mathbb{C}_q[y, z]$ so $\omega \in q^{\mathbb{Z}}$. If both $i, i' > 0$ we can reduce to the previous case by taking adjoints. Finally, if $i < 0$ and $i' > 0$ then we can reduce to the previous case by replacing $b$ by $a^m b$ for some sufficiently large $m$.
This proves that $\mathrm{CommSp}(\mathbb{C}[\overline{\mathbb{D}}_q/T_n]) \subseteq q^{\mathbb{Z}}$, and also that it is realized on monomials $a = y^jz^k$, $b = y^{j'}z^{k'} \in \mathbb{C}[\overline{\mathbb{D}}_q/T_n]$. Note also that it is closed under taking integer powers, by considering $a$ and $b^m$.

If $n$ is even or $\infty$, then Equation (4.1) forces $j \equiv k$ [2] and $j' \equiv k'$ [2]. Therefore, by Equation (5.1), $\mathrm{CommSp}(\mathbb{C}[\overline{\mathbb{D}}_q/T_n]) \subseteq q^{2\mathbb{Z}}$. Putting $a = F_q(\gamma^*\alpha) = yz^*, b = F_q(\gamma^*\gamma) = y^2$ gives $\omega = q^2$.

If $n$ is odd, we can put $a = F_q(\gamma^{n+1}\alpha^*) = y^{n+1}z$, $b = F_q(\gamma^{\frac{n+1}{2}}\gamma^{*\frac{n-1}{2}}\alpha^*) = y^n z$ and we get $\omega = q$. $\square$

## 6. QUANTIZED HOMOGENEOUS SPACES

Let $\Pi = \{\alpha_1, \dots, \alpha_r\}$ be the simple roots of $G$ and $\{\omega_1, \dots, \omega_r\}$ the associated fundamental weights. We write $(\cdot, \cdot)$ for the $G$-invariant bilinear form on $\mathfrak{h}^*$ with $(\alpha, \alpha) = 2$ for all short roots $\alpha$, and we use the standard notation $q_i = q^{(\alpha_i, \alpha_i)/2}$. Thus $q_i = q, q^2$ or $q^3$ depending on the length of $\alpha_i$. For every $\alpha_i \in \Pi$, there is a restriction map $\sigma_i : \mathbb{C}[G_q] \to \mathbb{C}[SU_{q_i}(2)]$.

The Poisson-Lie subgroups of $G$ are determined by pairs $(S, L)$ where $S$ is a subset of the simple roots for $G$ and $L$ is a subgroup of the lattice $P(S^c) = \mathbb{Z}\{\omega_i \mid \alpha_i \in S^c\}$. We refer to [NT12, Proposition 1.1] for the details. Fix such a subgroup $K^{S,L}$ and consider $\mathcal{X} = G/K^{S,L}$ and its quantized algebra of function $\mathbb{C}[\mathcal{X}_q]$.

**Lemma 6.1.** For every $\alpha_i \in S^c$ we have $\sigma_i(\mathbb{C}[\mathcal{X}_q]) = \mathbb{C}[SU_{q_i}(2)/T_{n_i}]$ for some $n_i \in \{1, 2, \dots, \infty\}$.

*Proof.* This follows from [NT12, Corollary 2.4]. Explicitly, $n_i$ is the generator of the subgroup of $\mathbb{Z}$ obtained by restricting the weights $\mu \in L$ to weights for $U_{q_i}(sl_2)$, i.e., $n_i\mathbb{Z} = \{(\mu, \alpha_i) \mid \mu \in L\}$. $\square$

Let $T$ be the maximal torus of $G$, and $T_L \subseteq T$ the subgroup annihilated by $L \subseteq P = \hat{T}$. Let $W^S \subset W$ be the set of $w \in W$ such that $w(\alpha_i) > 0$ for all $\alpha_i \in S$.

The irreducible representations of $\mathbb{C}[\mathcal{X}_q]$ are indexed by pairs $(w, t) \in W^S \times T/T_L$. These representations $\pi_{w,t}$ are described explicitly in the pre-amble to Theorem 2.2 of [NT12]. See also [SD99, Theorem 5.9]. We recall only that in the case where $w = s_i$ is the simple reflection in $\alpha_i \in S^c$, the associated representation is

$$ \pi_{s_i, 1} = \rho_q \circ \sigma_i : \mathbb{C}[\mathcal{X}_q] \to \mathbb{C}[\overline{\mathbb{D}}_{q_i}/T_{n_i}] \to \mathcal{B}(\ell^2(\mathbb{Z}_+)). \quad (6.1) $$

## 7. QUANTUM 2-CELL REPRESENTATIONS

**Definition 7.1.** An irreducible representation $\pi$ of $\mathbb{C}[\mathcal{X}_q]$ will be called a 2-cell representation if the norm closure of the image is an extension of an abelian $C^*$-algebra by the compacts—that is, if it fits into an exact sequence of $C^*$-algebras

$$ 0 \to \mathcal{K} \to \pi(\mathbb{C}(\mathcal{X}_q)) \to A \to 0 $$

with $A$ abelian and $\mathcal{K}$ being the compacts on a separable infinite dimensional Hilbert space.
**Example 7.2.** The standard representation $\rho_q$ of $\mathbb{C}[\mathrm{SU}_q(2)/T_n]$ is a 2-cell representation for any $n \in \{1,2,\dots,\infty\}$. To see this, let $I$ denote the $*$-ideal of $\mathbb{C}[\mathrm{SU}_q(2)]$ generated by $\gamma$ and put $I_n = I \cap \mathbb{C}[\mathrm{SU}_q(2)/T_n]$. Then $\overline{\rho_q(I_n)} = \mathcal{K}(\ell^2(\mathbb{Z}_+))$ since it contains $\rho_q(\alpha^*\gamma)$ which is a compact weighted shift with distinct nonzero weights. And putting $\gamma = 0$ in the relations (3.1) shows that $\mathbb{C}[\mathrm{SU}_q(2)/T_n]/I_n$ is abelian.

**Lemma 7.3.** The representation $\pi_{w,t}$ is a 2-cell representation if and only if $w$ is a simple reflection, i.e. $w = s_i$ for some $\alpha_i \in S^c$.

*Proof.* The representation $\pi_{w,t}$ is the representation $\pi_{w,1}$ twisted by a character, so the image of $\pi_{t,w}(\mathbb{C}[\chi_q])$ is independent of $t$. Therefore, we may take $t=1$.

If $w=1$, then $\pi_{1,1}$ is the counit.

If $w = s_i$ for $i \in S^c$, then by Lemma 6.1 $\pi_{w,1}(\mathbb{C}[\chi_q]) = \rho_{q_i}(\mathbb{C}[\mathrm{SU}_{q_i}(2)/T_{n_i}])$, so $\pi_w$ is a 2-cell representation by Example 7.2.

If $w \in W^S$ has Bruhat length two or more, then [NT12, Theorem 4.1] shows that $\pi_{w,1}(\mathbb{C}(\chi_q))$ has a nontrivial nonabelian quotient, so $\pi_{w,1}$ is not a 2-cell representation.

<figure><img src="image_2.png" /></figure>

## 8. PROOF OF THE MAIN THEOREM

*Proof of Theorem 1.1.* By Lemmas 5.2, 6.1 and 7.3, the greatest value, less than one, of the commutator spectrum of the image of any 2-cell representation of $\mathbb{C}[\chi_q]$ is $q^m$, for some $m \in \mathbb{N}$ independent of $q$. This is an invariant of $\mathbb{C}[\chi_q]$.

<figure><img src="image_3.png" /></figure>

## REFERENCES

[DAn24] Francesco D'Andrea. *Isomorphisms of quantum spheres*. 2024. arXiv: 2406.17288 [math.QA]. URL: https://arxiv.org/abs/2406.17288.

[Gis20] Olof Giselsson. "q-independence of the Jimbo-Drinfeld quantization". In: *Comm. Math. Phys.* 376.3 (2020), pp. 1737–1765. ISSN: 0010-3616. DOI: 10.1007/s00220-019-03660-9. URL: https://doi.org/10.1007/s00220-019-03660-9.

[HS03] Jeong Hee Hong and Wojciech Szymański. "Quantum lens spaces and graph algebras." English. In: *Pac. J. Math.* 211.2 (2003), pp. 249–263. ISSN: 1945-5844. DOI: 10.2140/pjm.2003.211.249.

[Krä08] Ulrich Krähmer. "On the non-standard Podleś spheres". English. In: *C*-algebras and elliptic theory II. Selected papers of the international conference, Bedlewo, Poland, January 2006. Basel: Birkhäuser, 2008, pp. 145–147. ISBN: 978-3-7643-8603-0.

[NT12] Sergey Neshveyev and Lars Tuset. "Quantized algebras of functions on homogeneous spaces with Poisson stabilizers". English. In: *Commun. Math. Phys.* 312.1 (2012), pp. 223–250. ISSN: 0010-3616. DOI: 10.1007/s00220-012-1455-6.
[NT13] Sergey Neshveyev and Lars Tuset. *Compact quantum groups and their representation categories*. Vol. 20. Cours Spécialisés [Specialized Courses]. Société Mathématique de France, Paris, 2013, pp. vi+169. ISBN: 978-2-85629-777-3.

[Pod87] P. Podleś. "Quantum spheres". English. In: *Lett. Math. Phys.* 14 (1987), pp. 193–202. ISSN: 0377-9017. DOI: 10.1007/BF00416848.

[Pod95] Piotr Podleś. "Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups". English. In: *Commun. Math. Phys.* 170.1 (1995), pp. 1–20. ISSN: 0010-3616. DOI: 10.1007/BF02099436.

[SD99] Jasper V. Stokman and Mathijs S. Dijkhuizen. "Quantized flag manifolds and irreducible *-representations". English. In: *Commun. Math. Phys.* 203.2 (1999), pp. 297–324. ISSN: 0010-3616. DOI: 10.1007/s002200050613.

[Soi91] Ya. S. Soibelman. "The algebra of functions on a compact quantum group, and its representations". English. In: *Leningr. Math. J.* 2.1 (1991), pp. 161–178. ISSN: 1048-9924.

[Sto03] Jasper V. Stokman. "The quantum orbit method for generalized flag manifolds". English. In: *Math. Res. Lett.* 10.4 (2003), pp. 469–481. ISSN: 1073-2780. DOI: 10.4310/MRL.2003.v10.n4.a6.

[Wor87] S. L. Woronowicz. "Compact matrix pseudogroups". In: *Comm. Math. Phys.* 111.4 (1987), pp. 613–665. ISSN: 0010-3616. URL: http://projecteuclid.org/euclid.cmp/1104159726.