Akshay Bhuva, Surajit Biswas, Bipul Saurabh

Topological invariance of quantum homogeneous spaces of  type B and D

In this article,  we study two families of quantum homogeneous spaces, namely,  $SO_q(2n+1)/SO_q(2n-1)$, and $SO_q(2n)/SO_q(2n-2)$. By applying a two-step Zhelobenko branching rule, we show that the $C^*$-algebras $C(SO_q(2n+1)/SO_q(2n-1))$, and $C(SO_q(2n)/SO_q(2n-2))$ are generated by the entries of the first and the last rows of the fundamental matrix of the quantum groups $SO_q(2n+1)$, and $SO_q(2n)$, respectively.  We then construct  a chain of short exact sequences, and using that, we  compute $K$-groups of these spaces with explicit generators.
Invoking  homogeneous $C^*$-extension theory, we show $q$-independence of some  intermediate $C^*$-algebras arising as the middle $C^*$-algebra of these  short exact sequences. As a consequence, we get the $q$-invariance of $SO_q(5)/SO_q(3)$ and $SO_q(6)/SO_q(4)$.

AMS Subject Classification No.: 58B34, 46L80, 19K33

Keywords.  Homogeneous extension, $m$-torsioned quantum double suspension , corona algebra.

§ INTRODUCTION

Let $G$ be a semisimple compact  Lie group with complexified Lie algebra $\mathfrak{g}$.  Fix $ 0<q<1$.
The algebra of functions $C(G_q)$ on its  $q$-deformation $G_q$ is defined as the
enveloping $C^*$-algebra of the Hopf $*$-algebra generated by
matrix coefficients of all finite-dimensional representations of Quantized universal enveloping algebra (QUEA)  $U_q(g)$.  It turns out that if $H$ is a closed  Poisson Lie subgroup of $G$, then  $H_q$ is a quantum subgroup of $G_q$. The  $C^*$-algebra
$C(G_q/H_q)$ underlying the quotient space $G_q/H_q$
is a $C^*$-subalgebra of $C(G_q)$ generated by matrix elements of certain finite dimensional representations
of $U_q(g)$.   One of the main problems in noncommutative geometry (NCG) is to see how the theory of quantum groups and their quotient spaces fits under Connes formulation of NCG (see [7] for details).  Thus, it becomes necessary to understand the $C^*$-algebra underlying these spaces.  The direct approach of exploring the operators obtained as images of the generators of $C(G_q/H_q)$ under a faithful representation, as mentioned in ([14], [17]), seems to be complicated. Other possible approaches could be to see whether the given $C^*$- algebra can be associated with a graph, groupoid, or semigroup or it can be obtained by applying some noncommutative operations on a simple, maybe a classical space. Many articles have investigated this (see [9], [22])).  However, most discuss $SU_q(n)$ and its homogeneous spaces. The $C^*$-algebra of quotient spaces of other types is much less explored.   To give a glimpse of the situation, the very first question of proving whether $C(SO_q(N)/SO_q(N-2))$ is given by a finite set of generators and relations  has not been answered yet. Therefore, it is worthwhile to investigate such spaces.  In this article, we take up homogeneous spaces of type $B$ and type $D$ and explore their topological properties using the tools of extension theory.

$C^*$-extension theory has its origin in the work of
Brown, Douglas, and Fillmore ([3]), where the authors  classified all essentially normal operators  acting on an infinite dimensional separable Hilbert space  with essential spectrum $X$ up to essentially unitarily equivalence  by  proving that such an operator $A$ can be identified,  precisely,  by the set of indices  of the Fredholm operators $A-\lambda I$, where $I$ is the identity operator and $\lambda \in \bbc \setminus X$. Later they converted this classification problem to the classification of all essential extension of $C(X)$ by compact operators as any essential normal operator with essential spectrum $X$ gives rise to  an essential extension

\[
0 \rightarrow \clk \rightarrow C^*(\{N,\clk\}) \rightarrow C(X) \rightarrow 0.
\]

Kasparov ([12])  extended this concept by considering a group   $Ext(A,B)$  of stable unitary equivalence classes of essential   $C^*$-algebra extensions of $A$ by $B \otimes \clk$, where $A$ is nuclear and separable, and $B$ is separable.
However, if $B\neq \clk$, then $Ext(A,B)$  remains silent regarding any information  about unitary equivalence classes of such extensions, and therefore, two elements in the same class
may have non-isomorphic middle $C^*$-algebras.
For a nuclear $C^*$-algebra $A$ and a
finite dimensional compact metric space $Y$,
Pimsner, Popa and Voiculescu ([19]) constructed another group $Ext_{PPV}(Y,A)$
consisting of strong unitary equivalence classes of unital homogeneous
extensions of $A$ by $C(Y) \otimes \clk$.

We say that the $C^*$-algebra of a quotient space $G_q/H_q$  is  $q$-invariant  if for different values of $q \in (0,1)$,  $C(G_q/H_q)$ are isomorphic.
In ([9]),  Hong and Szymanski showed that the odd dimensional quantum sphere $C(S_q^{2n+1})=C(SU_q(n+1)/SU_q(n))$ can be obtained by applying  quantum double suspension (QDS)  operation to  $C(\bbbt)$ iteratively, and  as a result, one gets $q$-invariance of $C(S_q^{2n+1})$.
Chakraborty and Sundar [4] exploited this fact to construct good spectral triples of $C(S_q^{2n+1})$.  	Lance ([16]) proved $q$-invariance of $C(SO_q(3))$, which,  to the best of our understanding,   has a flaw.  The author  correctly established the following short exact sequence of $C^*$-algebras:

\[
\xi: 0 \rightarrow C(\bbbt)\otimes \clk \rightarrow C(SO_q(3)) \rightarrow C(\bbbt) \rightarrow 0.
\]

This extension can be equivalently  described in terms of its Busby invariant $\beta: C(\bbbt) \rightarrow Q(C(\bbbt) \otimes \clk)$. However, the author argues that $Q(C(\bbbt) \otimes \clk)$ is just $Q(\clk)$ again, so the extension is still specified up to
strong equivalence by an index,
which is not true. One way to see this is by the Kunneth theorem, it follows that  the group $\mbox{Ext}(C(\bbbt), C(\bbbt))=KK^1(C(\bbbt), C(\bbbt))$ is isomorphic to $\bbz^2$, not $\bbz$, and hence all extensions can't be distinguished by an integer as claimed in [16].  Though flawed, the argument clearly suggests the need for a group, which is based on  unitary equivalence classes of essential extensions.  In ([21]), Saurabh  used such a  group, namely,  $Ext_{PPV}(\bbbt,C(S_0^{2\ell+1}))$ group, showed that $K_0$-group is the complete invariant of the middle $C^*$-algebras of  extensions in the group, and thus,  proved $q$-invariance of quantum quaternion spheres.

The paper is organized as follows.
In Sect.$~2$, we begin with an overview of the quantum group $SO_q(N)$ using FRT approach as given in [13].  We discuss irreducible representations of $C(SO_q(N))$ using the dual pairing between  $U_q(\mathfrak{so}_N)$ and $\mathcal{O}(SO_q(N))$.   In Sect.$~3$, we recall the $U_q(\mathfrak{so}_N)$-module structure on the  quotient space $SO_q(N)/SO_q(N-2)$. Applying  the Zhelobenko branching rule in two steps, we  obtain the multiplicity of each co-representations of $SO_q(N)$ occuring in $C(SO_q(N)/SO_q(N-2))$. Using that, we establish that the quotient space  $C(SO_q(N)/SO_q(N-2))$ is generated by the matrix entries of the first and the last row of the generating fundamental matrix of $SO_q(N)$.	 Furthermore, we list out all of its irreducible representations explicity, and  obtain its faithful representation. In the next section, we computed the $K$-groups with explicit generators. It is worth mentioning that the $K$-groups of $C(SO_q(N)/SO_q(N-2))$ is known, thanks to the $KK$-equivalence with its classical counterpart (see [17]).  However, here we obtain their generators explicitly, which could be helpful in many situations, for example, in finding $K$-theory-$K$-homolgy pairing through index computation. In Sect. 5, we rectify Lance's argument [16] and prove the \( q \)-invariance of \( C(SO_q(3)) \) using homogeneous \( C^* \)-extension theory. We also apply this theory to prove the \( q \)-invariance of the \( C^* \)-algebras \( C(SO_q(5)/SO_q(3)) \) and \( C(SO_q(6)/SO_q(4)) \).

Let $\mathbb{T}$ denote the set of complex numbers whose modulus is $1$, and let $q$ be a real number lying in the interval $(0,1)$. Define $\mathbb{N}_{0}=\mathbb{N}\cup \{0\}$. The standard bases of the Hilbert spaces $\ell^2(\mathbb{N}_0)$ and
$\ell^2(\mathbb{Z})$ will be denoted by $\{e_n : n \in \mathbb{N}_0\}$ and $\{e_n : n \in \mathbb{Z}\}$ respectively. The length of a Weyl word $w$ is denoted by $\ell(w)$. The number operator $e_n\mapsto ne_n$ is denoted by $N$. The letter $S$ is for the left shift operator  $e_n\longmapsto e_{n-1}$. The $C^{\ast}$-subalgebra of $\cll(\ell^2(\mathbb{N}_0))$ generated by $S$ is denoted by $\scrt$.
Let $\sigma: \scrt\longrightarrow\mathbb{C} $
be the homomorphism for which $\sigma(S)=1$. The multiplier algebra and the corona algebra of a $C^*$-alebra $A$ is denoted  by $M(A)$ and   $Q(A)$, respectively.

§ PRELIMINARIES

We begin by recalling some key aspects of the compact quantum group $C(SO_q(N))$ as detailed in <cit.>.

§.§ The $C^*$-algebra $C(SO_q(N))$

In this subsection, we provide a brief overview of the Hopf algebra structure associated with the compact quantum group $SO_q(N)$ as introduced in <cit.>. We begin by introducing key notations. Let $N\geq 3$, and for $1\leq i,j,m,n\leq N$ we define
i'=    N+1-i,
ρ_i  =  N/2-i  if  i < i',
ρ_i' = -ρ_i  if  i ≤i',
C_j^i =  δ_ij' q^-ρ_i,

(q-q^-1)(δ_jmδ_in- C_i^jC_n^m)      i>m, q^δ_ij-δ_ij^'δ_imδ_jn       i≤m, ,
where $\delta_{ij}$ represents the Kronecker delta function. Let $A(R)$ be the unital associative algebra generated by $v_j^i$, $i,j=1,2,\ldots, N$, subject to the following relations:
i,j,s,t=1,2,…, N.

The matrices $(\!(v_j^i)\!)$ and $(\!(C_j^i)\!)$ are denoted as $V$ and $C$, respectively. Define $J$ as the two-sided ideal of $A(R)$ generated by the entries of the matrices $VCV^t C^{-1}-I$ and $CV^t C^{-1}V-I$. Let $\mathcal{O}\left(O_q(N)\right)$ denote the quotient algebra $A(R)/J$. The Hopf $*$-algebra structure on $\mathcal{O}(O_{q}(N))$ comes from the following maps.

\begin{align*}
:\Delta(v_{l}^{k}) = \sum_{i=1}^{N}v_{i}^{k} \otimes v_{l}^{i},~~~~~\bullet\text{Counit}:
\epsilon(v_{l}^{k}) = \delta_{kl}, \\& \bullet\text{Antipode}:
S(v_l^k) = q^{\rho_k-\rho_l}v_{l'}^{k'}, ~~~~~~~~~~~~~~\bullet\text{Involution}:
(v_{l}^{k})^* = q^{\rho_k - \rho_l}v_{l'}^{k'}.
\end{align*}

Let $\mathcal{D}_q$ be the quantum determinant of the matrix $V$ for the quantum group $O_q(N)$ <cit.>. Denote the quotient $\mathcal{O}(O_q(N))/\langle \mathcal{D}_q - 1\rangle$ of $\mathcal{O}(O_q(N))$ by the two-sided ideal $\langle\mathcal{D}_q - 1\rangle$ as $\mathcal{O}(SO_q(N))$. The Hopf $*$-algebra structure on $\mathcal{O}(SO_q(N))$ is induced from $\mathcal{O}(O_q(N))$. In $\mathcal{O}\left(SO_q(N)\right)$, the relation $V^*=CV^t C^{-1}$ leads to the following:

\begin{equation}\label{Relation 2.2}
VV^* = V^*V = I.
\end{equation}

The algebra $\mathcal{O}\left(SO_q(N)\right)$ becomes a normed $*$-algebra with the norm defined by

$$\|a\|=\sup\left\{\|\pi(a)\|: \pi \text{ is a representation of } \mathcal{O}\left(SO_q(N)\right)\right\} \text{ for } a\in \mathcal{O}(SO_q(N)).$$

With relation <ref>, we can conclude that $\left\|v_j^i\right\|\leq 1$. This implies that for all $a\in\mathcal{O}\left(SO_q(N)\right)$, $\|a\| < \infty$. We denote the completion of $\mathcal{O}\left(SO_q(N)\right)$ as $C\left(SO_q(N)\right)$. The pair $\left(C\left(SO_q(N)\right),\Delta\right)$ forms a compact quantum group known as a $q$-deformation of the group $SO(N)$. In particular, we define $SO_q(2)$ to be the circle group $\mathbb{T}$, and $SO_q(1)$ to be the trivial group containing only the identity element. For details, please see <cit.>.

§.§ Dual paring between Hopf algebras

In this subsection, we recall from [13] the dual paring between $U_{q}(\mathfrak{g})$ and $\mathcal{O}(G_q)$.
Consider $N=2n+1$, $2n$, $2$, which correspond to the algebras $U_{q^{1/2}}(\mathfrak{so}_{2n+1})$, $U_{q}(\mathfrak{so}_{2n})$, and $U_{q}(\mathfrak{sl}_2)$, respectively.

There exist unique nondegenerate dual paring between $U_{q^{1/2}}(\mathfrak{so}_{2n+1})$  and $\mathcal{O}(SO_q(2n+1))$, $U_{q}(\mathfrak{so}_{2n})$  and $\mathcal{O}(SO_q(2n))$,  $U_{q}(\mathfrak{sl}_2)$ and $\mathcal{O}(SL_q(2)$
such that

\begin{align}\label{eqpair}
\left\langle f,v_j^i\right\rangle=t_{ij}(f), \quad i,j=1,2,\hdots N,
\end{align}

where $t_{ij}(f)$ be the matrix entries of $T_1(f)$, where $T_1$ is the vector representation of
$U_{q^{1/2}}(\mathfrak{so}_{2n+1})$, $U_{q}(\mathfrak{so}_{2n})$, $U_{q}(\mathfrak{sl}_2)$, respectively.
We will explicitly describe $T_1$ for these three cases and determine the pairing. Let $E_i,~ F_i, ~K_i$
and $K_i^{-1}$ be generators of  $U_{q^{1/2}}(\mathfrak{so}_{2n+1})$, $U_{q}(\mathfrak{so}_{2n})$, and $U_{q}(\mathfrak{sl}_2)$, where $i=1,\hdots,n$. For more details, please refer to <cit.>.
Define $I_{i,j}$ as a $N \times N$
matrix with $1$ in the $(i,j)^{th}$ position and $0$ elsewhere, and $D_{j}$ as a diagonal matrix with $q$ in the $(j,j)^{th}$
position and $1$ elsewhere on the diagonal.

* For the QUEA $U_{q^{1/2}}(\mathfrak{so}_{2n+1})$, we have

\[ \left.
\begin{array}{rcl}
T_1(K_i) &=& D_i^{-1}D_{i+1}D_{2n-i+1}^{-1}D_{2n-i+2}, \nonumber\\
T_1(E_i) &=& I_{i+1,i}-I_{2n-i+2,2n-i+1},\nonumber\\
T_1(F_i) &=& I_{i,i+1}-I_{2n-i+1,2n-i+2},\nonumber
\end{array} \right\}\quad  \mbox{for} \quad i \in \{1,2,\hdots,n-1\},
\]

and for $i=n$,

\[
T_1(K_n) = D_n^{-1}D_{n+2},~~
T_1(E_n) = c(I_{n+1,n}-q^{1/2}I_{n+2,n+1}),~~
T_1(F_n) = c(I_{n,n+1}-q^{-1/2}I_{n+1,n+2}),
\]

where $c=(q^{1/2}+q^{-1/2})^{1/2}$.

* For the QUEA $U_{q}(\mathfrak{so}_{2n})$, one has

\[ \left.
\begin{array}{rcl}
T_1(K_i) &=& D_i^{-1}D_{i+1}D_{2n-1}^{-1}D_{2n-i+1}. \nonumber\\
T_1(E_i) &=& I_{i+1,i}-I_{2n-i+1,2n-i}.\nonumber\\
T_1(F_i) &=& I_{i,i+1}-I_{2n-i,2n-i+1}.\nonumber
\end{array} \right\}\quad  \mbox{for} \quad i \in \{1,2,\hdots,n-1\}
\]

and for $i=n$

\[
\begin{array}{rcl}
T_1(K_n) = D_{n-1}^{-1}D_n^{-1}D_{n+1}D_{n+2},
~T_1(E_n) = -I_{n+2,n}+I_{n+1,n-1},
~T_1(F_n) = -I_{n,n+2}+I_{n-1,n+1}. \nonumber
\end{array}
\]

* For $U_{q}(\mathfrak{sl}_{2})$, we have
T_1(K) =   (
q^-1     0

0    q

),    T_1(E) =  (
0     0

1    0

),     T_1(F) =  (
0     1

0    0

We will utilize these pairings to write down irreducible representations of $\mathcal{O}(\text{SO}_q (N))$, which can be extended to $C(\text{SO}_q (N))$ to obtain elementary representations of $C(\text{SO}_q (N))$.

§.§ Irreducible Representation of $C(SO_{q}(N))$

Let $\Pi$ denote the set  $\{\alpha_1, \alpha_2, \ldots, \alpha_n\}$ consisting of simple roots of $\mathfrak{so}_{N}$. To keep the notations simple, we denote the root $\alpha_i$ as $i$ and the reflection $s_{\alpha_i}$ defined by the root $\alpha_i$ as $s_i$. The Weyl group $W_n$ of $\mathfrak{so}_{N}$ can be represented as the group generated by the reflections $\{s_i : 1 \leq i \leq n\}$.

Elementary representation of $C(SO_{q}(N))$: For $1\leq i \leq n$, let $d_i=\left\langle \alpha_i, \alpha_i\right\rangle$ and $q_i=q^{d_i}$. Let $K$, $E$, and $F$ be the standard generators of $U_{q_i}(\mathfrak{sl}_2)$. Let  $\Psi_i : U_{q_i}(\mathfrak{sl}_2) \longrightarrow U_{q}(\mathfrak{so}_N)$ be a homomorphism given on generators by,

$$\Psi_i(K)=K_i,~~\Psi_i(E)=E_i,~~\Psi_i(F)=F_i .$$

By duality, there exist a surjective homomorphism

Ψ_i^* :C(SO_q(N)) ⟶C(SU_q_i(2))

given by

⟨f, Ψ_i^*(v_l^k)⟩= ⟨Ψ_i(f), v_l^k⟩,

where $\left\langle\cdot,\cdot\right\rangle$ given by equation $\ref{eqpair}$.  Consider the matrix $\left( \begin{smallmatrix}
u_1^1 & u_2^1\\
u_1^2 & u_2^2
\end{smallmatrix}\right)$  whose entries are generators of $\mathcal{O}(SU_{q}(2))$. Let $\pi$ represent the following representation of $C(SU_{q}(2))$ on $\ell^2(\mathbb{N}_0)$:

\[
\pi(u_l^k)=\begin{cases}
\sqrt{1-q^{2N+2}}S & \mbox{ if } k=l=1,\cr
S^{*}\sqrt{1-q^{2N+2}} & \mbox{ if } k=l=2,\cr
-q^{N+1} & \mbox{ if } k=1,~l=2,\cr
q^N & \mbox{ if } k=2,~l=1.\cr           \end{cases}
\]

For each $i=1,2,\hdots,n$, define a map $\pi_{s_{i}} = \pi \circ \Psi_{i}^{*}$ of
$C(SO_{q}(N))$. Each $\pi_{s_{i}}$ is an irreducible elementary representation of
$C(SO_{q}(N)).$  For $t=(t_{1},t_{2},\hdots ,t_{n}) \in \bbbt^{n}$, define the one dimensional representation  $\tau_t: C(SO_{q}(N)) \longrightarrow \bbc $ by

\[
\tau_{t}(v_j^i)=\begin{cases}
\overline{t_{i}}\delta_{ij} & \mbox{ if } i \leq n,\cr
t_{N+1-i}\delta_{ij} & \mbox{ if } i > n.\cr
\end{cases}
\]

For any two representations $\phi_1$ and $\phi_2$ of $C(SO_q (N))$, define a representation $\phi_1 \ast \phi_2 :=(\phi_1 \otimes \phi_2)\circ\Delta$. For $w\in W_n$ such that $s_{i_1}s_{i_2}\cdots s_{i_n}$ is a reduced form for $w$ and $t\in\mathbb{T}^n$, we define a representation $\pi_{t,w}$ by $\tau_t\ast\pi_{s_{i_1}}\ast\pi_{s_{i_2}}\ast\cdots\ast\pi_{s_{i_n}}$. When $t=1$, we denote the representation $\pi_{t,w}$ by $\pi_{w}$.

Let $t\in\mathbb{T}^n$ and $w\in W_n$. Then the representation $\pi_{t,w}$ of $C(SO_q(N))$ is irreducible. Moreover, two representations, $\pi_{t_1,w_1}$ and $\pi_{t_2,w_2}$,  are equivalent if and only if $t_1=t_2$ and $w_1=w_2$.

Also, we have a homomorphism  $\chi_{w}:C(SO_q(N))\longrightarrow C(\mathbb{T}^n)\otimes \scrt^{\otimes \ell(w)}$ such that
$$\chi(w) (a)(t) = \pi_{t,w }(a), \mbox{ for all } a\in C(SO_q(N)).$$

Let $v$ be the longest element in the Weyl group $W_n$. Then the homomorphism $\chi_{v}:C(SO_q(N))\longrightarrow C(\mathbb{T}^n)\otimes \scrt^{\otimes \ell(\vartheta)}$ is  faithful.
We omit  proof of the above Theorem.
The quotient space $C(SO_{q}(N)/SO_{q}(N-2))$
In this section, our aim is to prove that the $C^*$-algebra $C(SO_{q}(N)/SO_{q}(N-2))$ is the $C^*$-subalgebra of $C(SO_q(N))$ generated by the elements $\left\{v_m^1, v_m^{N} :
m \in \{1,2,\ldots,N\}\right\}$.

Let $\vartheta_n$ denote the longest word in the Weyl group $W_n$. We realize $W_{n-1}$, the Weyl group of $\mathfrak{so}_{N-2}$, as a subgroup of $W_n$ generated by simple reflections $s_2, s_3, \ldots, s_n$. Also, noting that the longest word $\vartheta_{n-1}$ in $W_{n-1}$ is a subword of $\vartheta_n$.
We define a mapping $\eta_N:~ C(SO_{q}(N))\rightarrow C(SO_{q}(N-2))$ as follows:

\begin{align*}
& \mbox{ if } i \neq 1 \mbox{ or } N, \mbox{ or } j \neq 1 \mbox{ or } N, \cr
\delta_{ij}, & \mbox{ otherwise } \cr
\end{cases}
\end{align*}

where $u_{j}^{i}$ are generators of $C(SO_{q}(N-2))$, and $\Delta\eta_N = (\eta_N\otimes \eta_N)\Delta$. To show that $\eta_N$ is a $C^{*}$-epimorphism, we'll proceed with odd and even cases.

* Case I: Consider $N=2n+1$.

We view $\chi_{\vartheta_n}(C(SO_{q}(2n+1)))$ as a $C^{\ast}$-subalgebra of $C(\mathbb{T}^{n})\otimes \scrt^{\otimes n^{2}}$. Let $\phi:=1^{\otimes n-1}\otimes \mathrm{ev}_{1}\otimes 1^{\otimes (n-1)^{2}}\otimes \sigma^{\otimes (2n-1)}$ be the homomorphism from $C(\mathbb{T}^{n})\otimes \scrt^{\otimes n^{2}}$ to $\chi_{\vartheta_n}(C(SO_{q}(2n+1)))$. Restricting $\eta_{2n+1}$ to $\phi$ on $ \chi_{\vartheta_n}(C(SO_{q}(2n+1)))$, we have

\[
\eta_{2n+1}(\chi_{\vartheta_{n}}(v_{j}^{i}))=\begin{cases}
\chi_{\vartheta_{n-1}}(u_{j}^{i}),
& \mbox{ if } i \neq 1 \mbox{ or } 2n+1, \mbox{ or } j \neq 1 \mbox{ or } 2n+1, \cr
\delta_{ij}, & \mbox{ otherwise } \cr
\end{cases}
\]

the image of the restriction map equal to $\chi_{\vartheta_{n-1}}(C(SO_{q}(2n-1)))$.

* Case II: Take $N=2n$.

Define $\eta_{2n}$ as the restriction of $1^{\otimes n-1}\otimes \mathrm{ev}_{1}\otimes 1^{\otimes (n^2-3n+3)}\otimes \sigma^{\otimes (2n-2)}$ to $\chi_{\vartheta_{n}}(C(SO_{q}(2n-1)))$, which is contained in $C(\mathbb{T}^{n})\otimes \scrt^{\otimes (n^{2}-n-1)}$. Note that $\vartheta_{n}$ is the longest word in the Weyl group $W_n$ associated with $\mathfrak{so}_{2n}$.

\[
\eta_{2n}(\chi_{\vartheta_{n}}(v_{j}^{i}))=\begin{cases}
\chi_{\vartheta_{n-1}}(u_{j}^{i}),
& \mbox{ if } i \neq 1 \mbox{ or } 2n, \mbox{ or } j \neq 1 \mbox{ or } 2n, \cr
\delta_{ij}, & \mbox{ otherwise } \cr
\end{cases}
\]

where $u_{j}^{i}$ are generators of $C(SO_{q}(2n-2))$.

The quotient space $C(SO_{q}(N)/SO_{q}(N-2))$ is defined by

\[
C(SO_{q}(N)/SO_{q}(N-2)) = \left\{a\in C(SO_{q}(N)) : (\eta_N\otimes \mathrm{id})\Delta(a) = I\otimes a\right\}.
\]

We define $\mathfrak{A}^N$ to be the $*$-algebra generated by $\left\{v_m^1,v_m^{N} : m \in \{1,2,\hdots N\}\right\}$. Using dual paring, we  define following action on $\mathfrak{A}^N$.

* For $N=2n+1$,
$U_{q^{1/2}}(\mathfrak{so}_{2n+1})$-module structure: The pairing $\left\langle\cdot,\cdot\right\rangle$ given by $\ref{pair}$ induces a $U_{q^{1/2}}(\mathfrak{so}_{2n+1})$-module structure on  $\mathfrak{A}^{2n+1}$ which is as follows:
fv=(1⊗⟨  f ,· ⟩)Δ(v)= ⟨f,  v_(2)⟩v_(1) ;    v ∈𝔄^2n+1,  f ∈U_q^1/2(𝔰𝔬_2n+1),

where $\Delta(v)=\sum  v_{(1)} \otimes v_{(2)}$ in Sweedler notation.

* For $N=2n$, $U_{q}(\mathfrak{so}_{2n})$-module structure: Replace $\left\langle\, \cdot\,,\cdot\,\right\rangle $ given in <ref> by second pair in $\ref{pair}$ will give action on  $\mathfrak{A}^{2n}$ .

Let $a = v_{N-1}^1$, $b = v_{N-1}^{N}$, $c = v_{N}^1$, and $d = v_{N}^{N}$. For $n>2$, utilizing the defined module structure, we get the following:

*  For $x\in\{a,b\}$ and $y\in\{c,d\}$,

\[
K_i(x)= \begin{cases}
q^{-1}x &\text{ if } i=1,\\
qx &\text{ if } i=2,\\
x &\text{ if } i\geq 3,
\end{cases} \quad\text{and}\quad K_i(y)=\begin{cases}
qy &\text{ if } i=1,\\
y &\text{ if } i\geq 2.
\end{cases}
\]

*  For $y\in\{c,d\}$, $E_i(y)=0$ for each $i\in\mathbb{N}$, and

\[
E_i(a)= \begin{cases}
-c &\text{ if } i=1,\\
0 &\text{ if } i\geq 2,
\end{cases} \quad\text{and}\quad E_i(b)=\begin{cases}
-d &\text{ if } i=1,\\
0 &\text{ if } i\geq 2.
\end{cases}
\]

*  Moreover, using the relation $\Delta^n E_1 = E_1\otimes K_1^{\otimes n} + 1\otimes E_1\otimes K_1^{\otimes (n-1)} + \cdots + 1^{\otimes (n-1)}\otimes E_1\otimes K_1 + 1^{\otimes n}\otimes E_1$, we get

\begin{align*}
E_1(b^nc^n) &=A_1^0b^{n-1}c^{n}d,\\
E_1(ab^{n-1}c^{n-1}d) &=A_1^1b^{n-1}c^nd+A_2^1ab^{n-2}c^{n-1}d^2,\\
\cdots &  \cdots\cdots\\
\end{align*}

where $A_1^{0}=-q^{-2n+1}[n]_q$, $A_1^n=-q[n]_q$, $A_1^i=-q^{n-i+1}[i]_q$, $A_2^i=-q^{3i-2n+1}[n-i]_q$ for $i\in\{1,2,\cdots,n-1\}$ and $[a]_q:=\frac{q^a-q^{-a}}{q-q^{-1}}$.

For $N\neq 2,~4$, we consider an n-tuple of  integers of the form $(\lambda_1,\lambda_2,0,\hdots,0)$ where $\lambda_1\geq\lambda_2\geq 0$.  For $N=4$, we consider 2-tuple  of the  form $(\lambda_1,\lambda_2)$  with $\lambda_1\geq |\lambda_2|$ and for $N=2$, we consider $\lambda_1$, where $\lambda_1,\lambda_2\in\mathbb{Z}$. We define a vector $u$ in $\mathfrak{A}^N$ as the highest weight vector with highest weight $(\lambda_1,\lambda_2,0,\cdots,0)$ if it satisfies the following conditions:
K_i(u)   =   q^r_i u,

E_i(u)   =   0,        i ∈{1,, n},
where $r_i$ is determined as follows:

1|c|Case     1|c|$N=2n$     1|c|$N=2n+1$

$n=1$    $\lambda_1$      $r_1=2\lambda_1$

$n=2$     $r_1=\lambda_1-\lambda_2$, $r_2=\lambda_1+\lambda_2$     $r_1=\lambda_1-\lambda_2$, $r_2=2\lambda_2$

$n=3$     $r_1=\lambda_1-\lambda_2$, $r_2=\lambda_2$, $r_3=\lambda_2$     $r_1=\lambda_1-\lambda_2$, $r_2=\lambda_2$, $r_3=0$

$n>3$     $r_1=\lambda_1-\lambda_2$, $r_2=\lambda_2$, $r_i=0$ for $i>2$     $r_1=\lambda_1-\lambda_2$, $r_2=\lambda_2$, $r_i=0$ for $i>2$

For $N>4$, there exist $\lambda_1-\lambda_2+1$ linearly independent highest weight vectors in $\mathfrak{A}^N$ with highest weight $(\lambda_1,\lambda_2,0,\hdots,0)$ with $\lambda_1\geq\lambda_2\geq 0$.
Define $r = \lambda_1 - \lambda_2$.
Consider the element $\omega_N$ of the Weyl group of $\mathfrak{so}_N$ defined by

\[
\omega_N= \begin{cases}
s_1 s_2 \cdots s_{\left\lfloor\frac{N}{2}\right\rfloor -1} s_{\left\lfloor\frac{N}{2}\right\rfloor} s_{\left\lfloor\frac{N}{2}\right\rfloor -1}\cdots s_2 s_1, &\text{if } N \text{ is odd,}\\
s_1 s_2 \cdots s_{\left\lfloor\frac{N}{2}\right\rfloor -1} s_{\left\lfloor\frac{N}{2}\right\rfloor} s_{\left\lfloor\frac{N}{2}\right\rfloor -2}\cdots s_2 s_1, &\text{if } N \text{ is even,}
\end{cases}
\]

and consider the subword $\omega_N'$ of $\omega_N$ defined by

\[
\omega_N'= \begin{cases}
s_1 s_2 \cdots s_{\left\lfloor\frac{N}{2}\right\rfloor -1} s_{\left\lfloor\frac{N}{2}\right\rfloor} s_{\left\lfloor\frac{N}{2}\right\rfloor -1}\cdots s_2, &\text{if } N \text{ is odd,}\\
s_1 s_2 \cdots s_{\left\lfloor\frac{N}{2}\right\rfloor -1} s_{\left\lfloor\frac{N}{2}\right\rfloor} s_{\left\lfloor\frac{N}{2}\right\rfloor -2}\cdots s_2, &\text{if } N \text{ is even.}
\end{cases}
\]

It's then straightforward to observe that $\pi_{\omega'_N}(c)=0$. According to Observation $\ref{Observation 3}$ , we can choose nonzero constants $A_k$'s such that $E_1(b^{\lambda_2}c^{\lambda_2}+A_1ab^{\lambda_2-1}c^{\lambda_2-1}d+\cdots +A_{\lambda_2}a^{\lambda_2}d^{\lambda_2})=0$. Let $u_{\lambda_2}=b^{\lambda_2}c^{\lambda_2}+A_1ab^{\lambda_2-1}c^{\lambda_2-1}d+\cdots +A_{\lambda_2}a^{\lambda_2}d^{\lambda_2}$. Using representation of $C(SO_q(N))$, we have

\[
\pi_{\omega^{'}_N}(u_{\lambda_2})(e_0\otimes e_0\otimes \cdots \otimes e_0)=
\pi_{\omega^{'}_N}(A_{\lambda_2}a^{\lambda_2}d^{\lambda_2})(e_0\otimes e_0\otimes \cdots \otimes e_0)\neq 0
\]

which implies that $u_{\lambda_2}\neq 0$. Now, define
\begin{align*}
& \mbox{ if } N=5, \cr
c^id^{r-i}u_{\lambda_2}, & \mbox{ if } N>5, \cr
\end{cases}
\end{align*}

for $i\in\{0,\cdots,r\}$. Utilizing the actions of $K_i$ and $E_i$ computed in Observations <ref> and <ref>, we get that $x_i$'s are elements of $\mathfrak{A}^N$ with the highest weight $(\lambda_1,\lambda_2,0,\hdots,0)$.  Now, if we look at $n^{\text{th}}$ position of each term of the  $\pi_{\omega_N}(x_i)(e_0\otimes e_0\otimes \cdots \otimes e_0)$ then we can see  that one term has $n^{\text{th}}$ position $e_{2(r-i)}$ for $N=2n+1$ and $e_{r-i}$ for $N=2n$, while the other term has the $n^{\text{th}}$ position say $e_k$ where $k<2(r-i)$ for $N=2n+1$ and $k<(r-i)$ for $N=2n$. Hence $x_i$'s are linearly independent.

There exist $\lambda_1-|\lambda_2|+1$ linearly independent highest weight vectors in $\mathfrak{A}^4$ with highest weight $(\lambda_1,\lambda_2)$ with $\lambda_1\geq|\lambda_2|$.

We divide the proof in two cases. Firstly, consider the case where $\lambda_2 \geq 0$. It follows from the reasoning presented in Proposition $\ref{pp1}$.

Secondly, we tackle the case where $\lambda_2 < 0$. Here, we define $a = v_{2}^1$, $b = v_{2}^{4}$, $c = v_{4}^1$, $d = v_{4}^{4}$, and $r = \lambda_1 + \lambda_2$. Notably, the action of $E_1$ and $E_2$ on $\mathfrak{A}^4$ interchanges roles  in Observation <ref>, while the action of $K_1$ and $K_2$ is as follows:

\[K_1(a) = qa,~ K_1(b) = qb,~ K_1(c) = qc, ~K_1(d) = qd.\]

\[K_2(a) = q^{-1}a, ~K_2(b) = q^{-1}b, ~K_2(c) = c,~ K_2(d) = d.
\]

Define $x_i = c^id^{r-i}u_{-\lambda_2}$ for $i \in \{0,\hdots,r\}$, where $u_{-\lambda_2}$ is defined similarly to Proposition $\ref{pp1}$ but with the replacement of $\lambda_2$ by its negative sign. Utilizing a similar argument as presented in $\ref{pp1}$, we get that $u_{-\lambda_2}\neq 0$. By using the actions of $K_1$, $K_2$, and replacing $E_1$ by $E_2$  in Observation <ref> , it follows that $x_i$'s are highest weight vectors. Let $w_4=s_1s_2$ and using the representation of $C(SO_q(4))$, we find that $\pi_{w_4}(x_i)(e_0\otimes e_0) = Ce_{r-i-\lambda_2}\otimes e_{r-i}$, where $C$ is a non-zero constant. Hence, we conclude that the $x_i$'s are linearly independent.

The quotient space $C(SO_{q}(N)/SO_{q}(N-2))$ is the $C^*$-algebra generated by $\left\{v_i^1,v_i^{N} : i \in \{1,2,\hdots N\}\right\}$.
It can be verified using  $\eta_N$ and quotient space that both $v_{i}^{N}$ and $v_{i}^{1}=q^{\rho_{1}-\rho_{i}}(v_{N-i+1}^{N})^{*}$ belong to $C(SO_{q}(N)/SO_{q}(N-2))$ for $i=1,2,\hdots,N$. Consequently, we have the inclusion

\[
C(SO_{q}(N)/SO_{q}(N-2)) \supseteq C^*\left\{v_i^1,v_i^{N} : i \in \{1,2,\hdots,N\}\right\}.
\]

To establish the equality, we use the co-multiplication action on $C(SO_{q}(N)/SO_{q}(N-2))$ induced by the compact quantum group $C(SO_q(N))$:
C(SO_q(N)/SO_q(N-2))    ⟶    C(SO_q(N)/SO_q(N-2)) ⊗C(SO_q(N))

a    ⟼    Δa.

Using   <cit.> of Podles, we will get that
C(SO_q(N)/SO_q(N-2))= ⊕_α∈SO(N)⊕_i ∈I_α W_α,i
where $\alpha$ represents a finite-dimensional irreducible co-representation $u^{\alpha}$ of $C(SO_q(N))$, $ W_{\alpha, i}$ corresponds to
$u^{\alpha}$ for all $i \in I_{\alpha}$, and $I_{\alpha}$ denotes the multiplicity of $u^{\alpha}$.

\[
\cla= \oplus_{\alpha \in \widehat{SO(N)}}\oplus_{i \in I_{\alpha} }W_{\alpha,i}.
\]

To establish the claim, we aim to show that $\cla \subseteq C^*\left\{v_i^1,v_i^{N} : i \in \{1,2,\hdots,N\}\right\}$. There exist a one to one correspondence between  the finite-dimensional irreducible co-representations of $C(SO_q(N))$ and $n$-tuples of integers $\alpha_N=(\alpha_1,\hdots,\alpha_n)$ satisfying the following inequalities:

* For $N=2n+1$:

\[
\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_n\geq 0.
\]

* For $N=2n$:

\[
\alpha_1 \geq \alpha_2 \geq \cdots \geq |\alpha_n|.
\]

This $n$-tuple $\alpha_N$ is referred to as the highest weight of the corresponding representation and it is denoted by $V(\alpha_N)$. The restriction of $V(\alpha_N)$ to the subalgebra $U_q(\mathfrak{so}_{N-2})$ is isomorphic to a direct sum of irreducible finite-dimensional representations $V^{'}(\beta_{N-2})$, where $\beta_{N-2} =(\beta_1,\cdots , \beta_{n-1})$ is referred to as the highest weight of the representation $V^{'}(\beta_{N-2})$ , associated with $U_q(\mathfrak{so}_{N-2})$. This isomorphism includes certain multiplicities denoted as $n_{\alpha_N}(\beta_{N-2})$.
We use fact that multiplicity  $n_{\alpha_N}(\beta_{N-2})$ is the same as classical case.
Let $V(\alpha_N)$ be the irreducible representations of $SO(N)$. We apply the branching rule in a two-step process; more details can be found in $[25]$. In the first step, we restrict irreducible representation $V(\alpha_N)$ to  $SO(N-1)$ that decompose $V(\alpha_N)$ into disjoint sub-representations of $SO(N)$, denote it by $V(\eta)$. Note that each of the sub-representation occurring only once. In the second step, we again restrict sub-representation $V(\eta)$ to $SO(N-2)$ that decompose $V(\eta)$ into disjoint sub-representations of $SO(N-1)$ with multiplicity one.  Let's proceed with case by case.

* For $N=2n+1$:   $V(\alpha_{N})|_{SO(2n)}=\oplus_{\alpha_1 \geq \eta_1 \geq \cdots \geq \alpha_n \geq |\eta_n|} V(\eta)$.
We again restrict  $V(\eta)$ to $SO(2n-1)$, using branching rule, we have
$V(\eta)|_{SO(2n-1)}=\oplus_{ \eta_1 \geq \beta_1 \geq  \cdots \geq \beta_{n-1}\geq |\eta_n|} V(\beta_{N-2}).$ Therefore, we get

$$V(\alpha_N)|_{SO(2n-1)}=\oplus_{\alpha_1 \geq \beta_1 \geq \cdots \geq \beta_{n-1} \geq \alpha_n}n_{\alpha_{N}}(\beta_{N-2}) V(\beta_{N-2})$$

The multiplicity $n_{\alpha_N}(\beta_{N-1})$ is determined by the number of $n$-tuples of integers $(\gamma_1, \cdots \gamma_n)$ that satisfy the following inequalities:
α_1 ≥γ_1 ≥α_2 ≥γ_2 ≥⋯≥α_n ≥γ_n≥0,

γ_1 ≥β_1 ≥γ_2 ≥β_2 ⋯≥β_n-1 ≥γ_n≥0.

* For $N=2n$:  $V(\alpha_{N})|_{SO(2n-1)}=\oplus_{\alpha_1 \geq \eta_1 \geq \cdots \geq \eta_{n-1}\geq |\alpha_n| } V(\eta)$.
We again restrict  $V(\eta)$ to $SO(2n-2)$, using branching rule, we have
$V(\eta)|_{SO(2n-2)}=\oplus_{ \eta_1 \geq \beta_1 \geq  \cdots \eta_{n-1}\geq |\beta_{n-1}|} V(\beta_{N-2}).$ Therefore, we get

$$V(\alpha_N)|_{SO(2n-2)}=\oplus_{\alpha_1 \geq \beta_1 \geq \cdots \geq \beta_{n-1} \geq \alpha_n}n_{\alpha_{N}}(\beta_{N-2}) V(\beta_{N-2})$$

The multiplicity $n_{\alpha_N}(\beta_{N-1})$ is determined by the number of $n$-tuples of integers $(\gamma_1, \cdots \gamma_n)$ that satisfy the following inequalities:

\begin{align}\label{3.3}
\alpha_1 \geq \gamma_1 \geq \alpha_2 \geq \gamma_2 \geq \cdots \geq |\alpha_n |\geq \gamma_n,  \\
\gamma_1 \geq \beta_1 \geq \gamma_2 \geq \beta_2\geq \cdots  \geq |\beta_{n-1}| \geq \gamma_n. \label{3.4}
\end{align}

When we restrict a finite-dimensional irreducible representation of $U_q(\mathfrak{so}_{N})$ with the highest weight $\alpha_N$ to the subalgebra $U_q(\mathfrak{so}_{N-2})$, it contains the trivial representation if and only if $\beta_i=0$ for all $i$. This condition implies that $\alpha_i=0$ for $i\geq 3$. For the multiplicity, we take following cases:

* Case $N>4$: The multiplicity of the trivial representation is given by $\alpha_1-\alpha_2+1$, denoted as $n_{\alpha}(0)$. According to Theorem 1.7, $[20]$ by Podles,

\[
I_{\alpha}=n_{\alpha}(0)= \begin{cases}
\alpha_1-\alpha_2+1, & \quad \text{if} \quad \alpha_i=0 \quad \text{for all}\quad  i \geq 3, \\
0, & \quad \text{otherwise}.
\end{cases}
\]

It can be deduced from Proposition $\ref{pp1}$ that $\mathfrak{A}^N \subseteq C^*\left\{v_i^1,v_i^{N} : i \in \{1,2,\hdots N\}\right\}$ contains $\alpha_1-\alpha_2+1$ linearly independent highest weight vectors with the highest weight given by
$(\alpha_1,\alpha_2,0,\hdots,0)$. This establishes that, for each co-representation $\alpha$ of $C(SO_q(N))$, $\oplus_{j\in I_{\alpha}}W_{\alpha,j} \subseteq C^*\left\{v_i^1,v_i^{N} : i \in \{1,2,\hdots N\}\right\}$ and this shows that
$$\cla \subseteq C^*\left\{v_i^1,v_i^{N} : i \in \{1,2,\hdots N\}\right\}.$$
This completes the proof of the claim in this case.

* Case $N=4$: Using equations ($\ref{3.3})$ and ($\ref{3.4}$), we get that the multiplicity of the trivial representation is given by $\alpha_1-|\alpha_2|+1$. Now, using similar argument of above case and Proposition $\ref{pp2}$, we get the claim.

§.§ Irreducible Representation of $C(SO_{q}(2n+1)/SO_q(2n-1))$

Let $\omega_{k}$ represent the following element of the Weyl group of $\mathfrak{so}_{2n+1}$:

\[
\omega_k =\begin{cases}
I & \text{if } k =1, \\
s_1s_2\cdots s_{k-1} & \text{if } 2 \leq k \leq n+1,\\
s_1s_2\cdots s_{n-1}s_{n}s_{n-1}\cdots s_{2n-k+1} & \text{if } ~n+1 < k <2n+1,\\

\end{cases}
\]

Define $\phi_{t,w_k}$ as the restriction of $\pi_{t,w_k}$ to the quotient space $C(SO_{q}(2n+1)/SO_q(2n-1))$. Thus, we obtain an irreducible representation $\phi_{t,w_k}$ of $C(SO_{q}(2n+1)/SO_q(2n-1))$, where:

* For $2\leq k\leq n$,

\[
\begin{aligned}
\phi_{t,\omega_k}(v_1^{2n+1})&=\phi_{t,\omega_k}(v_2^{2n+1})=\cdots =\phi_{t,\omega_k}(v_{2n-k+1}^{2n+1})=0,
\phi_{t,\omega_k}(v_{2n-k+2}^{2n+1})u = tu.
\end{aligned}
\]

* For $n<k<2n+1,$

\[
\begin{aligned}
\phi_{t,\omega_k}(v_1^{2n+1})&=\phi_{t,\omega_k}(v_2^{2n+1})=\cdots =\phi_{t,\omega_k}(v_{2n-k}^{2n+1})=0,
\phi_{t,\omega_k}(v_{2n-k+1}^{2n+1})u= tu.
\end{aligned}
\]

This explicit description provides the irreducible representations satisfying these conditions. For $k=1$, define $\phi_{t,I}:C(SO_{q}(2n+1)/SO_q(2n-1))\rightarrow \bbc$ such that $\phi_{t,I}(v_j^{2n+1}) = t\delta_{1(2n-j+2)}$ for $j\in\{1,2,\hdots,2n+1\}$. The set $\left\{\phi_{t,I}: t \in \mathbb{T}\right\}$ gives  all one-dimensional irreducible representations of $C(SO_{q}(2n+1)/SO_q(2n-1))$. Furthermore, it meets the conditions $\phi_{t,I}(v_1^{2n+1})=\phi_{t,I}(v_2^{2n+1})=\cdots =\phi_{t,I}(v_{2n}^{2n+1})=0$ and $\phi_{t,I}(v_{2n+1}^{2n+1})u = tu$.

The collection $\left\{\phi_{t,\omega_k}: t \in \bbbt, 1\leq k < 2n+1 \right\}$ gives a complete list of irreducible representations of $C(SO_{q}(2n+1)/SO_q(2n-1))$.
To obtain a faithful representation of $C(SO_{q}(2n+1)/SO_q(2n-1))$, define

\begin{align*}
\phi_{\omega_k}: &~C(SO_{q}(2n+1)/SO_q(2n-1))\rightarrow C(\bbbt) \otimes \scrt^{\otimes \ell({w_k})}\\
&\phi_{\omega_k}(a)(t) = \phi_{t,\omega_k}(a) \quad \forall a \in  C(SO_{q}(2n+1)/SO_q(2n-1)).
\end{align*}

$\phi_{\omega_{2n}}$ is a faithful representation of $C(SO_{q}(2n+1)/SO_q(2n-1))$.
It is easy to see that any irreducible representation factors through $\phi_{\omega_{2n}}$ as, $\omega_k$ is a subword of $\omega_{2n}$.
This proves the claim.

We will illustrate these representations through diagrams, with further detailed provided in $[1]$. In these diagrams, each path from node $i$ on the left to node $j$ on the right represents an endomorphism acting on the Hilbert space  given at top of the diagram. Here, the arrows denote operators explicitly defined in  <cit.>. As an example, consider the case where $n=3$ and $w=s_1s_2s_3s_2$. The representation $\phi_{\omega}$ corresponds to Figure 1.

@[r]_   0.6ℓ^2(ℤ) ⊗
@[r]_ 0.6ℓ^2(ℕ_0) ⊗
@[r]_ 0.6ℓ^2(ℕ_0) ⊗
@[rr]_     0.6ℓ^2(ℕ_0)    ⊗

7 ∘@-[r]^t   ∘@-[r]^+@-[rd]^  -
∘@-[r]   ∘@-[rr]      ∘@-[r]
∘ 7

6 ∘@-[r]   ∘@-[r]_-@-[ru]_  -
∘@-[r]^+@-[rd]^  -
∘@-[rr]      ∘@-[r]^+@-[rd]^  -
∘ 6

5 ∘@-[r]   ∘@-[r]   ∘@-[r]_-@-[ru]_  -
∘@-[r]_-@-[ru]_  -
∘ 5

4 ∘@-[r]   ∘@-[r]   ∘@-[r]
∘@-[r]    ∘ 4

3 ∘@-[r]   ∘@-[r]
∘@-[r]@-[r]^+@-[rd]^  +   ∘@-[rr]_–@–>[urr]@-[uurr]^-        ∘@-[r]^+@-[rd]^  +
∘ 3

2 ∘@-[r]   ∘@-[r]^+@-[rd]   ∘@-[r]_-@-[ru]
∘@-[rr]      ∘@-[r]_-@-[ru]
∘ 2

1 ∘@-[r]   ∘@-[r]_-@-[ru]
∘@-[r]   ∘@-[rr]      ∘@-[r]
∘ 1

@[r]_    0.6ℓ^2(ℤ)⊗
@[r]_  0.6ℓ^2(ℕ_0)⊗
@[r]_  0.6ℓ^2(ℕ_0)⊗
@[r]_  0.6ℓ^2(ℕ_0)⊗
@[r]_  0.6ℓ^2(ℕ_0)⊗

8 ∘@-[r]^t   ∘@-[r]^+@->[rd]
∘@-[r]   ∘@-[r]   ∘@-[r]   ∘@-[r]
∘ 8

7 ∘@-[r]   ∘@-[r]_-@->[ru]
∘@-[r]   ∘@-[r]   ∘@-[r]^+@->[rd]
∘ 7

6 ∘@-[r]   ∘@-[r]   ∘@-[r]_-@->[ru]
∘@-[r]^+@->[dr]   ∘@-[r]@->[ddr]
∘ 6

5 ∘@-[r]   ∘@-[r]   ∘@-[r]
∘@-[r]_-@->[ur]   ∘@-[r]@-[ddr]
∘@-[r]    ∘ 5

4 ∘@-[r]   ∘@-[r]   ∘@-[r]
∘@-[r] @->[uur]	   ∘@-[r]    ∘ 4

3 ∘@-[r]   ∘@-[r]
∘@-[r]@-[r]^+@-[rd]   ∘@-[r]_-@-[ur]     ∘@-[r]@-[uur]   ∘@-[r]^+@-[rd]
∘ 3

2 ∘@-[r]   ∘@-[r]^+@-[rd]   ∘@-[r]_-@-[ru]
∘@-[rr]      ∘@-[r]_-@-[ru]
∘ 2

1 ∘@-[r]   ∘@-[r]_-@-[ru]
∘@-[r]   ∘@-[rr]      ∘@-[r]
∘ 1

§.§ Irreducible Representation of $C(SO_{q}(2n)/SO_q(2n-2))$

Let $\omega_{k}$ represent an element of the Weyl group of $\mathfrak{so}_{2n}$ as follows: for $k\leq n+1$, it follows the same case as $N=2n+1$, and if $n+1<k< 2n$,
it is defined as $s_1s_2\cdots s_{n-1}s_{n}s_{n-2}\cdots s_{2n-k}$.

Now, define $\psi_{t,w_k}$ as the restriction of $\pi_{t,w_k}$ to $C(SO_{q}(2n)/SO_q(2n-2))$. This provides an explicit description of irreducible representations of $C(SO_{q}(2n)/SO_q(2n-2))$ that satisfy the following conditions:

* For $2\leq k\leq n$,

\[
\begin{aligned}
\psi_{t,\omega_k}(v_1^{2n})&=\psi_{t,\omega_k}(v_2^{2n})=\cdots =\psi_{t,\omega_k}(v_{2n-k}^{2n})=0,
\psi_{t,\omega_k}(v_{2n-k+1}^{2n})u = tu.
\end{aligned}
\]

* For $n<k< 2n$,

\[
\begin{aligned}
\psi_{t,\omega_k}(v_1^{2n})&=\psi_{t,\omega_k}(v_2^{2n})=\cdots =\psi_{t,\omega_k}(v_{2n-k-1}^{2n+1})=0,
\psi_{t,\omega_k}(v_{2n-k}^{2n})u= tu.
\end{aligned}
\]

For $k=1$, define $\psi_{t,I}:C(SO_{q}(2n)/SO_q(2n-2))\rightarrow \bbc$ such that $\psi_{t,I}(v_j^{2n}) = t\delta_{1(2n-j+1)}$ for $j\in\{1,2,\hdots,2n\}$. The set $\left\{\psi_{t,I}: t \in \mathbb{T}\right\}$ gives all one-dimensional irreducible representations of $C(SO_{q}(2n)/SO_q(2n-2))$.

The collection $\left\{\psi_{t,\omega_k}: t \in \bbbt, 1\leq k < 2n \right\}$ gives a complete list of irreducible representations of $C(SO_{q}(2n)/SO_q(2n-2))$.
Let $t\in\mathbb{T}$, defining $
\psi_{\omega_k}: C(SO_{q}(2n)/SO_q(2n-2))\rightarrow C(\bbbt) \otimes \scrt^{\otimes \ell({w_k})}$, where $
\psi_{\omega_k}(a)(t) = \psi_{t,\omega_k}(a)$ for all $a \in  C(SO_{q}(2n+1)/SO_q(2n-1))$.
$\psi_{\omega_{2n-1}}$ is a faithful representation of $C(SO_{q}(2n)/SO_q(2n-2))$.

Similar to the preceding subsection, we will draw the diagram corresponding to the representation $\psi_{w}$, for more details given in <cit.>. For instance, consider the case where $n=4$ and $w=s_1s_2s_3s_4s_2$. The representation $\psi_{\omega}$ corresponds to Figure 2.

§ K-GROUPS OF THE QUOTIENT SPACES

This section deals with the computation of $K$-groups of the quotient spaces $C(SO_q(2n+1)/SO_q(2n-1))$, $C(SO_q(2n)/SO_q(2n-2))$, and certain intermediate $C^*$-algebras. To understand what is involved, it is instructive to understand the diagram associated with each representation, as given in the last section (see [1] for details).

§.§ K-groups of $C(SO_q(2n+1)/SO_q(2n-1))$

\[
\begin{cases}
C(\bbbt) & \mbox{ if } k=1, \cr
\eta_{\omega_k}(C(SO_q(2n+1)/SO_q(2n-1))) & \mbox{ if } 1<k \leq 2n. \cr
\end{cases}
\]

Observe that $B_k^{2n+1}$ is a $C^*$-subalgebra of $C(\bbbt) \otimes \scrt^{\otimes (k-1)}$.  Moreover,
$ \eta_{\omega_k}(v^{2n+1}_j)=0$ for $j>k$. Therefore, $B_k^{2n+1}$ is generated by $ \eta_{\omega_k}(v^{2n+1}_j)$'s with $j\leq k$. With a slight use of notation, we denote   $\eta_{\omega_k}(v^{2n+1}_j)$ by $x_j$.  Define the homomorphism

\[
\sigma:\scrt \rightarrow \bbc, \,\quad  S \mapsto 1.
\]

For $1< k \leq 2n$, let

$$\rho_k: C(\bbbt) \otimes \scrt^{\otimes (k-1)} \rightarrow C(\bbbt) \otimes \scrt^{\otimes (k-2)}$$

be the homomorphism given by

\[
\rho_k(a)=(1 \otimes 1^{\otimes (k-2)} \otimes \sigma)(a), \quad  \mbox{ for } a \in C(\bbbt) \otimes \scrt^{\otimes (k-1)}.
\]

It follows from the description of representations  of $C(SO_q(2n+1)/SO_q(2n-1))$ given in the previous section  (see [1] for details)   that $\rho_k(a) \in B_{k-1}^{2n+1}$ if $a \in B_k^{2n+1}$. This induces a homomorphism

$$\rho_k: B_k^{2n+1}\rightarrow B_{k-1}^{2n+1}$$
given by  the restriction of $\rho_k$ to the subalgebra $B_k^{2n+1}$.

Let  $1< k \leq 2n$. Denote by $I_k$ the kernel of the homomorphism $\rho_k$. Then
one has the following.

\[
\langle x_{k} \rangle & \mbox{ if } 1< k\leq n, \cr
\langle x_{n+1},x_{n+2} \rangle & \mbox{ if } k=n+1, \cr
\langle x_{k+1} \rangle & \mbox{ if } n+2 \leq  k \leq 2n. \cr
\end{cases}
\]

We prove the claim for  $1 < k\leq n$. The other cases follows by a similar argument. From the diagram  of $\eta_{\omega_k}$ given in the previous section (see [1]), it is easy to see that $x_k \in \ker \rho_k$, which futher implies that $I_k \subset \ker \rho_k$.  For the converse part,  let $\pi$ be an irreducible representation of $B_k^{2n+1}$ that vanishes on $I_k$.  It follows from Corollary <ref> that $\pi \cong \pi_{t,w}$, where $w \in W_n$ is a subword of $\omega_k$. Thus,  $\pi$ factors through $\rho_k$.  This proves  that  $\ker \rho_k \subset I_k$, hence the claim.

For $1 < k \leq 2n$, one has the following short exact sequence $\chi_{k}$ of $C^*$-algebras.

\[
\chi_{k}: \quad 0\longrightarrow C(\bbbt) \otimes \clk \xrightarrow{i} B_{k}^{2n+1}\xrightarrow{\rho_k}  B_{k-1}^{2n+1}\longrightarrow  0.
\]

Fix $1 < k \leq n$. Invoking  Lemma <ref>, it   is enough to show that
$$I_k=C(\bbbt) \otimes \clk(\ell^2(\bbn))^{\otimes (k-1)}.$$
First, observe that $x_k= t \otimes q^N \otimes \cdots q^N \in  C(\bbbt) \otimes \clk(\ell^2(\bbn_0))^{\otimes (k-1)}$, hence we have

$$I_k \subset C(\bbbt) \otimes \clk(\ell^2(\bbn_0))^{\otimes (k-1)}.$$

Further, note that for $1 \leq l \leq k-1$ we have

\[
x_l=t \otimes (q^N  )^{\otimes (l-1)}\otimes \sqrt{1-q^{2N}}S^* \otimes 1^{\otimes (k-l-1)}.
\]

One can verify  that

\[
x_l^jx_k^r1_{\{1\}}(x_k^*x_k)(x_l^*)^i=t^r \otimes p^{\otimes (l-1)}\otimes p_{ij}\otimes p^{\otimes (k-l-1)} \in I_k
\]

for $r,i,j \in \bbn$. By taking product of such elements over $l$, we can see that

\[
t^r \otimes p_{i_1j_1}\otimes p_{i_2j_2}\otimes \cdots \otimes p_{i_{k-1}j_{k-1}}\in I_k
\]

for all $i_1, j_1, i_2, j_2 \cdots i_{k-1},j_{k-1} \in \bbn$. Hence we have

\[
C(\bbbt) \otimes \clk(\ell^2(\bbn_0))^{\otimes (k-1)} \subset I_k.
\]

This settles the case for $1<k \leq n$. In other cases, the claim follows by a similar argument.

For $1 \leq k \leq 2n$, define $u_k= t \otimes p^{\otimes (k-1)} +1-1\otimes  p^{\otimes (k-1)}$.
Then one has
K_0(B_k^2n+1)   =
⟨[1]⟩≅     1 ≤k ≤n, ⟨[1]⟩⊕⟨[1 ⊗p^⊗(n) ⊗1^⊗(k-n-1)]⟩≅⊕/2     n+1 ≤k ≤2n,

K_1(B_k^2n+1)   =   ⟨[u_k] ⟩≅.

To prove the claim, we will apply induction on $k$. For $k=1$, it is clear. Assume the result to be true for $k-1$.
From the Lemma $\ref{shortexactseq}$,
we have the short exact sequence,

\[
0 \longrightarrow C(\bbbt) \otimes \clk \longrightarrow B_{k}^{2n+1}
\stackrel{\rho_{k}}{\longrightarrow} B_{k-1}^{2n+1}\longrightarrow 0.
\]

which gives rise to the following six-term sequence in $K$-theory.

\[
\def\labelstyle{\scriptstyle}
\xymatrix@C=35pt@R=35pt{
& K_0(C(\bbbt) \otimes \clk)\ar@{->}[r]
& K_0(B_{k}^{2n+1})\ar@{->}^{K_{0}(\rho_{k})}[r]
& K_0(B_{k-1}^{2n+1}))\ar@{->}^{\delta}[d] \\
&  K_1(B_{k-1}^{2n+1})\ar@{->}^{\partial}[u]\ar@{<-}^{K_{1}(\rho_{k})}[r]
&K_1(B_{k}^{2n+1})\ar@{<-}[r]       &  K_1(C(\bbbt) \otimes \clk) &
\]

To compute the $K$-groups from the above six-term exact sequence, we consider the three following cases separately.

Case (A) $1< k\leq n$:
Since $\rho_{k}(1) = 1$, it follows that $\delta([1]) = 0$.  Further, note that
the operator $\widetilde{Y} = t \otimes \underbrace{q^{N} \otimes q^{N} \otimes \cdots \otimes q^N}_{k-2} \otimes S^{*}$ is in $B_{k}^{2n+1}$ as
$\eta_{\omega_{k}}(x_{k-1}) - \widetilde{Y}$ lies in $C(\bbbt) \otimes \clk$. Define

\[
Y = 1_{\left\{1\right\}}(\widetilde{Y}^{*}\widetilde{Y})\widetilde{Y} +1-1_{\left\{1\right\}}(\widetilde{Y}^{*}\widetilde{Y}).
\]

Then $Y$ is an isometry such that $\rho_{k}(Y) = u_{k-1}$ and hence we have

\[
\partial ([u_{k-1}]) = [1-Y^{*}Y]-[1-YY^{*}]= [1 \otimes \underbrace{p \otimes p\otimes ...\otimes p}_{k-1}].
\]

Using this,  the claim follows by following the six term exact sequence.

Case (B) $k=n+1$:  In this case, the $\delta$ map is zero as  $\delta([1]) = 0$. However, $\partial$ map is not surjective as in the previous case. To see this, first note that $\widetilde{Z} = t \otimes \underbrace{q^{N} \otimes q^{N} \otimes \cdots }_{n-1} \otimes (S^{*})^2$ is in $B_{n+1}^{2n+1}$ as
$\eta_{\omega_{n}}(x_{n}) - \widetilde{Z}$ lies in $C(\bbbt) \otimes \clk$. Define

\[
Z = 1_{\left\{1\right\}}(\widetilde{Z}^{*}\widetilde{Z})\widetilde{Z} +1-1_{\left\{1\right\}}(\widetilde{Z}^{*}\widetilde{Z}).
\]

Then $Z$ is an isometry  $ 1 \otimes \underbrace{p \otimes \cdots \otimes p}_{n-1} \otimes (S^{*})^2$ such that $\rho_{n+1}(Y) = u_{n}$ and hence

\[
\partial ([u_{n}]) = [1-Z^{*}Z]-[1-ZZ^{*}]= [1 \otimes \underbrace{p \otimes \cdots \otimes p}_{n-1} \otimes (p+p_1)] =2 [1 \otimes \underbrace{p \otimes \cdots \otimes p}_{n}  ] .
\]

Therefore $K_0(i)( [1 \otimes \underbrace{p \otimes \cdots \otimes p}_{n}  ])$ is a nontrivial element and generates the torsion subgroup $\bbz/2\bbz$ of $K_0(C_{n+1})$.

Case (C) $n+2\leq  k\leq 2n$:  We assume that  $K_0(B_{k-1}^{2n+1})$ is generated by $[1]$ and $[1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-2)} ]$, which holds for $k=n+1$ case.  Using the description of representation of $B_{k}^{2n+1}$ (see the diagram),  we have

$$\rho_{k}(1) = 1 \,  \mbox{ and }  \,  \rho_{k}(1_{\{1\}}(\eta_{\omega_k}(x_{n+1}x_{n+1}^*)) = \sigma(1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-1)}) =1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-2)}.$$

Hence we get

\[
\delta([1]) = 0 \, \mbox{ and }\,  \delta ([1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-2)} ]) =0
\]

As in the first case, we get surjectivity of $\partial$. The claim now follows by chasing the six term exact sequence.

§.§ K-groups of $C(SO_q(2n)/SO_q(2n-2))$

\[
\begin{cases}
C(\bbbt) & \mbox{ if } k=1, \cr
\psi_{\omega_k}(C(SO_q(2n)/SO_q(2n-2))) & \mbox{ if } 1<k \leq 2n-1. \cr
\end{cases}
\]

Observe that 	$D_k^{2n}$ is a $C^*$-subalgebra of $C(\bbbt) \otimes \scrt^{\otimes (k-1)}$. Moreover,
$ \psi_{\omega_k}(v^{2n}_j)=0$ for $j>k$. Therefore, $D_k^{2n}$ is generated by $ \psi_{\omega_k}(v^{2n}_j)$'s with $j\leq k$. With a slight use of notation, we denote   $\psi_{\omega_k}(v^{2n}_j)$ by $y_j$.  Let
be the homomorphism defined in the previous subsection.
It follows from the diagram associated with a representaion of $D_k^{2n}$ that $\rho_k(a) \in D_{k-1}^{2n}$ if $a \in D_k^{2n}$. This induces a homomorphism
from $D_k^{2n}$  to $D_{k-1}^{2n}$  given by  the restriction of $\rho_k$ to the subalgebra $D_{k}^{2n}$. We continue to denote it by the same notation.

Let  $1< k \leq 2n-1$. Denote by $J_k$ the kernel of the homomorphism $\rho_k$. Then
one has

\[
\langle y_{k} \rangle & \mbox{ if } k\leq n, \cr
\langle y_{n+1},y_{n+2} \rangle & \mbox{ if } k=n+1, \cr
\langle y_{k+1} \rangle & \mbox{ if } n+2\leq k \leq 2n-1, \cr
\end{cases}
\]

The claim follows from a similar  computations given in Lemma <ref>.

For $1 < k \leq 2n-1, k\neq n+1$, one has the following short exact sequence $\xi_{k}$ of $C^*$-algebras.

\[
\xi_{k}: \quad 0\longrightarrow C(\bbbt) \otimes \clk \xrightarrow{i} D_{k}^{2n}\xrightarrow{\rho_k}  D_{k-1}^{2n}\longrightarrow  0.
\]

For $k=n+1$, one has

\[
\xi_{n+1}: \quad 0 \longrightarrow C(SU_q(2)) \otimes \clk \xrightarrow{i} D_{n+1}^{2n}\xrightarrow{\rho_{n+1}}  D_{n}^{2n}\longrightarrow  0.
\]

We will prove the claim for $k=n+1$. The remaining cases follow along the same line as given in Lemma <ref>. By Lemma <ref>, it remains to show that the ideal  $J_{n+1}$generated by $y_{n+1}=t \otimes (q^{N})^{\otimes (n-2)} \otimes \sqrt{1-q^{2N}}S^* \otimes q^N$ and $y_{n+2}=t \otimes (q^{N})^{\otimes n} $ is isomorphic to $C(SU_q(2)) \otimes \clk$.  For that, we will simply interchange the second and $(n-1)$th-tensor component. By doing so, we get   $y_{n+1}= t \otimes \sqrt{1-q^{2N}}S^* \otimes (q^{N})^{\otimes n-1}  \in  C(SU_q(2))  \otimes \clk(\ell^2(\bbn_0))^{\otimes (n-1)}$, hence we have

$$J_k \subset C(SU_q(2))  \otimes \clk(\ell^2(\bbn_0))^{\otimes (n-1)}.$$

Further, note that

\[
y_{n+2}1_{\{1\}}(y_{n+2}^*y_{n+2})=t\otimes p^{\otimes (n)}  \quad  \mbox{ and } \quad
y_{n+1}y_{n+1}^*-y_{n+2}^*y_{n+2}=1\otimes 1 \otimes  p^{\otimes (n-2)}.
\]

Hence we have

\[
t\otimes p^{\otimes (n)} , 1 \otimes 1  \otimes p^{\otimes (n-1)},   t \otimes \sqrt{1-q^{2N}}S^* \otimes (p)^{\otimes n-1} \in J_k.
\]

This shows that

$$ C(SU_q(2)) \otimes  p^{\otimes (n-1)}   \in J_k.$$

By applying  $y_1,y_2, \cdots y_{n-1},$ and their adjoints appropriately, one can verify that

$$ C(SU_q(2)) \otimes  p_{i_1j_1} \otimes  p_{i_{n-1}j_{n-1}}\in J_k$$

for all $i_1, \cdots i_{n-1}, j_1, \cdots j_{n-1}\in \bbn_0$. This proves the claim.

The class $[1\otimes p]$ represents the trivial element $[0]$ in $K_0(C(SU_q(2)))$. As a consequence, $[1 \otimes p^{\otimes n}]$ is equal to $[0]$ in $K_0(C(SU_q(2))\otimes \clk)$.
We have

\[
0 \rightarrow C(\bbbt) \otimes \clk \xrightarrow{i} C(SU_q(2)) \xrightarrow{1 \otimes \sigma} C(\bbbt)\rightarrow 0.
\]

This induces the following six term exact sequence.

\[
\def\labelstyle{\scriptstyle}
\xymatrix@C=35pt@R=35pt{
& K_0(C(\bbbt)\otimes \clk)\ar@{->}[r]
& K_0(C(SU_q(2)))\ar@{->}^{K_{0}(1\otimes \sigma)}[r]
& K_0(C(\bbbt)))\ar@{->}^{\delta}[d] \\
&  K_1(C(\bbbt))\ar@{->}^{\partial}[u]\ar@{<-}^{K_{1}(1\otimes \sigma)}[r]
&K_1(C(SU_q(2)))\ar@{<-}[r]       &  K_1(C(\bbbt) \otimes \clk) &
\]

Here $(1 \otimes \sigma)(t \otimes S^*)=t$, one has

\[
\partial([t])=[1 \otimes p] \in K_0(C(\bbbt)\otimes \clk).
\]

Hence $i([1 \otimes p])=[0] \in K_0(C(SU_q(2)))$. Now using  Künneth theorem for the tensor product of $C^{*}$-algebra (see [2]), the claim follows.
For $1 \leq k \leq 2n-1$, define $u_k= t \otimes p^{\otimes (k-1)} +1-1\otimes  p^{\otimes (k-1)}$.
Then one has
K_0(D_k^2n)   =
⟨[1]⟩≅     1 ≤k ≤n, ⟨[1]⟩⊕⟨[1 ⊗p^⊗(n) ⊗1^⊗(k-n-1)]⟩≅⊕     n+1 ≤k ≤2n-1,

K_1(D_k^2n)   =   ⟨[u_k] ⟩≅.

The claim is true for $k=1$ as $D_1^{2n}=C(\bbbt)$. Assume the result to be true for $k-1$. To prove the claim for $k$, we split the proof in three cases.

Case (A) $1< k\leq n$:  In this case, the proof is similar to the proof of case $(A)$ in Theorem <ref>.

Case (B) $k=n+1$:
From the Lemma $\ref{shortexactseq1}$,
we have the short exact sequence,

\[
\xi_{n+1}: \quad 0\longrightarrow C(SU_q(2)) \otimes \clk \xrightarrow{i} D_{n+1}^{2n}\xrightarrow{\rho_{n+1}}  D_{n}^{2n}\longrightarrow  0.
\]

which gives rise to the following six-term sequence in $K$-theory.

\[
\def\labelstyle{\scriptstyle}
\xymatrix@C=35pt@R=35pt{
& K_0(C(SU_q(2)) \otimes \clk)\ar@{->}[r]
& K_0(D_{n+1}^{2n})\ar@{->}^{K_{0}(\rho_{n+1})}[r]
& K_0(D_{n}^{2n}))\ar@{->}^{\delta}[d] \\
&  K_1(D_{n}^{2n})\ar@{->}^{\partial}[u]\ar@{<-}^{K_{1}(\rho_{n+1})}[r]
&K_1(D_{n+1}^{2n})\ar@{<-}[r]       &  K_1(C(SU_q(2)) \otimes \clk) &
\]

Identifying $C(SU_q(2)) \otimes \clk)$ with its image under the injective map $i$, we have

$$K_0(C(SU_q(2)) \otimes \clk)=\langle [1 \otimes p ^{\otimes (n-2)}\otimes 1 \otimes p]\rangle, \, \,
K_1(C(SU_q(2)) \otimes \clk )= \langle [u_{n+1}] \rangle.$$

Moreover, from the induction hypothesis, we have

\[
K_0(D_{n}^{2n})=	\langle[1]\rangle \cong \bbz, \quad \, \, K_1(D_{n}^{2n})= \langle [u_n] \rangle\cong \bbz.
\]

Since $\rho_{n+1}(1)=1$, we get   $\delta([1]) = 0$, and hence  the $\delta$ map is zero. To compute $\partial$, note that
$\widetilde{W} = t \otimes \underbrace{q^{N} \otimes q^{N} \otimes \cdots }_{n-2} \otimes S^*$ is in $D_{n+1}^{2n}$ as
$\psi_{\omega_{n}}(x_{n}) - \widetilde{W} \in i(C(SU_q(2)) \otimes \clk)$. Define

\[
W = 1_{\left\{1\right\}}(\widetilde{W}^{*}\widetilde{W})\widetilde{W} +1-1_{\left\{1\right\}}(\widetilde{W}^{*}\widetilde{W}).
\]

Then $W$ is an isometry  $ 1 \otimes \underbrace{p \otimes \cdots \otimes p}_{n-1} \otimes S^*$ such that $\rho_{n+1}(W) = u_{n}$. By Lemma <ref>, we have

\[
\partial ([u_{n}]) = [1-W^{*}W]-[1-WW^{*}]= [1 \otimes \underbrace{p \otimes \cdots \otimes p}_{n}  ]=[0] .
\]

Hence $\partial$ is the zero map. With these facts in hand, the  claim now follows by simply chasing  the six term exact sequence of $K$-groups.

Case (C) $n+2\leq  k\leq 2n-1$:  Using the induction hypothesis,  it follows that $K_0(D_{k-1}^{2n})$ is generated by $[1]$ and $[1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-2)} ]$.  Using the diagram  associated with  $\psi_{\omega_k}$ of $D_k^{2n}$,  we have

$$\rho_{k}(1) = 1 \,  \mbox{ and }  \,  \rho_{k}(1_{\{1\}}(\psi_{\omega_k}(x_{n+1}x_{n+1}^*)) = \rho_{k}(1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-1)}) =1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-2)}.$$

Hence we get

\[
\delta([1]) = 0 \, \mbox{ and }\,  \delta ([1 \otimes p^{\otimes (n) }\otimes 1^{\otimes (k-n-2)} ]) =0.
\]

As in case $(B)$, we get surjectivity of $\partial$. The claim now follows by following the six term exact sequence.

Topological Invariance

In this section, we prove the main result of this paper.  we assume the terminologies related to homogeneous $C^*$-extension theory that are used in  [21] without any mention. However, we quickly recall some definitions to make the statement of the results accesible to the reader. For a detailed treatment, we refer the reader to [19].  Two homogeneous $C^*$-extensions  $\tau_1$ and $\tau_2$ of $A$ by $Q(C(Y) \otimes \clk)$  are said to be strongly unitarily equivalent   if there exists unitary $U \in M(C(Y) \otimes \clk)$  such that $[U]\tau_1(a)[U^*]=\tau_2(a)$ fro all $a \in A$.  We denote it by $\tau_1 \sim_{su} \tau_2$. The strongly unitarily equivalence class of a $C^*$-extension $\tau$ is denoted by $[\tau]_{su}$.

§.§ $q$-invariance of $C(SO_q(5)/SO_q(3))$

We introduce the following notational conventions, used henceforth. For \(1 \leq k \leq 2n\), we denote \(B_k^{2n+1}\) to specify its \(q\)-parametrization as \(B_k^{2n+1}(q)\), and the generators \(x_j\) for \(1 \leq j \leq k\) as \(x_{j,q}\). The limit of \(x_{j,q}\) as \(q \to 0\) will be denoted by \(x_{j,0}\). The \(C^*\)-algebra generated by \(\{x_{j,0}: 1 \leq j \leq k\}\) is denoted by \(B_k^{2n+1}(0)\). First, for each \(1 \leq k \leq n+1\), we show the \(q\)-invariance of the intermediate subalgebras \(B_k^{2n+1}(q)\) of \(C(SO_q(2n+1)/SO_q(2n-1))\). Our idea follows the approach of [21].

For $1 < k \leq 2n$, the short exact sequence

\[
\chi_{k}: \quad 0\longrightarrow C(\bbbt) \otimes \clk \xrightarrow{i} B_k^{2n+1}(q) \xrightarrow{\rho_{k+1}}  B_{k-1}^{2n+1}(q) \longrightarrow  0.
\]

is a unital homogeneous extension of $B_{k-1}^{2n+1}$ by
$C(\bbbt)\otimes \clk$.
Since  $B_{k+1}^{2n+1}$ is unital, the given extension is unital.
Let $\tau^k: B_{k}^{2n+1} \rightarrow Q(\bbbt)$ be its Busby invariant.  For $t_0 \in \bbbt$,
define $\tau_{t_0}^k : B_{k}^{2n+1} \rightarrow Q$ to be $ev_{t_0} \circ \tau^k$.  Assume that
$J_{t_0}=\ker(\tau_{t_0}^k)$. We need to prove that $J_{t_0}=\left\{0\right\}$ for all $t_0 \in \bbbt$. From the diagram of all representations of $C(SO_q(2n+1))$ described in Section $3$ of [1], we have the following.

Case 1: $k\neq n.$
τ_t_0(y_k^k)    =    t_0[ q^N ⊗⋯⊗q^N_(k-1) ⊗√(1-q^2N)S^*]

τ_t_0(y_k^k(y_k^k)^*)    =    t_0[ q^2N ⊗⋯⊗q^2N_(k-1) ⊗(1-q^2N)]=[ q^2N ⊗⋯⊗q^2N_(k-1) ⊗1]

τ_t_0(y_k^k1_{y_k^k(y_k^k)^*=1})    =   t_0[p⊗⋯⊗p_ (k-1)  ⊗√(1-q^2N)S^*]
=t_0[p⊗⋯⊗p_ (k-1)  ⊗S^*].

Case 2: $k= n.$
τ_t_0(y_k^k)    =    t_0[ q^N ⊗⋯⊗q^N_(k-1) ⊗√(1-q^2N)(S^2)^*]

τ_t_0(y_k^k(y_k^k)^*)    =    t_0[ q^2N ⊗⋯⊗q^2N_(k-1) ⊗(1-q^2N)]=[ q^2N ⊗⋯⊗q^2N_(k-1) ⊗1]

τ_t_0(y_k^k1_{y_k^k(y_k^k)^*=1})    =   t_0[p⊗⋯⊗p_ (k-1)  ⊗√(1-q^2N)(S^2)^*]
=t_0[p⊗⋯⊗p_ (k-1)  ⊗(S^2)^*].

First, observe that in both cases,  $y_k^k \notin J_{t_0}^k$. Thus, the only primitive ideals that contains $J_{t_0}^k$ are  maximal ideals $I_t$, and hence

$$J_{t_0}^k =\cap_{I_t \subset J_{t_0}^k } I_t= I_F(\bbbt)\otimes \clk$$
for some closed
subset $F$ of $\bbbt$ where $I_F(\bbbt)$ is the closed ideal  of all continuous functions on $\bbbt$ vanishing on $F$.
Define the homomorphisms
$$\eta_1: C(\bbbt) \rightarrow Q(\ell^2(\bbn_0)); \quad \bbt \mapsto [S^*],\quad \mbox{ and } \quad \eta_2: C(\bbbt) \rightarrow Q(\ell^2(\bbn_0)); \quad \bbt \mapsto  [(S^*)^2].$$

Both the  maps $\eta_1$ and $\eta_2$ are injective as the spectrum of $[S^*]$  and $ [(S^2)^*] $ are $\bbbt$.
Hence  in both cases, we have

$$\tau_{t_0}(f(t)\otimes p^{\otimes (k-1)} ) \neq 0$$

for any nonzero function $f$ on $\bbbt$, which further implies that $F=\bbbt$ and $J_{t_0}=\left\{0\right\}$.

Although the folllowing result is  proved in [9], we give a different proof here  to fill the gap   in the argument given in [16].
For $q, q^{\prime} \in (0,1)$, the $C^*$-algebra $C(SO_q(3)$ is isomorphic to  $C(SO_{q^{\prime}}(3)$.
From [16], we have the following short exact sequence;

\[
\eta_q: \quad 0\longrightarrow C(\bbbt) \otimes \clk \xrightarrow{i} C(SO_q(3)) \rightarrow C(\bbbt) \longrightarrow  0.
\]

We denote the corresponding Busby invariant by the same notation $\eta_q$. From the same argument as given  in  Lemma <ref>, it follows that   $\eta_q$ is a homogeneous $C^*$-extension.  Hence

$$[\eta_q]_{su}\in \mbox{Ext}_{\mbox{PPV}}(\bbbt,C(\bbbt)) \mbox{  for all }  q\in (0,1).$$

From Lemma $3.4$ of [21], it follows that  $[\eta_q]_{su} =[\phi_m]_{su}$ for some $m \in \bbz$.
Comparaing the $K_0$ groups of the middle $C^*$-algebras, one can conclude that

$$ \mbox{ either }
[\eta_q]_{su} =[\phi_{ 2}]_{su} \, \mbox{ or } \,[\eta_q]_{su} =[\phi_{ -2}]_{su} \mbox{ for all } q \in (0,1).$$
Since the middle $C^*$-algebras $A_2$ and $A_{-2}$ of the extensions $[\phi_{ 2}]_{su}$ and $[\phi_{ -2}]_{su}$, respectively, are the same (see [21]), it follows that

$$C(SO_q(3))\cong A_2 \mbox{ for all } q \in (0,1).$$

This proves the claim.

Let $q \in (0,1)$ and  $1 \leq k \leq n$. Then one has
$B_{k}^{2n+1}(q) =B_{k}^{2n+1}(0)$.

From Figure 1, observe that $B_k^{2n+1}(q) = \pi_k\left(C(S_q^{2k+1})\right)$, where $\pi_k$ is the faithful representation of $C(S_q^{2k+1})$ described in [18]. Since $B_k^{2n+1}(0) = \pi_k(C(S_0^{2k+1}))$, the lemma follows from <cit.>.

For all $q\in (0,1)$, we have $B_{n+1}^{2n+1}(q) = B_{n+1}^{2n+1} (0)$.

The proof follows from a long but  straightforward computation using the diagram associated with the representations of $B_{n+1}^{2n+1}(q)$ and continuous functional calculus.

We recall that a nuclear, separable, unital $C^*$-algebra $A$ has the homotopy invariance property if for every finite-dimensional $X$ and $[\tau]\in \operatorname{Ext}_{PPV}(X\times [0,1],A)$, the condition $i_0^*[\tau] = 0$ implies $i_1^*[\tau] = 0$, where $i_t: X \rightarrow X\times [0,1]$ is the injection $i_t(x) = (x,t)$ <cit.>.

For each $q \in (0,1)$ and $1 \leq k \leq 2n$, the $C^*$-algebra $B_k^{2n+1}(q)$ is nuclear and has the homotopy invariance property.

The $C^*$-algebra $B_1^{2n+1}(q) = C(\mathbb{T})$, $C(\mathbb{T}) \otimes \mathcal{K}$ and its unitization are nuclear and quasidiagonal, hence possessing the homotopy invariance property <cit.>. Using the short exact sequence for $\chi_k$, $1 < k \leq 2n$, we then inductively show that $B_k^{2n+1}(q)$, for $1 \leq k \leq 2n$, are also nuclear and have the homotopy invariance property.

For each \(q, q' \in (0,1)\), the \(C^*\)-algebras \(B_{n+2}^{2n+1}(q)\) and \(B_{n+2}^{2n+1}(q')\) are isomorphic.

Let \(\tau_q: B_{n+1}^{2n+1}(q) = B_{n+1}^{2n+1}(0) \to Q(\mathbb{T})\) be the Busby invariant corresponding to the short exact sequence \(\chi_{n+1}\) mentioned in Lemma <ref>. To show that \(B_{n+2}^{2n+1}(q)\) and \(B_{n+2}^{2n+1}(q')\) are isomorphic, it suffices to show that the equivalence classes \([\tau_q]\) and \([\tau_{q'}]\) are equal in \( \mathrm{Ext}_{\mathrm{PPV}}(\mathbb{T}, B_{n+1}^{2n+1}(q)) \).

We claim that \(\tau_q\) and \(\tau_{q'}\) are homotopic. Since \(\tau_t\) is a homomorphism for \(t \in [q, q']\) and considering the topology of point-norm convergence as mentioned in [19], it suffices to show that \(t \mapsto \tau_t\) is continuous. If a sequence \(\{t_m\} \subseteq [q, q']\) converges to \(t\), then for each generator \(x_{j,t} \in B_{n+1}^{2n+1}(t)\), \(1 \leq j \leq n+1\),

\begin{align*}
\|\tau_{t_m}(x_{j,t}) - \tau_t(x_{j,t})\| &\leq \|\tau_{t_m}(x_{j,t}) - \tau_{t_m}(x_{j,t_m})\| + \|\tau_{t_m}(x_{j,t_m}) - \tau_t(x_{j,t})\|\\
&\leq \|x_{j,t} - x_{j,t_m}\| + \|x_{j,t_m} - x_{j,t}\|_{C(\mathbb{T})\otimes \mathscr{T}^{\otimes (n+1)}}
\end{align*}

which implies that \(\tau_{t_m}(x_{j,t}) \to \tau_t(x_{j,t})\) in norm. Hence, \(\tau_q\) and \(\tau_{q'}\) are homotopic.

By Lemma <ref>, \(B_{n+1}^{2n+1}(q)\) is a nuclear \(C^*\)-algebra with the homotopy invariance property. Thus, by <cit.>, \([\tau_q] = [\tau_{q'}]\).

For each $q, q' \in (0,1)$, the $C^*$-algebras $C(SO_q(5)/SO_q(3))$ and $C(SO_{q'}(5)/SO_{q'}(3))$ are isomorphic.

§.§ $q$-invariance of $C(SO_q(4)/SO_q(2))$

For \(1 \leq k \leq 2n-1\), we denote the \(q\)-parametrization of \(D_k^{2n}\) by \(D_k^{2n}(q)\), a convention that will be used henceforth. For $k\in\{1,2,3\}$, the $C^*$-algebra $D^4_k(q)$  is generated by $\{X_{l,q}, Y_{l,q}: l=0,1,\ldots, k-1\}$, where

\begin{align*}
X_{l,q}&= t\otimes \left(\sqrt{1-q^{2N+2}} S^*\right)^{\otimes l} \otimes \left(q^N\right)^{\otimes (k-1-l)}, \\
Y_{l,q}&= t\otimes \left(q^N\right)^{\otimes l} \otimes \left(\sqrt{1-q^{2N+2}}S^*\right)^{\otimes (k-1-l)}.
\end{align*}

Replace $\sqrt{1-q^{2N+2}}S^*$ and $q^N$ with their limits as $q\rightarrow 0$, i.e., $S^*$ and $p=|e_0\rangle \langle e_0|$, respectively, in each $X_{l,q}$ and $Y_{l,q}$. Denote the resulting operators as $X_{l,0}$ and $Y_{l,0}$, and the resulting $C^*$-algebra generated by $\{X_{l,0}, Y_{l,0}: l=1,2,\ldots, k\}$ as $D^4_k(0)$.

For all $q \in (0, 1)$ and $k \in \{1,2,3\}$,  we have

$$ D^4_k(q)= D^4_k(0).$$

Thus,  the $C^*$-algebras $D^4_k(q)$ are $q$-invariant.

Note that $D^4_1(q) = D_1(0) = C(\mathbb{T})$ is $q$-invariant. We now prove $D^4_3(q) = D^4_3(0)$. The proof for $D^4_2(q) = D^4_2(0)$ will follow similarly.
To show $D^4_3(0)\subseteq D^4_3(q)$, consider $1\otimes q^{2N}\otimes q^{2N} = Y_{2,q}^* Y_{2,q} \in D^4_3(q)$, hence $1\otimes p\otimes p \in D^4_3(q)$. Therefore, for each $m\in\mathbb{N}_0$, $t^m\otimes p\otimes p = Y_{2,q}^m (1\otimes p\otimes p) \in D^4_3(q)$. Taking adjoints gives $t^m\otimes p\otimes p\in D^4_3(q)$ for all $m\in \mathbb{Z}$.

Next, for $i_1,j_1, i_2,j_2\in\mathbb{N}_0$ and $m\in\mathbb{Z}$, we have

\begin{align*}
&Y_{1,q}^{j_2} X_{1,q}^{j_1}\left(t^{m + i_1 + i_2 -j_1 -j_2}\otimes p\otimes p\right)\left(X_{1,q}^*\right)^{i_1}\left(Y_{1,q}^*\right)^{i_2}\\
&= \sqrt{\prod_{l=1}^{i_1}(1-q^{2l+2})\prod_{l=1}^{j_1}(1-q^{2l+2})\prod_{l=1}^{i_2}(1-q^{2l+2})\prod_{l=1}^{j_2}(1-q^{2l+2})} \cdot t^m\otimes p_{j_1 i_1}\otimes p_{j_2 i_2},
\end{align*}

which yields $t^m\otimes p_{j_1 i_1}\otimes p_{j_2 i_2}\in D^4_3(q)$, hence $C(\mathbb{T})\otimes \mathcal{K}\left(\ell^2(\mathbb{N}_0)\right)\otimes\mathcal{K}\left(\ell^2(\mathbb{N}_0)\right)\in D^4_3(q)$.
Since $(1-\sqrt{1-q^{2N+2}})S^*, q^N\in \mathcal{K}\left(\ell^2(\mathbb{N}_0)\right)$, we have

\[
t\otimes S^*\otimes q^N = t\otimes (1-\sqrt{1-q^{2N+2}})S^*\otimes q^N + X_{1,q} \in D^4_3(q),
\]

which implies for each $k\in\mathbb{N}_0$,

\[
t\otimes S^*\otimes q^{(2k+1)N}= (t\otimes S^*\otimes q^N)((t\otimes S^*\otimes q^N)^*(t\otimes S^*\otimes q^N))^k\in D^4_3(q),
\]

thus $t\otimes S^*\otimes p\in D^4_3(q)$. Similarly, $t\otimes p\otimes S^* \in D^4_3(q)$. Also, since $|X_{2,q}|=1\otimes \sqrt{1-q^{2N+4}}^{\otimes 2}$ is invertible, we have $t\otimes S^*\otimes S^* = X_{2,q}|X_{2,q}|^{-1}\in D^4_3(q)$, concluding $D^4_3(0)\subseteq D^4_3(q)$.

For the reverse inclusion, consider $Y_{2,0}\in D^4_3(0)$, implying $t^m\otimes p\otimes p \in D^4_3(0)$ for each $m\in\mathbb{Z}$.
For $i,j\in\mathbb{N}_0$, $m\in\mathbb{Z}$, we have

\begin{align*}
t^m\otimes p_{ji}\otimes p &= X_{1,0}^j(t^{m+i-j}\otimes p\otimes p)(X_{1,0}^*)^i \in D^4_3(0), \\
t^m\otimes p\otimes p_{ji} &= Y_{1,0}^j(t^{m+i-j}\otimes p\otimes p)(Y_{1,0}^*)^i \in D^4_3(0).
\end{align*}

For $i_1,j_1,i_2,j_2\in\mathbb{N}_0$ with $i_1\geq i_2$ and $j_1\geq j_2$, we get

$$t^m\otimes p_{j_1 i_1}\otimes p_{j_2 i_2} = X_{2,0}^{j_2}(t^{m+i_2-j_2}\otimes p_{(j_1-j_2)(i_1-i_2)}\otimes p)(X_{2,0}^*)^{i_2} \in D^4_3(0).$$

Similarly, for $i_1,j_1,i_2,j_2\in\mathbb{N}_0$ with $i_1\leq i_2$ and $j_1\leq j_2$, we get

$$t^m\otimes p_{j_1 i_1}\otimes p_{j_2 i_2} = X_{2,0}^{j_1}(t^{m+i_1-j_1}\otimes p\otimes p_{(j_2-j_1)(i_2-i_1)})(X_{2,0}^*)^{i_1} \in D^4_3(0).$$

For $c_1,c_2> 0$, $m\in\mathbb{Z}$, we then have

$$t^m\otimes q^{c_1 N}\otimes q^{c_2 N} = \sum_{i,j=0}^{\infty} q^{c_1 i + c_2 j} t^m\otimes p_{ii}\otimes p_{jj} \in D^4_3(0).$$

In particular, $Y_{2,q}\in D^4_3(0)$. Using the binomial expansion for $\sqrt{1-q^{2N+2}}$, we get

$$X_{2,q}= \sum_{l,r=0}^{\infty} \left(\frac{q^{2l}(2l)!}{4^l(l!)^2(2l-1)}\right)\left(\frac{q^{2r}(2r)!}{4^r(r!)^2(2r-1)}\right)(1\otimes q^{2lN}\otimes q^{2rN})X_{2,0}\in D^4_3(0).$$

For each $i\in\mathbb{N}_0$, we have

$$t\otimes S^*\otimes p_{ii} = X_{2,0}^i X_{1,0}(X_{2,0}^*)^i + \sum_{l=0}^{i-1} t\otimes p_{(l+1)l}\otimes p_{ii} \in D^4_3(0),$$

implying $t\otimes S^*\otimes q^{cN} = \sum_{i=0}^{\infty} q^{ci}(t\otimes S^*\otimes p_{ii}) \in D^4_3(0)$ for each $c>0$. Thus,

$$X_{1,q}= -\sum_{l=0}^{\infty}\frac{q^{2l}(2l)!}{4^l(l!)^2(2l-1)}(1\otimes q^{2lN}\otimes q^{N/2})(t\otimes S^*\otimes q^{N/2})\in D^4_3(0).$$

Similarly, for each $i\in\mathbb{N}_0$, we have

$$ t\otimes p_{ii}\otimes S^* = X_{2,0}^i Y_{1,0}(X_{2,0})^i + \sum_{l=0}^{i-1} t\otimes p_{ii}\otimes p_{(l+1)l} \in D^4_3(0),$$

implying $t\otimes q^{cN}\otimes S^* \in D^4_3(0)$ for each $c>0$, and thus $Y_{1,q}\in D^4_3(0)$, yielding $D^4_3(q)=D^4_3(0)$.

The $C^*$-algebra $C\left(SO_q(4)/SO_q(2)\right)$ is $q$-independent.

For each $q, q' \in (0,1)$, the $C^*$-algebras $C(SO_q(6)/SO_q(4))$ and $C(SO_{q'}(6)/SO_{q'}(4))$ are isomorphic.

Similar to the proof of Lemma <ref>, one can show that \(D_{n+1}^{2n}(q)\) is nuclear and has the homotopy invariance property. Then, following the argument of Theorem <ref>, we obtain that for each \(q, q' \in (0,1)\), the \(C^*\)-algebras \(D_{n+2}^{2n}(q)\) and \(D_{n+2}^{2n}(q')\) are isomorphic. In particular, for \(n=3\), we obtain our result.

The third named author  would like to acknowledge the support from  NBHM grant 02011/19/2021/NBHM(R.P)/R&D II/8758.

[1]
Akshay Bhuva and Bipul Saurabh,
Computation of Gelfand-Kirillov dimension for B-type structures,
arXiv.2310.12163, 2023.

[2]
Bruce Blackadar,
$K$-theory for operator algebras, volume 5 of Mathematical Sciences Research Institute Publications,
Cambridge University Press, Cambridge, second edition, 1998.

[3]
L. G. Brown, R. G. Douglas and  P. A. Fillmore,
Extensions of $C^*$-algebras and $K$-homology,
Ann. of Math. (2) 105 (1977), no. 2, 265-324.

[4]
Partha Sarathi Chakraborty and S. Sundar,
$K$-groups of the quantum homogeneous space ${\rm SU}_q(n)/{\rm
Pacific J. Math., 252(2):275–292, 2011.

[5]
Partha Sarathi Chakraborty and   Bipul Saurabh,
Gelfand-Kirillov dimension of the quantized algebra of regular functions on homogeneous spaces,
Proc. Amer. Math. Soc. 147, no. 8, 3289-3302, 2019.

[6]
Partha Sarathi Chakraborty and   Bipul Saurabh,
Gelfand-Kirillov dimension of some simple unitarizable modules,
J. Algebra 514 , 199–218, 2018.

[7]
Alain Connes,
Noncommutative Geometry,
Academic Press, Inc., San Diego, CA, 1994.

[8]
Satyajit Guin and Bipul Saurabh,
Equivariant spectral triples for homogeneous spaces of the compact quantum group $U_q(2)$,
Math. Phys. Anal. Geom. 25 (2022), no. 3, Paper No. 21, 15 pp.

[9]
Jeong Hee  	Hong and  Wojciech Szymański,
Quantum spheres and projective spaces as graph algebras,
Comm. Math. Phys.  232(2002), no.1, 157-188,

[10]
Jeong Hee Hong and Wojciech Szymański,
Noncommutative balls and mirror quantum spheres,
J. Lond. Math. Soc. (2) 77 (2008), no. 3, 607-626.

[11]
Huaxin Lin,
Unitary equivalences for essential extensions of $C^*$-algebras,
Proc. Amer. Math. Soc. 137 (2009), no. 10, 3407-3420.

[12]
G. G. Kasparov,
The $K$-functors in the theory of extensions of $C^*$-algebras,
(Russian) Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 73-74.

[13]
Anatoli Klimyk and Konrad Schmüdgen,
Quantum groups and their representations,
Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997.

[14]
Leonid I. Korogodski and Yan S. Soibelman,
Algebras of functions on quantum groups. Part I, volume 56
of Mathematical Surveys and Monographs,
American Mathematical Society, Providence, RI, 1998.

[15]
D. Kucerovsky and P. W.  Ng,
The corona factorization property and approximate unitary equivalence,
Houston J. Math. 32 (2006), no. 2, 531-550.

[16]
E. Christopher	Lance,
The compact quantum group  $SO(3)_q$,
J. Operator Theory 40 (1998), no. 2, 295-307.

[17]
Sergey Neshveyev and Lars Tuset,
Quantized algebras of functions on homogeneous spaces with Poisson
Comm. Math. Phys., 312(1):223–250, 2012.

[18]
Pal, Arupkumar and Sundar, S,
Regularity and dimension spectrum of the equivariant spectral triple for the odd-dimensional quantum spheres,
J. Noncommut. Geom.4(2010), no.3, 389–439.

[19]
M. Pimsner, S. Popa and D. Voiculescu,
Homogeneous $C^*$-extensions of $C(X)\otimes \clk(\clh)$. I,
J. Operator Theory 1 (1979), no. 1, 55-108.

[20]
Piotr Podleś,
Symmetries of quantum spaces. Subgroups and quotient spaces of
quantum ${\rm SU}(2)$ and ${\rm SO}(3)$ groups,
Comm. Math. Phys., 170(1):1–20, 1995.

[21]
Bipul Saurabh,
Topological invariance of quantum quaternion spheres,
Pacific J. Math.  288 (2017), no. 2, 435-452.

[22]
Albert J. L. Sheu,
Compact quantum groups and groupoid $C^*$-algebras,
J. Funct. Anal., 144(2):371–393, 1997.

[23]
L. L. Vaksman and Ya. S. Soibelman,
An algebra of functions on the quantum group ${\rm SU}(2)$,
Funktsional. Anal. i Prilozhen., 22(3):1–14, 96, 1988.

[24]
L. L. Vaksman and Ya. S. Soibelman,
Algebra of functions on the quantum group ${\rm SU}(n+1),$ and
odd-dimensional quantum spheres,
Algebra i Analiz, 2(5):101–120, 1990.

[25]
D. P. Zhelobenko,
Classical groups. Spectral analysis of finite-dimensional representations,
Uspehi Mat. Nauk, 1 (103):27–120, 17, 1962.

Akshay Bhuva ()

Department of Mathematics,
Indian Institute Of Technology, Gandhinagar,
Palaj, Gandhinagar, 382055, INDIA

Surajit Biswas (,  )

Department of Mathematics,
Indian Institute of Technology, Gandhinagar,
Palaj, Gandhinagar 382055, India

Bipul Saurabh (,  )

Department of Mathematics,
Indian Institute of Technology, Gandhinagar,
Palaj, Gandhinagar 382055, India