Coverings of Quantum Groups

TL;DR

This paper explores the noncommutative generalization of coverings of topological groups, focusing on quantum groups, and investigates how these coverings differ from classical cases in structure and properties.

Contribution

It introduces a framework for understanding coverings of quantum groups, highlighting their weaker structural conditions compared to classical topological group coverings.

Findings

Coverings of quantum groups do not naturally inherit the quantum group structure.

A weaker condition than classical coverings applies to quantum group coverings.

The study advances the understanding of noncommutative topology and quantum algebra structures.

Abstract

It is known that any covering space of a topological group has the natural structure of a topological group. This article discusses a noncommutative generalization of this fact. A noncommutative generalization of the topological group is a quantum group. Also there is a noncommutative generalization of a covering. The combination of these algebraic constructions yields a motive to research the generalization of coverings of topological groups. In contrary to a topological group a covering space of a quantum group does not have the natural structure of the quantum group. However a covering space of a quantum group satisfies to a condition which is weaker than the condition of a covering space of a topological group.

Equations

$$(f\xi )(y)=f(y)\xi (p(y)),f\in C(\mathcal{Y}),\xi \in C(\mathcal{X}),$$

$${⟨a,b⟩}_{\tilde{A}}=\sum _{g\in H}g\left(a^{\ast },b\right)$$

$$\Delta \left(f\right)\left(g,h\right)=f\left(gh\right)\text{ for all }g,h\in G$$

$$f\left(\left(gh\right)k\right)=f\left(g\left(hk\right)\right)\text{ for all }f\in C\left(G\right),$$

$$A={\tilde{A}}^G=\left\{\tilde{a}\in \tilde{A}|g\tilde{a}=\tilde{a};\forall g\in G\right\}.$$

$$g\left(\tilde{a}m\right)=\left(g\tilde{a}\right)\left(gm\right)\text{ for any }\tilde{a}\in \tilde{A},g\in G.$$

$$\phi \left(gm\right)=g\phi \left(m\right)\text{ for any }m\in M,g\in G.$$

$$\left(a,g\right)\left(b,h\right)=\left(a\left(gb\right),gh\right).$$

$$\tilde{\Delta }\left(a\right)=\Delta \left(a\right)\text{ for any }a\in A.$$

$$\tilde{\Delta }\left(\tilde{a}\right)=\sum _{\iota \in I}{\tilde{b}}_{\iota }\otimes {\tilde{c}}_{\iota }$$

$$\begin{array}{r}\tilde{\Delta }\left(g\tilde{a}\right)=\sum _{\iota \in I}g{\tilde{b}}_{\iota }\otimes g{\tilde{c}}_{\iota }.\end{array}$$

$$\begin{array}{r}{\Delta }_L:\tilde{A}\to \tilde{A}\otimes A,\\ {\Delta }_R:\tilde{A}\to A\otimes \tilde{A}.\end{array}$$

$$\begin{array}{r}{\Delta }_L\left(\tilde{a}\right)=\sum _{\iota \in I}{\tilde{b}}_{\iota }\otimes c_{\iota },\\ {\Delta }_R\left(\tilde{a}\right)=\sum _{\iota \in I}{c}_{\iota }\otimes {\tilde{d}}_{\iota }\end{array}$$

$$\begin{array}{r}{\Delta }_L\left(g\tilde{a}\right)=\sum _{\iota \in I}g{\tilde{b}}_{\iota }\otimes c_{\iota },\\ {\Delta }_R\left(g\tilde{a}\right)=\sum _{\iota \in I}{c}_{\iota }\otimes g{\tilde{d}}_{\iota }.\end{array}$$

$$\begin{array}{r}{\Delta }_L\left(a\right)=\Delta \left(a\right),\\ {\Delta }_R\left(a\right)=\Delta \left(a\right).\end{array}$$

$${\Delta }_L:\tilde{A}\to \tilde{A}\otimes A$$

$$\begin{array}{r}{\alpha }^{\ast }\alpha +{\beta }^{\ast }\beta =1,\alpha {\alpha }^{\ast }+q^2\beta {\beta }^{\ast }=1,\\ \alpha \beta -q\beta \alpha =0,\alpha {\beta }^{\ast }-q{\beta }^{\ast }\alpha =0,\\ {\beta }^{\ast }\beta =\beta {\beta }^{\ast }.\end{array}$$

$$\begin{array}{r}\Delta (\alpha )=\alpha \otimes \alpha -q{\beta }^{\ast }\otimes \beta ,\\ \Delta (\beta )=\beta \otimes \alpha +{\alpha }^{\ast }\otimes \beta .\end{array}$$

$$\left\{1,q^2,q^4,q^6,...,0\right\}\subset \mathbb{C}.$$

$$f_{n}(t)=\left\{\begin{array}[]{c l}0&t\leq q^{2n+1}\\ 0&t\geq q^{2n-1}\\ 1&t=q^{2n}\end{array}\right..$$

$$\begin{split}Qe_{k}=q^{k}e_{k},\\ Se_{k}=\left\{\begin{array}[]{c l}e_{k-1}&k>0\\ 0&k=0\end{array}\right.,\end{split}$$

$$\begin{array}{r}\alpha \mapsto S\sqrt{1-Q^2}\otimes 1_{B\left({\ell }_2\left(\mathbb{Z}\right)\right)},\\ \beta \mapsto Q\otimes R.\end{array}$$

$$R_{\mathbb{R}}\left(\xi \right)=e^{2\pi ix}\xi ;\text{ where }{e}^{2\pi ix}\in C_b\left(\mathbb{R}\right)$$

$$\begin{array}{r}\alpha \mapsto S\sqrt{1-Q^2}\otimes 1_{B\left(L^2\left(\mathbb{R}\right)\right)},\\ \beta \mapsto Q\otimes R_{\mathbb{R}}.\end{array}$$

$$R_{\mathbb{R}}^{\frac{1}{n}}\left(\xi \right)=e^{\frac{2\pi ix}{n}}\xi .$$

$$\tilde{\beta }=\sum _{k=0}^{\infty }{\tilde{q}}^k{p}_k^{\alpha }\otimes R_{\mathbb{R}}^{\frac{1}{n}}\in B\left({\ell }^2\left({\mathbb{N}}^0\right)\otimes L^2\left(\mathbb{R}\right)\right)$$

$$M\left[\tilde{\beta }\right]=\underset{j=0}{\overset{n-1}{⨁}}C\left(S{U}_q\left(2\right)\right){\tilde{\beta }}^j.$$

$$\sum _{k=0}^{\infty }{z}_k{p}_k^{\alpha }{\left(R_{\mathbb{R}}^{\frac{1}{n}}\right)}^k\in M\left[\tilde{\beta }\right].$$

$$\begin{array}{r}{\tilde{\beta }}^j\alpha =\left(\sum _{k=0}^{\infty }{\tilde{q}}^{jk}{p}_k^{\alpha }\otimes {\left(R_{\mathbb{R}}^{\frac{1}{n}}\right)}^j\right)\left(S\sqrt{1-Q^2}\otimes 1\right)=\left(S\sqrt{1-Q^2}\otimes 1\right)\left(\sum _{k=0}^{\infty }{\tilde{q}}^{j\left(k+1\right)}{p}_k^{\alpha }\otimes {\left(R_{\mathbb{R}}^{\frac{1}{n}}\right)}^j\right)\end{array}$$

$$M\left[\tilde{\beta }\right]=C\left(S{U}_q\left(2\right)\right)\left[\tilde{\beta }\right],$$

Full text

Coverings of Quantum Groups

*Petr R. Ivankov

**e-mail: * monster.ivankov@gmail.com

It is known that any covering space of a topological group has the natural structure of a topological group. This article discusses a noncommutative generalization of this fact. A noncommutative generalization of the topological group is a quantum group. Also there is a noncommutative generalization of a covering. The combination of these algebraic constructions yields a motive to research the generalization of coverings of topological groups. In contrary to a topological group a covering space of a quantum group does not have the natural structure of the quantum group. However a covering space of a quantum group satisfies to a condition which is weaker than the condition of a covering space of a topological group.

1 Motivation. Preliminaries

In this article we discuss a noncommutative analog of the following proposition.

Proposition 1.1**.**

[6]** If $G$ is a topological group and $\pi:\widetilde{G}\to G$ is a covering, then for a covering space $\widetilde{G}$ one can introduce uniquely the structure of a topological group on $\widetilde{G}$ such that $\pi$ is a homomorphism and an arbitrary point $\widetilde{e}$ of the fibre over the unit $e$ of $G$ is the unit.

For this purpose we need noncommutative generalizations of following objects:

• •

Topological spaces,

• •

Coverings,

• •

Topological groups.

1.1 Generalization of topological objects

1.1.1 Noncommutative topological spaces

Gelfand-Naĭmark theorem [2] states the correspondence between locally compact Hausdorff topological spaces and commutative $C^{*}$-algebras.

Theorem 1.2**.**

[2]** (Gelfand-Naĭmark). Let $A$ be a commutative $C^{*}$-algebra and let $\mathcal{X}$ be the spectrum of A. There is the natural $*$-isomorphism $\gamma:A\to C_{0}(\mathcal{X})$.

So any (noncommutative) $C^{*}$-algebra may be regarded as a generalized (noncommutative) locally compact Hausdorff topological space.

1.1.2 Generalization of coverings

Following theorem gives a pure algebraic description of finite-fold coverings of compact spaces.

Theorem 1.3**.**

[8]** Suppose $\mathcal{X}$ and $\mathcal{Y}$ are compact Hausdorff connected spaces and $p:\mathcal{Y}\to\mathcal{X}$ is a continuous surjection. If $C(\mathcal{Y})$ is a projective finitely generated Hilbert module over $C(\mathcal{X})$ with respect to the action

[TABLE]

then $p$ is a finite-fold covering.

Definition 1.4**.**

If $A$ is a $C^{*}$- algebra then an action of a group $H$ is said to be *involutive * if $ga^{*}=\left(ga\right)^{*}$ for any $a\in A$ and $g\in H$. Action is said to be non-degenerated if for any nontrivial $g\in H$ there is $a\in A$ such that $ga\neq a$.

Following definition is motivated by the Theorem 1.3.

Definition 1.5**.**

[5] Let $A\hookrightarrow\widetilde{A}$ be an injective *-homomorphism of unital $C^{*}$-algebras. Suppose that there is a non-degenerated involutive action $H\times\widetilde{A}\to\widetilde{A}$ of finite group, such that $A=\widetilde{A}^{H}\stackrel{{\scriptstyle\text{def}}}{{=}}\left\{a\in\widetilde{A}~{}|~{}a=ga;~{}\forall g\in H\right\}$. There is an $A$-valued product on $\widetilde{A}$ given by

[TABLE]

and $\widetilde{A}$ is an $A$-Hilbert module. We say that $\left(A,\widetilde{A},H\right)$ is an unital noncommutative finite-fold covering if $\widetilde{A}$ is a finitely generated projective $A$-Hilbert module.

1.1.3 Generalization of topological groups

A compact quantum group can be regarded as a noncommutative analog of a compact topological group.

Definition 1.6**.**

[7] (Woronowicz) A compact quantum group is a pair $\left(A,\Delta\right)$, where $A$ is an unital $C^{*}$ -algebra and $\Delta:A\to A\otimes A$ is an unital *-homomorphism, called comultiplication, such that

1. (a)

$\left(\Delta\otimes\mathrm{Id}_{A}\right)\Delta=\left(\mathrm{Id}_{A}\otimes\Delta\right)\Delta$ as homomorphisms $A\to A\otimes A\otimes A$, (coassociativity); 2. (b)

The spaces $\left(A\otimes 1\right)\Delta A=\mathrm{span}\left\{\left(a\otimes 1\right)\Delta\left(b\right)~{}|~{}a,b\in A\right\}$ and $\left(1\otimes A\right)\Delta A$ are dense in $A\otimes A$ (cancellation property).

In this definition by the tensor product of $C^{*}$ -algebras we mean the minimal tensor product.

Following example shows that a compact topological group is a special case of a quantum group.

Example 1.7**.**

[7] Let $G$ be a compact group. Take $A$ to be the $C^{*}$-algebra $C\left(G\right)$ of continuous functions on $G$. Then $A\otimes A=C\left(G\times G\right)$, so we can define $\Delta$ by

[TABLE]

Coassociativity of $\Delta$ follows from associativity of the product in $G$. To see that the cancellation property holds, note that $\left(A\otimes 1\right)\Delta A$ is the unital $C^{*}$-subalgebra of $C\left(G\times G\right)$ spanned by all functions of the form $\left(g,h\right)\mapsto f_{1}\left(g\right)f_{2}\left(gh\right)$. Since such functions separate points of $G\times G$, the $C^{*}$-algebra $\left(A\otimes 1\right)\Delta A$ is dense in $C\left(G\times G\right)$ by the Stone-Weierstrass theorem. Any compact quantum group $\left(A,\Delta\right)$ with abelian $A$ is of this form. Indeed, by the Gelfand theorem, $A=C\left(G\right)$ for a compact space $G$. Then, since $A\otimes A=C\left(G\times G\right)$, the unital *-homomorphism $\Delta$ is defined by a continuous map $G\times G\to G$. Coassociativity means that

[TABLE]

whence $\left(gh\right)k=g\left(hk\right)$, so $G$ is a compact semigroup. If $gh=gk$, then $f_{1}\left(g\right)f_{2}\left(gh\right)=f_{1}\left(g\right)f_{2}\left(gk\right)$ for all $f_{1};f_{2}\in C\left(G\right)$. By the cancellation property the functions of the form $\left(g^{\prime},h^{\prime}\right)\mapsto f_{1}\left(g^{\prime}\right)f_{2}\left(g^{\prime}h^{\prime}\right)$ span a dense subspace of $C\left(G\times G\right)$. It follows that $f\left(g,h\right)=f\left(g,k\right)$ for all $f\in C\left(G\times G\right)$, whence $h=k$. Similarly, if $hg=kg$, then $h=k$. Thus $G$ is a semigroup with cancellation. In [7] it is proven that that any compact semigroup with cancellation is a group.

1.2 Finite Galois coverings

Here I follow to [1]. Let $A\hookrightarrow\widetilde{A}$ be an injective homomorphism of unital algebras, such that

• •

$\widetilde{A}$ is a projective finitely generated $A$-module,

• •

There is an action $G\times\widetilde{A}\to\widetilde{A}$ of a finite group $G$ such that

[TABLE]

Let us consider the category $\mathscr{M}^{G}_{\widetilde{A}}$ of $G-\widetilde{A}$ modules, i.e. any object $M\in\mathscr{M}^{G}_{\widetilde{A}}$ is a $\widetilde{A}$-module with equivariant action of $G$, i.e. for any $m\in M$ a following condition holds

[TABLE]

Any morphism $\varphi:M\to N$ in the category $\mathscr{M}^{G}_{\widetilde{A}}$ is $G$- equivariant, i.e.

[TABLE]

Let $\widetilde{A}\left[G\right]$ be an algebra such that $\widetilde{A}\left[G\right]\approx\widetilde{A}\times G$ as an Abelian group and a multiplication law is given by

[TABLE]

The category $\mathscr{M}^{G}_{\widetilde{A}}$ is equivalent to the category $\mathscr{M}_{\widetilde{A}\left[G\right]}$ of $\widetilde{A}\left[G\right]$ modules. Otherwise in [1] it is proven the equivalence between a category $\mathscr{M}_{A}$ of $A$-modules and the category $\mathscr{M}_{\widetilde{A}\left[G\right]}$. It turns out that the category $\mathscr{M}^{G}_{\widetilde{A}}$ is equivalent to the category $\mathscr{M}_{A}$.

2 Main result

From the Proposition 1.1, Theorem 1.3 and Example 1.7 it turns out the following lemma

Lemma 2.1**.**

Let $\left(A,\Delta\right)$ be a commutative compact quantum group, and let $\left(A,\widetilde{A},H\right)$ be a noncommutative finite-fold covering such that $\widetilde{A}$ is a commutative algebra. Following condition holds:

1. (i)

There is the natural structure $\left(\widetilde{A},\widetilde{\Delta}\right)$ of the compact quantum group, such that

[TABLE] 2. (ii)

Operation $\Delta$ is $H$-equivariant, i.e. from

[TABLE]

it turns out that for any $g\in H$ following condition holds

[TABLE]

Proof.

Indeed this lemma is an algebraic interpretation of the topological Proposition 1.1. ∎

The Lemma 2.1 is not true in general, there is a counterexample described in the Section 3. However any quantum group satisfies to a following theorem.

Theorem 2.2**.**

Let $\left(A,\Delta\right)$ be a quantum group. Let $\left(A,\widetilde{A},H\right)$ be a noncommutative finite-fold covering projection. There are natural $A$-bimodule morphisms

[TABLE]

such that following conditions hold:

1. (i)

Above morphisms are $H$-equivariant, i.e. for any $g\in H$ from

[TABLE]

it turns out that

[TABLE] 2. (ii)

If $a\in A$ then

[TABLE]

Proof.

(i) If we apply to $\Delta:A\to A\otimes A$ a functor $\widetilde{A}\otimes_{A}-$ then we have

[TABLE]

From the Section 1.2 it follows that $\Delta_{L}$ is left $H$-equivariant. Similarly one can construct $\Delta_{R}$.

(ii) Follows from the definition of functors $\widetilde{A}\otimes_{A}-$ and $-\otimes_{A}\widetilde{A}$ and from that the *-homomorphism $A\hookrightarrow\widetilde{A}$ is injective. ∎

Remark 2.3**.**

The statement of Theorem 2.2 is weaker than the statement of the Lemma 2.1. In fact the Theorem 2.2 describes a left and right action of the group $G$ on the quotient group $\widetilde{G}/H\approx G$.

3 Counterexample

The counterexample of the Lemma 2.1 is discussed here.

3.1 Noncommutative quantum $SU(2)$ group

Let $q$ be a real number such that $0<q<1$. A quantum group $C\left(SU_{q}(2)\right)$ is an universal $C^{*}$-algebra algebra generated by two elements $\alpha$ and $\beta$ satisfying following relations:

[TABLE]

The structure of the quantum group on $C\left(SU_{q}\left(2\right)\right)$ is given by

[TABLE]

From $C\left(SU_{1}\left(2\right)\right)\approx C\left(SU\left(2\right)\right)$ it follows that $C\left(SU_{q}(2)\right)$ can be regarded as a noncommutative deformation of $SU(2)$. It is proven in [9] that the spectrum of $\beta\beta^{*}$ is the discrete set

[TABLE]

If $n\in\mathbb{N}^{0}$ and $f_{n}:\mathbb{R}\to\mathbb{R}$ is a continuous function such that

[TABLE]

then $p^{\alpha}_{n}=f_{n}\left(\beta\beta^{*}\right)\in C\left(SU_{q}(2)\right)$ is a projection. Let $Q,S\in B\left(\ell_{2}\left(\mathbb{N}^{0}\right)\right)$ be given by

[TABLE]

and let $R\in B\left(\ell_{2}\left(\mathbb{Z}\right)\right)$ be given by $e_{k}\mapsto e_{k+1}$. There is a faithful representation $C\left(SU_{q}\left(2\right)\right)\to B\left(\ell_{2}\left(\mathbb{N}^{0}\right)\otimes\ell_{2}\left(\mathbb{Z}\right)\right)$ [9] given by

[TABLE]

If $R_{\mathbb{R}}\in B\left(L^{2}\left(\mathbb{R}\right)\right)$ is given by

[TABLE]

then similarly to (3) one has a representation $C\left(SU_{q}\left(2\right)\right)\to B\left(\ell^{2}\left(\mathbb{N}^{0}\right)\otimes L^{2}\left(\mathbb{R}\right)\right)$ given by

[TABLE]

3.2 Finite-fold coverings

If $R^{\frac{1}{n}}_{\mathbb{R}}\in B\left(L^{2}\left(\mathbb{R}\right)\right)$ is given by

[TABLE]

then $\left(R^{\frac{1}{n}}_{\mathbb{R}}\right)^{n}=R_{\mathbb{R}}$. If $\widetilde{q}=\sqrt[n]{q}$ and

[TABLE]

then $\widetilde{\beta}^{n}=\beta$. Denote by $C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$ a $C^{*}$-subalgebra of $B\left(\ell^{2}\left(\mathbb{N}^{0}\right)\otimes L^{2}\left(\mathbb{R}\right)\right)$ generated by $C\left(SU_{q}\left(2\right)\right)\bigsqcup\left\{\widetilde{\beta}\right\}$. Denote by $M\left[\widetilde{\beta}\right]\subset C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$ a free module left $C\left(SU_{q}\left(2\right)\right)$ module given by

[TABLE]

If $j\in\{0,...,n-1\}$, $j\in\mathbb{N}^{0}$ then from $p^{\alpha}_{k}\in C\left(SU_{q}\left(2\right)\right)$ it follows that $p^{\alpha}_{k}\widetilde{\beta}^{j}=p^{\alpha}_{k}\widetilde{q}^{-k}\left(R^{\frac{1}{n}}_{\mathbb{R}}\right)^{j}\in M\left[\widetilde{\beta}\right]$, hence $p^{\alpha}_{k}\left(R^{\frac{1}{n}}_{\mathbb{R}}\right)^{j}\in M\left[\widetilde{\beta}\right]$. Moreover if $\left\{z_{k}\in\mathbb{C}\right\}_{k\in\mathbb{N}^{0}}$ then from $\lim_{k\to\infty}z_{k}=0$ it turns out

[TABLE]

Following conditions hold:

[TABLE]

From $\lim_{k\to\infty}\widetilde{q}^{j\left(k+1\right)}=0$ it turns out $\widetilde{\beta}^{j}\alpha$ lies in $M\left[\widetilde{\beta}\right]$. Similarly we have $\widetilde{\beta}^{j}\alpha^{*}\in\left[\widetilde{\beta}\right]$ it follows that

[TABLE]

i.e. $C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$ is a finitely generated free $C\left(SU_{q}\left(2\right)\right)$-module

[TABLE]

There is the action of $\mathbb{Z}_{n}$ on $C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$ given by

[TABLE]

The above construction gives a following result.

Theorem 3.1**.**

[5]** The triple $\left(C\left(SU_{q}\left(2\right)\right),C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right],\mathbb{Z}_{n}\right)$ is an unital noncommutative finite-fold covering.

3.3 The structure of the covering algebra

From the above construction it follows that

[TABLE]

Direct calculations shows that

[TABLE]

Above relations coincide with (4) it follows that there is a $*$-isomorphism given by

[TABLE]

i.e. the covering algebra $C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$ is *-isomorphic to the base algebra $C\left(SU_{q}\left(2\right)\right)$.

3.4 Symmetry and grading

Let $\mathcal{A}\subset SU_{q}\left(2\right)$ is a dense subalgebra which is generated by $\alpha,\alpha^{*},\beta,\beta^{*}$ as an abstract algebra.

Theorem 3.2**.**

[9]** The set of all elements of the form

[TABLE]

where $k,m,n=0,1,2,\dots,~{}k^{\prime}=1,2,\dots$ forms a basis in $\mathcal{A}$: any element of $\mathcal{A}$ can be written in the unique way as a finite linear combination of elements (6).

From the above theorem there is an action of $U\left(1\right)$ on $\mathcal{A}$ given by

[TABLE]

where $g\in U\left(1\right)$ and $\varphi_{\mathbb{C}^{\times}}:U\left(1\right)\to\mathbb{C}^{\times}$ the natural homomorphism from $U\left(1\right)$ to the multiplicative group of complex numbers. There is a $\mathbb{Z}$-grading

[TABLE]

such that $a\in\mathcal{A}_{j}$ is equivalent to

[TABLE]

It turns out

[TABLE]

Let $\left(C\left(SU_{q}\left(2\right)\right),C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right],\mathbb{Z}_{n}\right)$ be a covering projection. From $\widetilde{\beta}^{n}=\beta$ and (5) it follows that there is the natural $\mathbb{Z}$-grading on $C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$ given by

[TABLE]

where subscripts mean the grading.

3.5 Contradiction

Suppose that there is a structure of quantum group $\left(C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right],\widetilde{\Delta}\right)$ which satisfies to the Lemma 2.1. From $\beta=\widetilde{\beta}^{n}$, (2), and the condition (i) of the Lemma 2.1 it turns out

[TABLE]

Denote by

[TABLE]

The $\mathbb{Z}$-grading on $C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$ induces the natural $\mathbb{Z}\times\mathbb{Z}$ grading on $D=C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]\otimes C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$. Clearly

[TABLE]

where subscripts $(n,0)$ and $(0,n)$ mean grading. Suppose that

[TABLE]

where

[TABLE]

Let $j_{\text{max}}\in\mathbb{Z}$ be a maximal number such that there is $k\in\mathbb{Z}$ which satisfy to the condition $a_{j_{\text{max}},k}\neq 0$. The inequality $j_{\text{max}}>1$ contradicts with (7) because right part of (7) does not contain summands in $D_{\left(nj_{\text{max}},k\right)}$. Similarly one can prove that the minimal value $j_{\text{min}}$ of $j$ such that $a_{j_{\text{min}},k}\neq 0$ satisfies to an inequality $j_{\text{min}}\geq 0$. Using the same arguments one can prove that if $a_{jk}\neq 0$ then $0\geq k\geq 1$. In result one has

[TABLE]

If $a_{00}\neq 0$ then $\left(\widetilde{\Delta}\left(\widetilde{\beta}\right)\right)^{n}\bigcap D^{(0,0)}\neq 0$ and from this contradiction it turns out $a_{00}=0$. Similarly $a_{11}=0$. Following condition holds

[TABLE]

hence $a_{01}^{n}=\alpha^{*}\otimes\widetilde{\beta}^{n}$, $a_{01}^{n}=\alpha^{*}\otimes\widetilde{\beta}^{n}$. Otherwise

[TABLE]

where $na_{10}a_{01}^{n-1}\in D^{(1,n-1)}$, $r^{\prime}\notin D^{(1,n-1)}$. From $a_{10}^{n}a_{01}^{n}\neq 0$ it turns out $a_{10}a_{01}^{n-1}\neq 0$ hence $r\neq 0$. It follows that

[TABLE]

This contradiction proves that the quantum group $\left(C\left(SU_{q}\left(2\right)\right),\Delta\right)$ and the finite-fold noncommutative covering $\left(C\left(SU_{q}\left(2\right)\right),C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right],\mathbb{Z}_{n}\right)$ projection do not satisfy to the Lemma 2.1.

Remark 3.3**.**

From 3.3 it follows the *-isomorphism $C\left(SU_{q}\left(2\right)\right)\xrightarrow{\approx}C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$, hence there is a structure of quantum group on $C\left(SU_{q}\left(2\right)\right)\left[\widetilde{\beta}\right]$. However in contrary to the commutative case this structure does not naturally follow from the structure of the quantum group $\left(C\left(SU_{q}\left(2\right)\right),\Delta\right)$ and the noncommutative finite-fold covering projection

[TABLE]

4 Conclusion

There is a set of geometrical statements which have noncommutaive generalizations, e.g. in [4] it is proven a noncommutative analog of the theorem about a covering projection of a Riemannian manifold. The described in the Section 3 counterexample proves that the analogy between coverings of topological groups and quantum groups is not full. However coverings of quantum groups satisfy to the Theorem 2.2 which is weaker than the Lemma 2.1 about coverings of commutative quantum groups.