\documentclass{amsart} % amslatex_2 \usepackage{amscd} \usepackage{amssymb} \DeclareFontFamily{OMS}{cmsy}{% \fontdimen16\font=3pt \fontdimen17\font=3pt} \setlength{\headheight}{1.5\baselineskip} \makeatletter \renewcommand{\subsection}{\@startsection{subsection}{2}{\z@}% {\baselineskip}{0.5\baselineskip}{\bfseries}} \def\l@section{\def\@tocpagenum##1{\hss{\mdseries\itshape ##1}}% \@tocline{1}{8pt}{0pc}{}{\bfseries}} \def\l@subsection{\def\@tocpagenum##1{} \@tocline{2}{0pt}{3pc}{5pc}{}} \makeatother \def\contentsname{Table of Contents} \def\dj{d\kern-.30em\raise1.25ex\vbox{\hrule width .3em height .03em}} \def\Dj{D\rlap{\kern-.70em\raise0.75ex \vbox{\hrule width .3em height .03em}}} \newenvironment{pf}{\proof[\proofname]}{\endproof} \def\restr{\restriction} \def\cal{\mathcal} \def\Bbb{\mathbb} \newcommand{\ADP}{\cal{G}(P)} \newcommand{\AD}{\cal{G}} \newcommand{\SC}{S_c} \newcommand{\WC}{\Omega_c} \newcommand{\hor}{\mathfrak{hor}} \newcommand{\grten}{\mathbin{\widehat{\otimes}}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\id}{\mathrm{id}} \newcommand{\Ad}{\varpi} \newcommand{\lie}{\mathrm{lie}} \def\e{\epsilon} \def\k{\kappa} \def\bla#1{$(${\it #1\/{}}$)$} \def\U{\mathrm{U}} \def\SU{\mathrm{SU}} \def\Mor{\mathrm{Mor}} \newcommand{\Sum}{{\displaystyle\sum}} \newcommand{\ver}{\mathfrak{ver}} \newcommand{\con}{\mathfrak{con}} \newcommand{\inv}{i\!\hspace{0.8pt}n\!\hspace{0.6pt}v} \newtheorem{lem}{Lemma}[section] \newtheorem{pro}[lem]{Proposition} \newtheorem{thm}[lem]{Theorem} \theoremstyle{definition} \newtheorem{defn}{Definition}[section] \numberwithin{equation}{section} \renewcommand{\thepage}{\ifnum\value{page}=1 \else\arabic{page}\fi} \parskip=0pt plus3pt \begin{document} \title[Quantum Principal Bundles]{Quantum Principal Bundles and\\ Corresponding Gauge Theories} \author{micho {\Dj}UR{\Dj}EVICH} \address{Instituto de Matematicas, UNAM, Area de la Investigacion Cientifica, Circuito Exterior, Ciudad Universitaria, M\'exico DF, CP 04510, MEXICO} \def\thefootnote{} \footnote{Journal of Physics A: Mathematical \& General {\bf 30} 2027--2054 (1997)} \medskip \begin{abstract} A generalization of classical gauge theory is presented, in which compact quantum groups play the role of the internal symmetry groups. All considerations are performed in the framework of a noncommutative-geometric formalism of locally trivial quantum principal bundles over classical smooth manifolds. Quantum counterparts of classical gauge bundles, and classical gauge transformations, are introduced and investigated. A natural differential calculus on quantum gauge bundles is constructed and analyzed. Kinematical and dynamical properties of corresponding gauge theories are discussed. A particular attention is given to the purely quantum phenomenas appearing in the formalism, and their physical interpretation. An example with quantum $\SU(2)$ group is considered. \end{abstract} \maketitle \bigskip \tableofcontents \newpage \section{Introduction} Gauge field theory is one of the fundamental theoretical tools for the development of unifying models of elementary particles and their interactions. The basic idea incorporated in the gauge formalism is that of the local symmetry, so that the internal symmetry transformations can be performed independently in various space-time points. A consistent formulation of such a theory is possible by introducing fields of a special nature--gauge fields. These fields appropriately `compensate' effects of arbitrary local symmetry transformations of the standard matter fields. The group of local tranformations is infinite-dimensional. In particular, as a consequence of this high-level symmetry, gauge theory is always renormalizable \cite{sf}. The simplest example of a gauge theory is given by classical and quantum electrodynamics. In this case the internal symmetry group is simply $\U(1)$. Electrodynamics is included in a non-Abelian gauge theory unifying electromagnetic and weak interactions. The internal symmetry group for electroweak interactions is given by $\U(1)\times\SU(2)$. Furthermore, the physics of quarks and strong interactions is, at least phenomenologically, includable into the same conceptual framework. Finally, general relativity theory can be viewed as a special case of a gauge theory, associated to the Poincare group. On the other hand, gauge theory is a paradigmic example of the interplay between fundamental physics and differential geometry \cite{dv,egu}. The appropriate differential geometric framework is given by principal bundles \cite{hus,kn}. The physical space-time plays the role of the base manifold of the bundle, and the structure group is identified with the group describing internal symmetries. Gauge fields are interpreted as connection forms, while the matter fields are sections of the appropriate associated vector bundles. The infinite-dimensional group of gauge transformations is identified with the group of vertical automorphisms of the bundle. If the gauge theory really reflects something fundamental in Nature, then it is plausible to assume that the basic picture holds also at the level of ultra-small distances (the Planck scale). However, the classical smooth manifold description of space-time is not appropriate at this level. Such a thinking naturally leads to the assumption that general philosophy of gauge theory should be independent of the nature of the underlying space-time. Noncommutative differential geometry \cite{c,c-book} essentially enlarges the classical idea of geometric space, introducing the concept of a quantum space. Informally speaking, a quantum space shows `quantum fluctuations' of geometry and it is not understandable in the standard way--as a collection of points equipped with an additional structure. Quantum spaces are described by the appropriate non-commutative *-algebras, the elements of which are interpretable as `smooth functions' over these spaces. There exist non-trivial reasons \cite{c-book} to beleive that noncommutative geometry is capable to provide the appropriate description of the space-time at the Planck scale, and to overcome deep mathematical inconsistencies figuring in the standard formulation of quantum field theory. Paralelly with generalizing the concept of space, noncommutative geometry opens a possibility of extending the concept of symmetry--via quantum groups. Geometrically, quantum groups are quantum spaces endowed with a group structure and they are included into the framework of Hopf algebras \cite{abe}. It is therefore natural to look for the appropriate noncommutative-geometric generalization of standard gauge theory. A natural mathematical framework for such a theory should be given by {\it quantum principal bundles}, where quantum groups play the role of structure groups and quantum spaces are the counterparts of base manifolds. Such a general formalism was built in papers \cite{d1,d2,d-cl}. A short presentation is given in \cite{d-ixt}. Conceptually similar, but technically different formulation of quantum principal bundles was presented in \cite{bm2}. In this study we shall discuss properties of a gauge theory based on quantum principal bundles. We shall assume here that the space-time is described by classical geometry, and use locally-trivial quantum principal bundles over classical manifolds \cite{d1} accordingly. However, general compact matrix quantum groups \cite{w2} will play the role of entities describing local symmetries. This paper is a first step towards a `fully quantum' gauge theory \cite{d-qgf} in which the space-time will be quantum, too. There exist several reasons why it is plausible to consider separately the special case of the theory over a classical space-time. Firstly, it is technically easier to deal with classical base manifolds, where the concept of locality is clearly defined and computations can be performed in local trivializations, as in the standard theory. In particular, the formalism gives an alternative (but equivalent) way to perform calculations in the standard gauge theory (and surprisingly, such calculations are simpler than the conventional ones). In general quantum context it is not possible to introduce the concept of a local trivialization, and all considerations should be performed `globally'. On the other hand, it turns out that various global components of the fully quantum formalism of quantum principal bundles \cite{d2} are essentially the same as in the special theory \cite{d1}, being independent of the nature of the base manifold. Finally, in noncommutative geometry the concept of space and the concept of symmetry are completely independent. Therefore it would be interesting to see what is the separate contribution of the idea of local quantum group symmetry to the structure of the corresponding gauge theory. As we shall see, introducing quantum groups in the game leads to various `purely quantum' phenomenas. The enlarged concept of symmetry opens, in principle, a possibility of further unifying standard particle multiplets into quantum group `supermultiplets' and it is natural to expect that the theory possesses the same nice properties as its classical-geometric counterpart. Let us outline the contents of the paper. In the next section, a preparatory material is collected. At first, we fix the notation and introduce in the game relevant quantum group entities. Secondly, we present the most important ideas and results of \cite{d1}, which will be used in the main considerations. As we have mentioned, the starting point for all constructions will be a quantum principal $G$-bundle $P$ over a smooth manifold $M$, playing the role of space-time, while $G$ is a compact matrix quantum (structure) group \cite{w2} representing `local symmetries' of the system. In Section 3 a quantum analogue of the gauge bundle will be constructed and investigated. This quantum bundle (over $M$) will be denoted by $\ADP$. Various quantum counterparts of gauge transformations are naturally associated to $\ADP$. Further, a differential calculus on the bundle $\ADP$ will be constructed, by combining the standard differential calculus on $M$ (based on differential forms) with an appropriate differential calculus on the quantum group $G$. This calculus on $\ADP$ is relevant in situations in which quantum counterparts of gauge transformations act on entities related to differential calculus on the principal bundle $P$, as connection forms for example. It is important to mention that there exist two natural inequivalent ways of introducing quantum counterparts of gauge transformations. The first one is to translate into the quantum context the idea that gauge transformations are vertical automorphisms of the principal bundle $P$. This approach leads to a standard group (of gauge transformations of $P$). The same group will be obtained if we consider counterparts of sections of the bundle $\ADP$. However, it turns out that such a concept of a gauge transformation does not describe gauge-like phenomenas related to the quantum nature of the space $G$. Namely, because of the inherent geometrical inhomogeneity of quantum groups, every quantum principal bundle $P$ over $M$ is completely determined by its classical part $P_{cl}$ (interpretable as the set of points of $P$). The classical part is an ordinary principal $G_{cl}$-bundle over $M$, where $G_{cl}$ is a group (the classical part of $G$) interpretable as consisting of points of $G$. We shall prove that gauge transformations of $P$ are in a natural bijection with standard gauge transformations of $P_{cl} $. Further, we shall prove that the set of points of $\ADP$ coincides, in a natural manner, with the standard gauge bundle $\AD(P_{cl})$. The second approach to gauge transformations is in some sense indirect. The main idea is to construct the `action' of the bundle $\ADP$ on $P$ (generalizing the classical situation). This approach does not meet geometrical obstructions. In classical geometry, the mentioned action naturally contains all the information about gauge transformations. Section 4 is devoted to the formulation and kinematical and dynamical analysis of quantum group gauge theories, in the framework of quantum principal bundles. Gauge fields will be geometrically represented by connections on $P$. Internal degrees of freedom of such gauge fields are determined by fixing a bicovariant first-order differential *-calculus \cite{w3} on the structure quantum group $G$. In this paper we shall deal with a unique differential calculus on $G$ which can be characterized as the minimal bicovariant differential calculus compatible, in appropriate sense, with the geometrical structure on the bundle $P$. If we start from this calculus on the group then it is possible to built natural differential calculi on bundles $P$ and $\ADP$ which are always `locally trivialized' when bundles $P$ (and therefore $\ADP$) are locally trivialized. Dynamical properties of the gauge theory will be determined after fixing an appropriate lagrangian. In analogy with the classical gauge theory, we shall consider lagrangians which are quadratic functions of the curvature form. We shall compute the corresponding equations of motion. Symmetry properties of the introduced lagrangian will be analyzed. We shall prove the invariance of the lagrangian under the action of the (ordinary) group of gauge transformations of $P$. Further, it turns out that the lagrangian is invariant, in an appropriate sense, under the natural action of $\ADP$ on $P$. This corresponds to the full gauge invariance of the lagrangian in the classical theory. In Section 5 everything will be illustrated on a conceptually simple but technically highly non-trivial example in which $G$ is the quantum $\SU(2)$ group. The most important observation is that the corresponding gauge theory is {\it essentially different} from the classical $\SU(2)$ gauge theory, and does not reduce to the classical theory when the deformation parametar $1-\mu$ tends to zero. This is caused by the fact that the minimal admissible bicovariant calculus does not respect the classical limit. Namely, a detailed analysis \cite{d1} shows that for $\mu\in (-1,1)\setminus\{0\}$ the space of left-invariant elements (playing the role of the dual space of the corresponding Lie algebra) of the mentioned minimal calculus is infinitely dimensional, and can be naturally identified with the algebra of polynomial functions on a quantum 2-sphere \cite{p}. Hence, the corresponding gauge fields possess infinitely many internal degrees of freedom, in contrast to the classical case. Finally, in Section 6 concluding remarks are made. The paper ends with an Appendix, in which some technical properties related to the minimal admissible bicovariant calculus on the quantum $\SU(2)$ group are collected. \section{ Mathematical Background} Let $G$ be a compact matrix quantum group \cite{w2}. We shall denote by $\cal{A}$ the *-algebra of `polynomial functions' on $G$, and by $\phi\colon \cal{A}\rightarrow \cal{A}\otimes \cal{A},$ $\e\colon \cal{A}\rightarrow \Bbb{ C}$ and $\k\colon \cal{A}\rightarrow \cal{A}$ the coproduct, counit and the antipode respectively. The symbols $a^{(1)} \otimes \dots \otimes a^{(n)}$ will be used for the result of an $(n-1)$-fold coproduct of an element $a\in \cal{A}$ (so that $\phi (a)=a^{(1)} \otimes a^{(2)} )$. Let $G_{cl}$ be the classical part \cite{d1} of $G$. Explicitly, $G_{cl}$ is consisting of *-characters (nontrivial multiplicative linear hermitian functionals) of $\cal{A}$. The Hopf algebra structure on $\cal{A}$ naturally induces the group structure on $G_{cl},$ such that \begin{align*} gg'&=(g\otimes g')\phi \\ g^{-1} &=g\k, \end{align*} for each $g,g'\in G_{cl}$. The counit $\e\colon \cal{A}\rightarrow \Bbb{C}$ is the neutral element of $G_{cl}$. We shall assume that the (complex) Lie algebra $\lie(G_{cl})$ is realized \cite{d1} as the space of linear functionals $X\colon \cal{A}\rightarrow \Bbb{C}$ satisfying $$X(ab)=\e(a)X(b)+\e(b)X(a),$$ for each $a,b\in \cal{A}$. Let $\Gamma$ be a first-order differential calculus over $G$. This means \cite{w3} that $\Gamma$ is a bimodule over $\cal{A}$ endowed with a differential $d\colon \cal{A}\rightarrow \Gamma$ such that elements of the form $a\,db$ linearly generate $\Gamma$. Let $$\Gamma^{\otimes} =\sideset{}{^\oplus}\sum_{k\geq 0} \Gamma^{\otimes k}$$ be the tensor bundle algebra \cite{w3} built over $\Gamma$. Let $$\Gamma^{\wedge} =\sideset{}{^\oplus}\sum_{k\geq 0} \Gamma^{\wedge k}$$ be the universal differential envelope (\cite{d1}--Appendix B) of $\Gamma$. The algebra $\Gamma^{\wedge}$ can be obtained from $\Gamma^{\otimes}$ by factorising through the ideal $S^{\wedge} \subseteq\Gamma^{\otimes}$ generated by the elements of the form $$Q=\sum_ida_i \otimes_{\cal{A}} db_i,$$ where $a_i ,b_i \in \cal{A}$ satisfy $\Sum_i a_i db_i =0$. In particular, the differential $d\colon\Gamma^{\wedge}\rightarrow\Gamma^{\wedge}$ extends $d\colon\cal{A}\rightarrow\Gamma,$ in a natural manner. Let us assume that $\Gamma$ is left-covariant \cite{w3} and let $\ell_{\Gamma} \colon \Gamma\rightarrow \cal{A}\otimes \Gamma$ be the left action of $G$ on $\Gamma$. Let $\Gamma_{\inv}$ be the space of left-invariant elements of $\Gamma$ (playing the role of the dual space of the Lie algebra of $G$) and let $\pi\colon \cal{A}\rightarrow \Gamma_{\inv}$ be the canonical projection map, given by $$\pi(a)=\k(a^{(1)} )da^{(2)} .$$ This map is surjective and $\cal{R}=\ker(\e)\cap \ker(\pi)$ is the right $\cal{A}$-ideal which canonically \cite{w3} corresponds to $\Gamma$. The space $\Gamma_{\inv}$ possesses a natural right $\cal{A}$-module structure, which will be denoted by $\circ $. Explicitly $$\pi(a)\circ b=\pi\bigl[\bigl(a-\e(a)1\bigr)b\bigr] $$ for each $a,b\in \cal{A}$. Let us now assume that $\Gamma$ is bicovariant, and let $\wp_{\Gamma}\colon \Gamma\rightarrow \Gamma\otimes \cal{A} $ be the right action of $G$ on $\Gamma$. The `adjoint' action $\ad\colon \cal{A}\rightarrow \cal{A}\otimes \cal{A}$ of $G$ on $G$ is given by $$\ad(a)=a^{(2)}\otimes \k(a^{(1)})a^{(3)}.$$ The space $\Gamma_{\inv}$ is right-invariant, that is $\wp_{\Gamma}(\Gamma_{\inv})\subseteq\Gamma_{\inv} \otimes \cal{A}$. The corresponding restriction $\Ad\colon \Gamma_{\inv} \rightarrow \Gamma_{\inv} \otimes \cal{A}$ is interpretable as the adjoint action of $G$ on $\Gamma_{\inv} $. Explicitly $\Ad$ is characterized by $$\Ad\pi=(\pi\otimes \id)\ad.$$ The actions $\ell_{\Gamma}$ and $\wp_{\Gamma}$ can be naturally extended to the grade preserving homomorphisms $\wp_{\Gamma}^{\wedge,\otimes} \colon \Gamma^{\wedge,\otimes} \rightarrow \Gamma^{\wedge,\otimes} \otimes \cal{A}$ and $\ell_{\Gamma}^{\wedge,\otimes} \colon \Gamma^{\wedge,\otimes} \rightarrow \cal{A}\otimes \Gamma^{\wedge,\otimes}$ (their restrictions on $\cal{A}$ coincide with $\phi$). The symbol $\grten$ will be used for the graded tensor product of graded-differential algebras. The coproduct $\phi$ admits the unique extension $\widehat{\phi}\colon \Gamma^{\wedge}\rightarrow \Gamma^{\wedge} \grten \Gamma^{\wedge}$ which is a homomorphism of graded-differential algebras \cite{d1}. In particular, $$ \widehat{\phi}(\xi)=\ell_{\Gamma}(\xi)+\wp_{\Gamma}(\xi)$$ for each $ \xi\in \Gamma$. The antipode $\k$ admits the unique extension $\widehat{\k}\colon\Gamma^\wedge\rightarrow \Gamma^\wedge$, which is graded-antimultiplicative and satisfies $\widehat{\k}d=d\widehat{\k}$. Let us denote by $\Gamma_{\inv}^{\otimes}$ and $\Gamma_{\inv}^{\wedge}$ subalgebras of left-invariant elements of $\Gamma^{\otimes}$ and $\Gamma^{\wedge}$ respectively. We have $$\Gamma_{\inv}^{\otimes}= \sideset{}{^\oplus}\sum_{k\geq 0} \Gamma_{\inv}^{\otimes k}\qquad \Gamma_{\inv}^{\wedge} = \sideset{}{^\oplus}\sum_{k\geq 0} \Gamma_{\inv}^{\wedge k} , $$ where $\Gamma_{\inv}^{\otimes k}$ and $\Gamma_{\inv}^{\wedge k}$ consist of left-invariant elements from $\Gamma^{\otimes k}$ and $\Gamma^{\wedge k}$ respectively. The space $\Gamma_{\inv}^{\otimes k}$ is actually the tensor product of $k$-copies of $\Gamma_{\inv}$. The following natural isomorphism holds $$\Gamma_{\inv}^{\wedge}=\Gamma_{\inv}^{\otimes}/S_{\inv}^{\wedge},$$ where $S_{\inv}^{\wedge}$ is the left-invariant part of $S^{\wedge}$. This space is an ideal in $\Gamma_{\inv}^{\otimes}$ generated by elements of the form $$q=\pi(a^{(1)} )\otimes \pi(a^{(2)} ),$$ where $a\in \cal{R}$. All introduced spaces of the form $\Gamma_{\inv}^*$ are right-invariant. We shall denote by $\Ad^{*}$ the adjoint actions of $G$ on the corresponding spaces. The formula $$\vartheta\circ a=\k(a^{(1)} )\vartheta a^{(2)} $$ defines an extension of the right $\cal{A}$-module structure $\circ$ from $\Gamma_{\inv}$ to $\Gamma_{\inv}^{\wedge,\otimes}$. We have \begin{align*} 1\circ a&=\e(a)1 \\ (\vartheta\eta)\circ a&=(\vartheta\circ a^{(1)})(\eta\circ a^{(2)} ) \end{align*} for each $\vartheta,\eta\in \Gamma_{\inv}^{\wedge,\otimes}$ and $a\in \cal{A}$. The algebra $\Gamma_{\inv}^{\wedge} \subseteq\Gamma^{\wedge}$ is $d$-invariant. The differential $d\colon \Gamma_{\inv}^{\wedge} \rightarrow \Gamma_{\inv}^\wedge$ is explicitly determined by $$d\pi(a)=-\pi(a^{(1)})\pi(a^{(2)}).$$ If $\Gamma$ is *-covariant then the *-involution $ *\colon \Gamma\rightarrow \Gamma$ is naturally extendible from $\Gamma$ to $\Gamma^{\wedge ,\otimes}$ (such that for each $\vartheta,\eta\in \Gamma^{\wedge ,\otimes}$ we have $(\vartheta\eta)^* =(-)^{\partial\vartheta\partial\eta}\eta^* \vartheta^* )$. Algebras $\Gamma_{\inv}^{\wedge} ,\Gamma_{\inv}^{\otimes} \subseteq\Gamma^{\wedge ,\otimes}$ are *-invariant. We have $$(\vartheta\circ a)^* =\vartheta^* \circ \k(a)^*$$ for each $a\in \cal{A}$ and $\vartheta\in \Gamma_{\inv}^{\wedge,\otimes}$. Explicitly, the *-involution on $\Gamma_{\inv}$ is determined by $$ \pi(a)^* =-\pi[\k(a)^* ].$$ The map $\widehat{\phi}$, as well as the left and the right actions of $G$ on $\Gamma^{\wedge ,\otimes}$ are *-preserving, in a natural manner. Let $M$ be a compact smooth manifold. By definition \cite{d1} a {\it quantum principal $G$-bundle} over $M$ is a triplet $P=(\cal{B},i,F)$ where $\cal{B}$ is a (unital) *-algebra, consisting of appropriate `functions' on $P$, while $F\colon \cal{B}\rightarrow \cal{B}\otimes \cal{A}$ and $i\colon S(M)\rightarrow \cal{B}$ are (unital) *-homomorphisms, intrepretable as the dualized right action of $G$ on $P,$ and the dualized projection of $P$ on $M$. Further, the bundle $P$ is {\it locally trivial} in the sense that for each $x\in M$ there exists an open set $U\subseteq M$ such that $x\in U,$ and a *-homomorphism $\pi_U \colon \cal{B}\rightarrow S(U)\otimes \cal{A}$ such that \begin{gather*} \pi_U i(f)=(f{\restr}_U )\otimes 1\\ \pi_U (\cal{B})\supseteq \SC (U)\otimes \cal{A}\\ (\id\otimes \phi)\pi_U =(\pi_U \otimes \id)F, \end{gather*} and such that $$\pi_U\bigl(i(f)b\bigr)=0\quad\Longrightarrow\quad i(f)b=0$$ for each $f\in \SC (U)$. Here $S$ and $\SC$ denote the corresponding *-algebras of complex smooth functions (with compact supports, respectively). The homomorphism $\pi_U$ is interpretable as the dualized trivialization of $P$ over $U$. Every pair $(U,\pi_U ),$ consisting of an open set $U\subseteq M$ and of a *-homomorphism $\pi_U \colon \cal{B}\rightarrow S(U)\otimes \cal{A},$ satisfying above conditions is called a local trivialization of $P$. A trivialization system for $P$ is a family $\tau=\bigl\{(U,\pi_U )\mid U\in\cal{U}\bigr\}$ of local trivializations of $P$, where $\cal{U}$ is a finite open cover of $M$. For each $k\in \Bbb{N}$ we shall denote by $N^k (\cal{U})$ the set of $k$-tuples $(U_1 ,\dots ,U_k )\in \cal{U}^k$ such that $U_1 \cap \dots \cap U_k\neq \emptyset$. The main structural result concerning quantum principal bundles is that there exists a natural correspondence between quantum principal $G$-bundles $P$ and classical principal $G_{cl}$-bundles $P_{cl}$ over $M.$ This corresponence can be described as follows. From a given trivialization system $\tau$ it is possible to construct the corresponding $G$-cocycle which is a system of *-automorphisms $\psi_{UV}$ of $S(U\cap V)\otimes \cal{A}$, where $(U,V)\in N^2 (\cal{U})$, realizing transformations between $(V,\pi_V )$ and $(U,\pi_U )$. Such systems of maps completely determine the bundle $P$. Explicitly, let us consider the *-algebra $$\Sigma (\cal{U})= \sideset{}{^\oplus}\sum_{U\in \cal{U}} \Bigl[S(U)\otimes \cal{A}\Bigr].$$ The algebra $\cal{B}$ is realizable as a subalgebra of $\Sigma (\cal{U}),$ consisting of elements $b\in \Sigma (\cal{U})$ satisfying $$(_U {\restr}_{U\cap V} \otimes \id)p_U (b)=\psi_{UV} (_V {\restr}_{U\cap V} \otimes \id)p_V (b) $$ for each $(U,V)\in N^2 (\cal{U}).$ Here, $p_U \colon \Sigma (\cal{U})\rightarrow S(U)\otimes \cal{A}$ are coordinate projections. In terms of this realization we have $$ \pi_U =p_U{\restr}\cal{B},$$ for each $U\in \cal{U}.$ However, it turns out that $G$-cocycles are in a natural bijection with standard $G_{cl}$-cocycles (over $\cal{U}$), which are systems of smooth maps $g_{UV} \colon U\cap V\rightarrow G_{cl}$ satisfying $$g_{UV}g_{VW}(x)=g_{UW}(x),$$ for each $(U,V,W)\in N^3(\cal{U})$ and $x\in U\cap V\cap W$ (in particular $g_{UV}^{-1}=g_{VU}$). The correspondence is established via the following formula $$\psi_{UV}(\varphi\otimes a)=\varphi g_{VU}(a^{(1)})\otimes a^{(2)}.$$ Here, maps $g_{UV}$ are understood as *-homomorphisms $g_{UV}\colon \cal{A}\rightarrow S(U)$, in a natural manner. On the other hand, $G_{cl}$-cocycles determine, in the standard manner, principal $G_{cl}$-bundles $P$ over $M.$ The bundle $P_{cl}$ is interpretable as the `classical part' of $P.$ The elements of $P_{cl}$ are in a natural bijection with *-characters of $\cal{B}.$ The correspondence $P\leftrightarrow P_{cl}$ has a simple geometrical explanation. The `transition functions' $\psi_{UV}$ are, at the geometrical level, vertical `diffeomorphisms' of $(U\cap V)\times G.$ Therefore they preserve the geometrical structure of $(U\cap V)\times G.$ In particular, they must preserve the classical part $(U\cap V)\times G_{cl}$ consisting of points of $(U\cap V)\times G.$ Moreover, transition diffeomorphisms are completely determined by their restrictions on $(U\cap V)\times G_{cl} ,$ because of the right covariance. The corresponding `restrictions' are precisely transition functions for the classical bundle $P_{cl}.$ So far about the structure of quantum principal bundles. For each (nonempty) open set $U\subseteq M$ let $\Omega (U)$ be the graded-differential *-algebra of differential forms on $U.$ In developing a differential calculus over quantum principal bundles it is natural to assume that the calculus is fully compatible with the geometrical structure on the bundle, such that all local trivializations of the bundle locally trivialize the calculus too (a precise formulation of this condition is given in \cite{d1}--Section~3). It turns out that this condition completely fixes the calculus on the bundle (if the calculus on the structure quantum group is fixed). However, the condition implies certain restrictions on a possible differential calculus $\Gamma$ over $G.$ Namely, all retrivialization maps $\psi_{UV}$ must be extendible to differential algebra automorphisms $\psi^{\wedge}_{UV}\colon\Omega(U\cap V)\grten \Gamma^{\wedge}\rightarrow \Omega(U\cap V)\grten \Gamma^{\wedge} .$ Differential calculi $\Gamma$ satisfying this condition are called admissible. If $\Gamma$ is left-covariant then it is admissible iff $$(X\otimes \id)\ad(\cal{R})=\{0\}$$ for each $X\in \lie(G_{cl} ).$ This fact implies that there exists the minimal admissible left-covariant calculus $\Gamma.$ This calculus is based on the right $\cal{A}$-ideal $\widehat{\cal{R}}\subseteq \ker(\e)$ consisting of all elements $a\in \ker(\e)$ satisfying $$(X\otimes \id)\ad(a)=0,$$ for each $X\in lie(G_{cl} ). $ Moreover, we have $\ad(\widehat{\cal{R}})\subseteq\widehat{\cal{R}}\otimes \cal{A}$ and $\k(\widehat{\cal{R}})^* =\widehat{\cal{R}}$, which implies \cite{w3} that $\Gamma$ is bicovariant and *-covariant respectively. In the following, $\Gamma$ will be this minimal admissible (bicovariant *-) calculus. Let $\Omega(P)$ be the graded-differential *-algebra representing differential calculus on $P$ (constructed by combining differential forms on $M$ with the universal envelope $\Gamma^{\wedge}$ of $\Gamma).$ Explicitly, let us consider the direct sum $$\Sigma^{\wedge}(\cal{U})= \sideset{}{^\oplus}\sum_{U\in\cal{U}} \Bigl[\Omega(U)\grten \Gamma^{\wedge} \Bigr].$$ Then $\Omega(P)$ can be viewed as a graded-differential subalgebra consisting of elements $w\in \Sigma^{\wedge} (\cal{U})$ satisfying $$(_U{\restr}_{U\cap V}\otimes\id)p_U (w)=\psi^{\wedge}_{UV}(_V{\restr}_{U\cap V}\otimes\id)p_V (w) $$ for each $(U,V)\in N^2 (\cal{U})$. Here $p_U \colon \Sigma^{\wedge} (\cal{U})\rightarrow \Omega(U)\grten\Gamma^{\wedge}$ are corresponding coordinate projections. As a differential algebra, $\Omega(P)$ is generated by $\cal{B}=\Omega^0 (P).$ For every local trivialisation $(U,\pi_U )$ of $P$ there exists the unique differential algebra homomorphism $\pi^{\wedge}_U \colon \Omega(P)\rightarrow \Omega(U)\grten \Gamma^{\wedge}$ extending $\pi_U$ (in fact $\pi^{\wedge}_U =p_U {\restr}\Omega(P)). $ The map $i\colon S(M)\rightarrow \cal{B}$ admits a natural extension $i^{\wedge} \colon \Omega(M)\rightarrow \Omega(P),$ which is interpretable as the `pull back' of differential forms on $M$ to $P$. We have $$\pi^{\wedge}_Ui^{\wedge}(w)=(w{\restr}_U)\otimes 1.$$ The right action $F\colon \cal{B}\rightarrow \cal{B}\otimes \cal{A}$ is (uniquely) extendible to a differential algebra homomorphism $\widehat{F}\colon \Omega(P)\rightarrow \Omega(P)\grten \Gamma^{\wedge}$, imitating the corresponding pull back map. The formula $$ F^{\wedge} =(\id\otimes \Pi )\widehat{F}$$ determines a *-homomorphism $F^{\wedge} \colon \Omega(P)\rightarrow \Omega(P)\otimes \cal{A}$ interpretable as the (dualized) right action of $G$ on $\Omega(P)$. Here $\Pi\colon\Gamma^\wedge\rightarrow\cal{A}$ is the projection map. Let $\ver(P)$ be the graded-differential *-algebra obtained by factorizing $\Omega(P)$ through the (differential *-ideal) generated by elements of the form $d[i(f)]$. The elements of $\ver(P)$ play the role of `verticalized' differential forms on $P$ (in classical geometry, entities obtained by restricting the domain of differential forms to the Lie algebra of vertical vector fields on the bundle). At the level of graded vector spaces, there exists a natural isomorphism $$\ver(P)\cong \cal{B}\otimes \Gamma^{\wedge}_{\inv} .$$ Let $\pi_v \colon \Omega(P)\rightarrow \ver(P)$ be the corresponding projection map. In terms of the above identifications, the differential *-algebra structure on $\ver(P)$ is specified by \begin{gather*} (q\otimes \eta)(b\otimes \vartheta)=\sum_k qb_k \otimes (\eta\circ a_k )\vartheta \\ (b\otimes \vartheta)^* =\sum_k b^*_k \otimes (\vartheta^* \circ a^*_k)\\ d_v (b\otimes \vartheta)=\sum_k b_k \otimes \pi(a_k )\vartheta+b\otimes d\vartheta \end{gather*} where $F(b)=\Sum_k b_k\otimes a_k$. Another important algebra naturally associated to $\Omega(P)$ is a graded *-subalgebra $\hor(P)\subseteq\Omega(P) $ representing horizontal forms. By definition, $\hor(P)$ consists of forms $w\in \Omega(P)$ with the property $$\pi^{\wedge}_U (w)\in \Omega(U)\otimes \cal{A},$$ for each local trivialization $(U,\pi_U )$. Equivalently, $$\hor(P)=(\widehat{F})^{-1}\Bigl\{\Omega(P)\otimes\cal{A}\Bigr\}.$$ The algebra $\hor(P)$ is invariant under the right action of $G,$ in other words $$F^{\wedge} \bigl(\hor(P)\bigr)\subseteq\hor(P)\otimes \cal{A}.$$ Let $\psi(P)$ be the space of all linear maps $\varphi\colon \Gamma_{\inv} \rightarrow \Omega(P)$ satisfying $$ (\varphi\otimes \id)\Ad=F^{\wedge}\varphi. $$ This space is naturally graded (the grading is induced from $\Omega(P)$). The elements of $\psi (P)$ are quantum counterparts of pseudotensorial forms on the bundle with coefficients in the structure group Lie algebra (relative to the adjoint representation). The space $\psi (P) $ is closed with respect to compositions with $d\colon \Omega(P)\rightarrow \Omega(P).$ Let $\tau (P)\subseteq\psi (P)$ be the subspace consisting of $\hor(P)$-valued maps. This space is imaginable as consisting of the corresponding tensorial forms. There exists a natural *-involution on $\psi (P)$. It is given by $$\varphi^* (\vartheta)=\varphi(\vartheta^* )^* .$$ The space $\tau (P)$ is *-invariant. Tensorial forms possess the following local representation: $$\pi^{\wedge}_U \varphi(\vartheta)=(f^U \otimes \id)\Ad(\vartheta),$$ where $f^U \colon\Gamma_{\inv} \rightarrow \Omega(U)$ is a linear map. For the purposes of this paper the most important topic of the theory of quantum principal bundles is the formalism of connections. By definition, a connection on $P$ is every pseudotensorial hermitian $1$-form $\omega$ satisfying $$\pi_v \omega(\vartheta)=1\otimes \vartheta $$ for each $\vartheta\in \Gamma_{\inv}$. The above formula is the quantum counterpart for the classical condition that connections map fundamental vector fields into their generators. Connections form a real affine space $\con(P)$. In local terms, connections possess the following representation $$\pi^{\wedge}_U \omega(\vartheta)=(A^U \otimes \id)\Ad(\vartheta)+1_U \otimes \vartheta, $$ where $A^U \colon \Gamma_{\inv} \rightarrow \Omega(U)$ is a $1$-form valued hermitian linear map (playing the role of the corresponding gauge potential). The curvature operator can be described as follows. Let us fix a map $\delta\colon \Gamma_{\inv} \rightarrow \Gamma_{\inv} \otimes \Gamma_{\inv}$ which intertwines the corresponding adjoint actions and such that if $$\delta(\vartheta)=\sum_k \vartheta^1_k \otimes \vartheta^2_k $$ then $$\delta(\vartheta^* )=-\sum_k (\vartheta^2_k )^* \otimes (\vartheta^1_k )^* \qquad d\vartheta=\sum_k\vartheta^1_k\vartheta^2_k.$$ Every such a map will be called an {\it embedded differential}. Further, for each pair of linear maps $\varphi,\psi$ on $ \Gamma_{\inv}$ with values in an arbitrary algebra $\Omega$ let $\langle \varphi,\psi\rangle\colon \Gamma_{\inv} \rightarrow \Omega$ be a map given by $$\langle \varphi,\psi\rangle(\vartheta)=\sum_k\varphi(\vartheta_k^1) \psi(\vartheta_k^2).$$ By construction, if $\varphi,\psi \in \psi (P)$ then $\langle \varphi,\psi\rangle\in \psi (P),$ too. Finally, the curvature $R_\omega$ of a connection $\omega $ can be defined as $$ R_\omega =d\omega-\langle \omega,\omega\rangle . $$ The above formula corresponds to the structure equation in the classical theory. It turns out that $R_\omega$ is a tensorial $2$-form. Locally, in terms of the corresponding gauge potentials we have $$\pi^{\wedge}_U R_\omega (\vartheta)=(F^U \otimes \id)\Ad(\vartheta),$$ where $$ F^U =dA^U -\langle A^U ,A^U \rangle . $$ For each open set $U\subseteq M$ the symbol $\otimes_U$ will be used for the tensor product over $S(U)$. Similarly, the symbol $\grten_U$ will denote the graded tensor product of graded-differential *-algebras containing $\Omega(U)$ as their subalgebra. \section{ Quantum Gauge Bundles } This section is devoted to generalizations of the most important aspects of the concept of gauge transformations, in the framework of the formalism of quantum principal bundles. The main geometrical object that will be constructed is the quantum gauge bundle, a noncommutative-geometric counterpart of the gauge bundle of the classical theory. \subsection{Classical Picture} In order to present motivations for constructions of this section let us assume for a moment that $G$ is an ordinary compact Lie group, and let $P$ be a (classical) principal bundle over $M.$ By definition, gauge transformations of $P$ are vertical automorphisms of this bundle. In other words, gauge transformations are diffeomorphisms $\psi \colon P\rightarrow P$ satisfying \begin{align*} &\pi_M\psi =\pi_M \\ &\psi (pg)=\psi (p)g, \end{align*} for each $p\in P$ and $g\in G,$ where $(p,g)\mapsto pg$ is the right action of $G$ on $P$ and $\pi_M\colon P\rightarrow M$ is the projection map. Equivalently, gauge transformations are interpretable as (smooth) sections of the gauge bundle $\ADP,$ which is the bundle associated to $P,$ with respect to the adjoint action of $G$ onto itself. The equivalence between two definitions is established via the following formula $$ \psi (p)=pf(p), $$ where $f\colon P\rightarrow G$ is a smooth equivariant function in the sense that $$ f(pg)=g^{-1} f(p)g, $$ for each $p\in P$ and $g\in G.$ Such functions are in a natural correspondence with sections of the corresponding associated bundle $\ADP$. For each $x\in M$ the fiber $G_x =\pi_M^{\sharp-1} (x)$ over $x$ (where $\pi_M^\sharp\colon \ADP\rightarrow M$ is the projection map) possesses a natural Lie group structure. The group $G_x$ is isomorphic (generally non-invariantly) to $G.$ For a given $p\in \pi_M^{-1} (x)=P_x$ there exists a canonical diffeomorphism $G\leftrightarrow P_x$ defined by $g\leftrightarrow pg$, and a group isomorphism $G\leftrightarrow G_x $ given by $g\leftrightarrow \bigl[(p,g)\bigr]$. Here, $\ADP$ is understood as the orbit space of the right action $((p,g'),g)\mapsto (pg,g^{-1} g'g)$ of $G$ on $P\times G$ and $\bigl[\,\bigr]$ denotes the corresponding orbit. There exists a natural left action of $G_x$ on $P_x .$ In terms of the above identifications this action becomes the multiplication on the left. Collecting all these fiber actions together, we obtain a smooth map \begin{equation}\label{33}\beta^*_M \colon \ADP\times_M P\rightarrow P. \end{equation} With the help of $\beta^*_M$ the equivalence between gauge transformations $\psi$ and sections $\varphi\colon M\rightarrow \ADP$ can be described as follows \begin{equation}\label{34}\psi =\beta^*_M (\varphi\times_M \id). \end{equation} Moreover, the correspondence $\psi \leftrightarrow \varphi$ is an isomorphism between the group $\cal{G}$ of gauge transformations of $P,$ and the group $\Gamma(\ADP)$ of smooth sections of $\ADP.$ The group structure in fibers of $\ADP$ determine the following maps of bundles \begin{equation}\label{35}\begin{gathered} \text{\it the fibewise multiplication}\quad \phi^*_M \colon \ADP\times_M \ADP\rightarrow \ADP\\ \text{\it the unit section}\qquad\qquad\quad \e^*_M \colon M\rightarrow \ADP\\ \text{\it the fiberwise inverse}\qquad\quad \k^*_M\colon \ADP \rightarrow \ADP\end{gathered} \end{equation} At the dual level of function algebras \eqref{33} and \eqref{35} are represented by the corresponding $S(M)$-linear *-homomorphisms \begin{equation}\label{36}\begin{aligned} \phi_M &\colon S(\ADP)\rightarrow S(\ADP)\otimes_M S(\ADP)\\ \e_M &\colon S(\ADP)\rightarrow S(M) \\ \k_M &\colon S(\ADP)\rightarrow S(\ADP) \\ \beta_M &\colon S(P)\rightarrow S(\ADP)\otimes_M S(\ADP). \end{aligned} \end{equation} The structure of the gauge group is completely encoded in maps $\bigl\{\phi_M,\k_M,\e_M\bigr\}$. At the dual level, gauge transformations $\psi$ can be viewed as $S(M)$-linear *-automorphisms $\psi\colon S(P)\rightarrow S(P)$ intertwining the (dualized) right action of $G.$ Further, interpreted as sections of $\ADP,$ gauge transformations become, at the dual level, $S(M)$-linear *-homomorphisms $\varphi \colon S(\ADP)\rightarrow S(M).$ In this picture, the action of $\cal{G}$ on $S(\ADP)$ is given by $$(\varphi,f)\mapsto(\varphi\otimes \id)\beta_M (f).$$ The maps \eqref{36} are not suitable for considering situations in which gauge transformations act on differential forms. This can be easily `improved' by extending these maps to $\Omega(M)$-linear homomorphisms \begin{equation}\label{37}\begin{aligned} \widehat{\phi}_M &\colon \Omega(\ADP)\rightarrow \Omega(\ADP)\grten_M \Omega(\ADP) \\ \widehat{\e}_M &\colon \Omega(\ADP)\rightarrow \Omega(M) \\ \widehat{\k}_M &\colon \Omega(\ADP)\rightarrow \Omega(\ADP) \\ \widehat{\beta}_M &\colon \Omega(P)\rightarrow \Omega(\ADP) \grten_M \Omega(\ADP) \end{aligned} \end{equation} of graded-differential *-algebras. It is worth noticing that the above maps are unique, as graded-differential extensions. Actually these maps can be viewed as `pull backs' of \eqref{33} and \eqref{35}. \subsection{Quantum Consideration} The presented picture admits a direct noncommutative-geometric generalization. At first, we shall construct, starting from a quantum principal bundle $P$, the corresponding quantum gauge bundle $\ADP$. Then the counterparts of maps \eqref{36} will be introduced and analyzed. In analogy with the classical case we shall define gauge transformations as vertical automorphisms of the bundle $P$. It turns out that such gauge transformations of $P$ are in a natural bijection with ordinary gauge transformations of the classical part $P_{cl}$ of $P$. We shall also study various equivalent interpretations of gauge transformations. Finally, a canonical differential calculus on the bundle $\ADP$ will be constructed and analyzed. Let $G$ be a compact matrix quantum group, and let $P=(\cal{B},i,F)$ be a quantum principal $G$-bundle over $M$. Let us fix a trivialization system $\tau$ for $P$. For each $(U,V)\in N^2 (\cal{U})$ let us define a linear map $\xi_{UV}\colon S(U\cap V)\otimes\cal{A}\rightarrow S(U\cap V)\otimes \cal{A}$ by the following formula \begin{equation}\label{38} \xi_{UV}(\varphi\otimes a) =\varphi g_{UV}\bigl[\k(a^{(1)})a^{(3)}\bigr]\otimes a^{(2)}. \end{equation} \begin{lem}\label{lem:31} \bla{i} The maps $\xi_{UV}$ are $S(U\cap V)$-linear *-automorphisms and \begin{equation}\xi^{-1}_{UV}=\xi_{VU}.\label{xinv}\end{equation} \bla{ii} We have \begin{equation}\label{39}\xi_{UV}\xi_{VW}(\varphi)=\xi_{UW}(\varphi). \end{equation} for each $(U,V,W)\in N^3 (\cal{U})$ and $\varphi\in\SC(U{\cap} V{\cap} W)\otimes\cal{A}$. \smallskip \bla{iii} The diagrams \begin{gather*} \begin{CD} S(U\cap V)\otimes\cal{A}@>{\mbox{$\id\otimes\phi$}}>> \bigl[S(U\cap V)\otimes\cal{A} \bigr]\otimes_{U\cap V}\bigl[S(U\cap V)\otimes\cal{A}\bigr]\\ @V{\mbox{$\xi_{UV}$}}VV @VV{\mbox{$\xi_{UV}\otimes \xi_{UV}$}}V\\ S(U\cap V)\otimes\cal{A} @>>{\mbox{$\id\otimes\phi$}}> \bigl[S(U\cap V)\otimes\cal{A} \bigr]\otimes_{U\cap V}\bigl[S(U\cap V)\otimes\cal{A}\bigr] \end{CD}\\ \begin{CD} S(U\cap V)\otimes\cal{A}@>{\mbox{$\id\otimes \e$}}>> S(U\cap V)\\ @V{\mbox{$\xi_{UV}$}}VV @VV{\mbox{$\id$}}V\\ S(U\cap V)\otimes\cal{A}@>>{\mbox{$\id\otimes \e$}}> S(U\cap V) \end{CD}\qquad \begin{CD} S(U\cap V)\otimes\cal{A}@>{\mbox{$\id\otimes \k$}}>> S(U\cap V)\otimes\cal{A} \\ @V{\mbox{$\xi_{UV}$}}VV @VV{\mbox{$\xi_{UV}$}}V\\ S(U\cap V)\otimes\cal{A}@>>{\mbox{$\id\otimes \k$}}> S(U\cap V)\otimes\cal{A} \end{CD}\\ \begin{CD} S(U\cap V)\otimes\cal{A}@>{\mbox{$\id\otimes\phi$}}>> \bigl[S(U\cap V)\otimes\cal{A} \bigr]\otimes_{U\cap V}\bigl[S(U\cap V)\otimes\cal{A}\bigr]\\ @V{\mbox{$\psi_{UV}$}}VV @VV{\mbox{$\xi_{UV}\otimes \psi_{UV}$}}V\\ S(U\cap V)\otimes\cal{A} @>>{\mbox{$\id\otimes\phi$}}> \bigl[S(U\cap V)\otimes\cal{A} \bigr]\otimes_{U\cap V}\bigl[S(U\cap V)\otimes\cal{A}\bigr] \end{CD} \end{gather*} are commutative. \end{lem} \begin{pf} We have \begin{equation*} \begin{split} \xi_{UV} \xi_{VW} (\varphi\otimes a)&=\xi_{UV} \left(\varphi g_{VW}\bigl[\k(a^{(1)} )a^{(3)} \bigr]\otimes a^{(2)} \right)\\ &=\varphi g_{VW}\bigl[\k(a^{(1)} )a^{(5)}\bigr]g_{UV} \bigl[\k(a^{(2)})a^{(4)}\bigr]\otimes a^{(3)} \\ &=\varphi g_{UW}\bigl[\k(a^{(1)})a^{(3)}\bigr]\otimes a^{(2)} =\xi_{UW} (\varphi\otimes a),\end{split}\end{equation*} for each $(U,V,W)\in N^3 (\cal{U})$, $ \varphi\in \SC(U{\cap} V{\cap} W)$ and $a\in \cal{A}$. In particular, for $W=V$ this implies that the maps $\xi_{UV}$ are bijective and that \eqref{xinv} holds. The maps $\xi_{UV}$ are *-homomorphisms because of \begin{multline*} \xi_{UV} (\varphi^* \otimes a^* )=\varphi^* g_{VU} (a^{(1)*})g_{UV}(a^{(3)*} )\otimes a^{(2)*}\\ =\left[\varphi g_{VU}(a^{(1)} )g_{UV}(a^{(3)} )\right]^* \!\otimes a^{(2)*}=\xi_{UV}(\varphi\otimes a)^*, \end{multline*} and \begin{equation*} \begin{split} \xi_{UV} (\varphi\psi \otimes ab)&=\varphi\psi g_{VU}(a^{(1)} b^{(1)} )g_{UV}(a^{(3)} b^{(3)} )\otimes a^{(2)} b^{(2)} \\ &=\left[\varphi g_{VU}(a^{(1)} )g_{UV}(a^{(3)} )\otimes a^{(2)} \right]\left[\psi g_{VU}(b^{(1)} )g_{UV} (b^{(3)} )\otimes b^{(2)} \right]\\ &=\xi_{UV}(\varphi\otimes a)\xi_{UV}(\psi \otimes b). \end{split}\end{equation*} Finally, let us check commutativity of the above diagrams. We compute \begin{equation*}\begin{split} (\xi_{UV} \otimes \xi_{UV} ) (\id\otimes \phi)(\varphi\otimes a) &=\varphi g_{UV} \bigl(\k(a^{(1)} )a^{(3)}\k(a^{(4)} )a^{(6)} \bigr)\otimes a^{(2)} \otimes a^{(5)} \\ &=\varphi g_{UV} \bigl(\k(a^{(1)} )a^{(4)} \bigr)\otimes a^{(2)} \otimes a^{(3)}\\ &=(\id\otimes \phi)\xi_{UV} (\varphi\otimes a), \end{split} \end{equation*} and similarly \begin{multline*} (\xi_{UV} \otimes \psi_{UV} ) (\id\otimes \phi)(\varphi\otimes a)=\varphi g_{UV} \bigl(\k(a^{(1)} )a^{(3)} \bigr) g_{VU} (a^{(4)} ) \otimes a^{(2)} \otimes a^{(5)} \\ =\varphi g_{VU} (a^{(1)} )\otimes a^{(2)} \otimes a^{(3)}=(\id\otimes \phi)\psi_{UV}(\varphi\otimes a). \end{multline*} Finally, \begin{equation*} \begin{split} \xi_{UV} \bigl(\varphi\otimes \k(a)\bigr)&=\varphi g_{UV} \bigl(\k^2 (a^{(3)} )\k(a^{(1)} )\bigr)\otimes \k(a^{(2)} ) \\ &=\varphi g_{UV} \bigl(\k(a^{(1)} )a^{(3)} \bigr)\otimes \k(a^{(2)} )\\&=(\id\otimes \k)\xi_{UV} (\varphi\otimes a). \end{split} \end{equation*} Together with a trivial observation that $$ (\id\otimes \e)\xi_{UV}(\varphi\otimes a)=\varphi g_{UV}\bigl(\k(a^{(1)})a^{(2)}\bigr)=\e(a)\varphi, $$ this completes the proof. \end{pf} The (algebra of functions on the) quantum gauge bundle $\ADP$ can be now constructed as follows. Let $\cal{D}$ be the set of elements $q\in \Sigma (\cal{U})$ such that \begin{equation}\label{314} ({}_U {\restr}_{U\cap V} \otimes \id)p_U (q) =\xi_{UV} ({}_V {\restr}_{U\cap V} \otimes \id)p_V (q) \end{equation} for each $(U,V)\in N^2 (\cal{U})$. Clearly, $\cal{D}$ is a *-subalgebra of $\Sigma (\cal{U})$. The quantum space $\ADP$ corresponding to $\cal{D}$ plays the role of the bundle associated to the principal bundle $P,$ with respect to the adoint action of $G$ onto itself (represented by $\ad\colon \cal{A}\rightarrow \cal{A}\otimes \cal{A}$). The fact that $\ADP$ is a bundle over $M$ is established throught the existence of a *-monomorphism $j_M \colon S(M)\rightarrow \cal{D},$ playing the role of the dualized fibering of $\ADP$ over $M$. This map is defined by equalities \begin{equation}\label{315} p_U j_M (f)=(f{\restr} U)\otimes 1. \end{equation} \begin{defn}\label{defn:31} The pair $\ADP=(\cal{D},j_M )$ is called {\it the quantum gauge bundle} associated to $P$. \end{defn} We are going to introduce quantum counterparts of maps $\phi_M$, $\k_M$, $\e_M$ and $\beta_M $. For each $U\in \cal{U}$, let $\pi^\sharp_U\colon \cal{D}\rightarrow S(U)\otimes \cal{A}$ be the restriction of $p_U$ on $\cal{D}$. \begin{pro}\label{pro:32} \bla{i} There exist the unique linear maps $\phi_M \colon\cal{D}\rightarrow \cal{D}\otimes_M \cal{D},$ $\e_M \colon \cal{D}\rightarrow S(M)$, $\k_M \colon \cal{D}\rightarrow \cal{D}$ and $\beta_M \colon \cal{B}\rightarrow \cal{D}\otimes_M \cal{B}$ such that \begin{gather} (\pi^\sharp_U\otimes \pi^\sharp_U)\phi_M =(\id\otimes \phi)\pi^\sharp_U \label{316}\\ (\pi^\sharp_U\otimes\pi_U)\beta_M =(\id\otimes\phi)\pi^\sharp_U \label{317}\\ \pi^\sharp_U \k_M =(\id\otimes \k)\pi^\sharp_U \label{318}\\ {\restr}_U \e_M =(\id\otimes \e)\pi^\sharp_U, \label{319} \end{gather} for each $U\in \cal{U}$. Here, $S(U)\otimes \cal{A}\otimes \cal{A}$ and $\bigl(S(U)\otimes\cal{A}\bigr)\otimes_U\bigl(S(U)\otimes\cal{A}\bigr)$ are identified, in a natural manner. \smallskip \bla{ii} All maps are $S(M)$-linear. The maps $\phi_M$ ,$\e_M$ and $\beta_M$ are *-homomorphisms while $\k_M$ is antimultiplicative and \begin{equation}\label{320} \k_M \bigl[\k_M (f^* )^* \bigr]=f \end{equation} for each $f\in \cal{D}$. \end{pro} \begin{pf} The above equalities uniquely fix the values of maps $\phi_M$, $\e_M$, $\k_M$ and $\beta_M$ because the maps $\pi_U$ and $\pi^\sharp_U$ distinguish points of $\cal{B}$ and $\cal{D}$. Let us consider the algebra $$\Sigma^*(\cal{U})= \sideset{}{^\oplus}\sum_{U\in \cal{U}} S(U)\otimes \cal{A}\otimes \cal{A}.$$ Algebras $\cal{D}\otimes_M \cal{D}$ and $\cal{D}\otimes_M \cal{B}$ are understandable as subalgebras of $\Sigma^*(\cal{U})$. Let us consider maps $\phi_M \colon \Sigma (\cal{U})\rightarrow \Sigma^*(\cal{U})$, $\k_M\colon\Sigma(\cal{U})\rightarrow \Sigma (\cal{U})$ and $\e_M \colon\Sigma(\cal{U})\rightarrow S(\cal{U})$ defined by \begin{gather*} p^*_U\phi_M =(\id\otimes \phi)p_U\\ p_U\k_M =(\id\otimes \k)p_U\\ {\restr}_U\e_M =(\id\otimes \e)p_U , \end{gather*} where $p^*_U\colon \Sigma^*(\cal{U})\rightarrow S(U)\otimes \cal{A}\otimes \cal{A}$ are coordinate projections and $S(\cal{U})$ is the direct sum of algebras $S(U)$. It is easy to see that $\phi_M (\cal{B})\subseteq\cal{D}\otimes_M \cal{B}$, $\phi_M (\cal{D})\subseteq\cal{D}\otimes_M\cal{D}$, $\k_M (\cal{D})\subseteq\cal{D}$ and $\e_M (\cal{D})\subseteq S(M)$. Let us denote by $\bigl\{\phi_M, \beta_M, \k_M, \e_M\bigr\}$ the corresponding restrictions. By construction,\eqref{316}--\eqref{319} hold, maps $\beta_M$ ,$\phi_M$ and $\e_M$ are *-homomorphisms, $\k_M$ is antimultiplicative and \eqref{320} holds.\end{pf} The fibers of the bundle $\ADP$ possess a natural quantum group structure. Further, the bundle $\ADP$ acts on the bundle $P$, preserving fibers and the right action. This is a geometrical background for the next proposition. \begin{pro}\label{pro:33} The following identities hold \begin{gather} (\id\otimes\phi_M)\phi_M=(\phi_M\otimes \id)\phi_M\\ (\id\otimes F)\beta_M=(\beta_M\otimes \id)F\label{Fcov}\\ (\id\otimes\beta_M)\beta_M=(\phi_M\otimes \id)\beta_M\label{322}\\ (\id\otimes \e_M)\phi_M=(\e_M\otimes \id)\phi_M=\id \label{324} \\ (\e_M\otimes \id)\beta_M=\id \label{325} \\ m_M(\k_M\otimes \id)\phi_M=m_M(\id\otimes \k_M)\phi_M=j_M\e_M \label{326} \end{gather} where $m_M \colon \cal{D}\otimes_M \cal{D}\rightarrow \cal{D}$ is the multiplication map. \end{pro} \begin{pf} In terms of local trivializations, everything reduces to elementary algebraic properties of the coproduct, the counit, and the antipode. \end{pf} We pass to the analysis of gauge transformations, in this quantum framework. In analogy with classical geometry, these transformations will be defined as vertical automorphisms of the bundle. \begin{defn}\label{defn:32} {\it A gauge transformation} of the bundle $P$ is every $S(M)$-linear *-automorphism $\gamma\colon \cal{B}\rightarrow \cal{B}$ such that the diagram \begin{equation}\label{d1} \begin{CD} \cal{B} @>{\mbox{$F$}}>> \cal{B}\otimes\cal{A}\\ @V{\mbox{$\gamma$}}VV @VV{\mbox{$\gamma\otimes \id$}}V\\ \cal{B} @>>{\mbox{$F$}}> \cal{B}\otimes\cal{A} \end{CD} \end{equation} is commutative. \end{defn} The above diagram says that $\gamma$ intertwines the right action of $G$ on $P$, while the $S(M)$-linearity property ensures that $\gamma$ is a `vertical' automorphism of $P$. Obviously, gauge transformations form a subgroup $\cal{G}\subseteq \mbox{Aut}(\cal{B})$. \begin{pro}\label{pro:34} \bla{i} The formula \begin{equation}\label{327} f \leftrightarrow (f\otimes \id)\beta_M =\gamma \end{equation} establishes a bijection between $S(M)$-linear *-homomorphimsms $f\colon \cal{D}\rightarrow S(M)$ and gauge transformations $\gamma\in \cal{G}$. In terms of this correspondence, the map $\e_M$ corresponds to the neutral element in $\cal{G}$ while the product and the inverse in the gauge group are given by \begin{gather} f\k_M \leftrightarrow \gamma^{-1} \label{329}\\ (f'\otimes f)\phi_M \leftrightarrow \gamma\gamma'. \label{330} \end{gather} \bla{ii} Let $\gamma$ be an arbitrary gauge transformation. Then the map $\gamma_{cl}\colon P_{cl} \rightarrow P_{cl}$ defined by \begin{equation} \label{331} \gamma_{cl} (p)=p\gamma^{-1} \end{equation} is an ordinary gauge transformation of $P_{cl}$. Moreover, the above formula establishes an isomorphism between groups of gauge transformations of bundles $P$ and $P_{cl} . $ \end{pro} \begin{pf} Identity \eqref{Fcov} implies that a $S(M)$-linear homomorphism $\gamma\colon \cal{B}\rightarrow \cal{B}$ given by the right-hand side of \eqref{327} satisfies \eqref{d1}. Identity \eqref{325} ensures that $\e_M$ corresponds to the neutral element of $\cal{G}$. Let us consider an arbitrary gauge transformation $\gamma\in \cal{G}$. In terms of the trivialization system $\tau$ we have \begin{equation}\label{332} \pi_U \gamma(b)=\sum_{i} \varphi_i \gamma_U (a^{(1)}_i )\otimes a^{(2)}_i \end{equation} for each $U\in \cal{U}$. Here, $\pi_U (b)=\Sum_{i} \varphi_i \otimes a_i $ and $\gamma_U \colon U\rightarrow G_{cl}$ are smooth functions uniquely determined by $\gamma$ (understood here in the `dual' manner). We have \begin{equation}\label{333} \bigl[\gamma_V (a^{(1)}){\restr}_{U\cap V}\bigr] g_{VU}(a^{(2)})=g_{VU}(a^{(1)})\bigl[\gamma_U(a^{(2)}){\restr}_{U\cap V}\bigr] \end{equation} for each $a\in \cal{A}$ and $(U,V)\in N^2 (\cal{U})$. Conversely, if *-homomorphisms $\gamma_U \colon \cal{A}\rightarrow S(U)$ are given such that equalities \eqref{333} hold then formula \eqref{332} consistently determines a gauge transformation $\gamma$. Let us now consider a map $f\colon \Sigma (\cal{U})\rightarrow S(\cal{U})$ defined by $$ f=\sideset{}{^\oplus}\sum_{U\in\cal{U}} {f}_U ,$$ where ${f}_U \colon S(U)\otimes \cal{A}\rightarrow S(U)$ are maps given by ${f}_U (\varphi\otimes a)=\varphi \gamma_U (a)$. It is easy to see that if $b\in \cal{D}$ then $f(b)\in S(M)$ (where $S(M)$ is understood as a subalgebra of $S(\cal{U})$). Let us pass to the corresponding restriction $f\colon \cal{D}\rightarrow S(M)$. By construction \eqref{327} holds (it is evident in a local trivialization). Conversely, if $f\colon \cal{D}\rightarrow S(M) $ determines a gauge transformation $\gamma$ then $$f(b){\restr}_U =\sum_{i}\varphi_i \gamma_U(a_i ).$$ This easily follows from \eqref{327}. Let us check correspondences \eqref{329}--\eqref{330}. We have \begin{align*} &\left[(f\otimes f')\phi_M \otimes \id\right]\beta_M =(f\otimes \gamma')\beta_M =\gamma'\gamma, \\ & (f\k_M \otimes f)\phi_M =fm_M (\k_M \otimes \id)\phi_M =\e_M . \end{align*} Finally, the second statement easily follows from the definition of gauge transformations, and from the local expression \eqref{332} for them. \end{pf} A geometrical explanation of the statement ({\it ii\/}) is this. Gauge transformations, being diffeomorphisms of $P$ at the geometrical level, must preserve classical and quantum parts of $P$. On the other hand, because of the intertwining property, gauge transformations $\gamma$ are completely determined by their `restrictions' $\gamma_{cl}$ on $P_{cl}$, which correspond precisely to the standard gauge transformations of $P_{cl}$. The quantum gauge bundle $\ADP $ is also an inherently inhomogeneous geometrical object. This is a consequence of the inhomogeneity of $G$. The classical part of the bundle $\ADP$ (*-characters on $\cal{D}$) is naturally identificable with the ordinary gauge bundle of $P_{cl}$. In other words, $$(\ADP)_{cl}=\AD(P_{cl} ).$$ Let $f\colon \cal{D}\rightarrow S(M) $ be the *-epimorphism corresponding to $\gamma\in \cal{G}$. This map determines a section $f^*$ of the bundle $(\ADP)_{cl}$ as follows \begin{equation}\label{335} \left[f^* (x)\right](\varphi)=\left[f(\varphi)\right](x) \end{equation} where $x\in M$ and $\varphi\in \cal{D}$. In the framework of the correspondence \eqref{327}, the map $ f^*$ becomes the section corresponding to the gauge transformation $\gamma_{cl}$, in the classical manner. We pass to the construction and the study of differential calculus on the bundle $\ADP$. The calculus will be constructed by combining differential forms on the base manifold $M$ with a differential calculus on the quantum group $G$. This calculus will be based on the universal differential envelope $\Gamma^{\wedge}$ of the minimal admissible first-order bicovariant *-calculus $\Gamma$ over $G$. \begin{lem} \label{lem:35} \bla{i} For each $(U,V)\in N^2 (\cal{U})$ there exists the unique homomorphism $\xi^{\wedge}_{UV}\colon\Omega(U\cap V)\grten \Gamma^{\wedge} \rightarrow \Omega(U\cap V)\grten \Gamma^{\wedge}$ of (graded) differential algebras, extending the map $\xi_{UV}$. The map $\xi^{\wedge}_{UV}$ is *-preserving and bijective, and \begin{equation}\label{336} (\xi^{\wedge}_{UV})^{-1}=\xi^{\wedge}_{VU}. \end{equation} \bla{ii} We have \begin{equation}\label{337} \xi^{\wedge}_{UV} \xi^{\wedge}_{VW}(\varphi) =\xi^{\wedge}_{UW}(\varphi) \end{equation} for each $(U,V,W)\in N^3(\cal{U})$ and $\varphi\in\WC(U{\cap}V{\cap} W) \grten\Gamma^\wedge$. \smallskip \bla{iii} The diagrams \begin{gather*} \begin{CD} \Omega(U\cap V)\grten\Gamma^\wedge @>{\mbox{$\id\otimes \widehat{\phi}$}}>> \bigl[\Omega(U\cap V)\otimes\Gamma^\wedge \bigr]\grten_{U\cap V}\bigl[\Omega(U\cap V)\otimes\Gamma^\wedge\bigr]\\ @V{\mbox{$\xi_{UV}^\wedge$}}VV @VV{\mbox{$\xi_{UV}^\wedge \otimes \xi_{UV}^\wedge $}}V\\ \Omega(U\cap V)\grten\Gamma^\wedge @>>{\mbox{$\id\otimes \widehat{\phi}$}}> \bigl[\Omega(U\cap V)\otimes\Gamma^\wedge \bigr]\grten_{U\cap V}\bigl[\Omega(U\cap V)\otimes\Gamma^\wedge\bigr] \end{CD}\\ \begin{CD} \Omega(U\cap V)\grten\Gamma^\wedge @>{\mbox{$\id\otimes \e\Pi$}}>> \Omega(U\cap V)\\ @V{\mbox{$\xi_{UV}^\wedge$}}VV @VV{\mbox{$\id$}}V\\ \Omega(U\cap V)\grten\Gamma^\wedge @>>{\mbox{$\id\otimes \e\Pi$}}> \Omega(U\cap V) \end{CD}\qquad \begin{CD} \Omega(U\cap V)\grten\Gamma^\wedge @>{\mbox{$\id\otimes\widehat{\kappa}$}}>> \Omega(U\cap V)\grten\Gamma^\wedge\\ @V{\mbox{$\xi_{UV}^\wedge$}}VV @VV{\mbox{$\xi_{UV}^\wedge$}}V\\ \Omega(U\cap V)\grten\Gamma^\wedge @>>{\mbox{$\id\otimes\widehat{\kappa}$}}> \Omega(U\cap V)\grten\Gamma^\wedge \end{CD}\\ \begin{CD} \Omega(U\cap V)\grten\Gamma^\wedge @>{\mbox{$\id\otimes \widehat{\phi}$}}>> \bigl[\Omega(U\cap V)\otimes\Gamma^\wedge \bigr]\grten_{U\cap V}\bigl[\Omega(U\cap V)\otimes\Gamma^\wedge\bigr]\\ @V{\mbox{$\psi_{UV}^\wedge$}}VV @VV{\mbox{$\xi_{UV}^\wedge \otimes \psi_{UV}^\wedge $}}V\\ \Omega(U\cap V)\grten\Gamma^\wedge @>>{\mbox{$\id\otimes \widehat{\phi}$}}> \bigl[\Omega(U\cap V)\otimes\Gamma^\wedge \bigr]\grten_{U\cap V}\bigl[\Omega(U\cap V)\otimes\Gamma^\wedge\bigr] \end{CD} \end{gather*} are commutative. \end{lem} \begin{pf} The uniqueness of $\xi^{\wedge}_{UV}$ follows from the fact that $\WC(U\cap V)\grten \Gamma^{\wedge}$ is generated, as a differential algebra, by $\SC (U\cap V)\otimes \cal{A}$. The hermicity of $\xi^{\wedge}_{UV}$ follows from the fact that $*\xi^{\wedge}_{UV} *$ is a differential extension of the same map $*\xi_{UV} *=\xi_{UV}$. In a similar way it follows from Lemma~\ref{lem:31} that above diagrams are commutative, and that \eqref{336}--\eqref{337} hold. We prove the existence of $\xi^{\wedge}_{UV}$. The admissibility of $\Gamma$ and the universality of $\Gamma^{\wedge}$ imply that maps $g_{UV}$ admit the unique graded-differential (*-preserving) extensions $\widehat{g}_{UV} \colon\Gamma^{\wedge} \rightarrow \Omega(U\cap V)$. Now, the maps $f_{UV} \colon \Gamma^{\wedge} \rightarrow \Omega(U\cap V)\grten\Gamma^{\wedge} $ given by $$ f_{UV}(w)=\sum_i \bigl[\widehat{g}_{VU} (w^1_i )\otimes w^2_i\bigr]\bigl[\widehat{g}_{UV}(w_i^3)\otimes 1\bigr] ,$$ where $\Sum_i w^1_i \otimes w^2_i \otimes w^3_i =(\widehat{\phi}\otimes \id)\widehat{\phi}(w) =(\id\otimes \widehat{\phi})\widehat{\phi}(w ),$ are homomorphisms of differential *-algebras. Finally let $\xi^{\wedge}_{UV}$ be defined by $$ \xi^{\wedge}_{UV}(\alpha\otimes w)=\alpha f_{UV} (w).$$ It is evident that such defined maps are differential algebra homomoprhisms extending $\xi_{UV}$.\end{pf} Let $\Omega(\tau,\ADP)$ be the set of all elements $w\in\Sigma^{\wedge} (\cal{U})$ satisfying \begin{equation} \label{342} (_U {\restr}_{U\cap V} \otimes \id)p_U (w)=\xi^{\wedge}_{UV} (_V {\restr}_{U\cap V} \otimes \id)p_V (w), \end{equation} for each $(U,V)\in N^2 (\cal{U})$. It is clear that $\Omega(\tau,\ADP) $ is a graded- differential *-subalgebra of $\Sigma^{\wedge} (\cal{U}),$ and that $\Omega^0 (\tau,\ADP)=\cal{D}$. The elements of the algebra $\Omega(\tau,\ADP)$ play the role of differential forms on the bundle $\ADP$. This algebra is generated by $\cal{D},$ and in fact does not depend of a trivialization system $\tau$. More precisely, if $\eta$ is another trivialization system for $P$ then there exists (the unique) differential $ (*-)$ isomorphism $\Omega(\tau,\ADP)\leftrightarrow \Omega(\eta,\ADP)$ extending the identity map on $\cal{D}$. For this reason we shall simply write $\Omega(\tau,\ADP)=\Omega(\ADP)$. \begin{pro}\label{pro:36} \bla{i} The maps $\bigl\{\e_M, j_M, \beta_M, \phi_M\bigr\}$ admit unique extensions \begin{gather*} \phi^{\wedge}_M \colon\Omega(\ADP)\rightarrow \Omega(\ADP) \grten_M \Omega(\ADP)\\ \e^{\wedge}_M \colon \Omega(\ADP)\rightarrow \Omega(M)\\ j^{\wedge}_M \colon \Omega(M)\rightarrow \Omega(\ADP)\\ \beta^{\wedge}_M \colon \Omega(P)\rightarrow \Omega(\ADP) \grten_M \Omega(P), \end{gather*} which are homomorphisms of graded-differential algebras. \smallskip \bla{ii} The map $\k_M$ admits the unique extension $\k^{\wedge}_M \colon \Omega(\ADP)\rightarrow \Omega(\ADP)$ which is graded-antimultiplicative and satisfies \begin{equation}\label{343} \k^{\wedge}_M d=d\k^{\wedge}_M . \end{equation} \bla{iii} The following identities hold \begin{gather} (\phi^{\wedge}_M \otimes \id)\phi^{\wedge}_M =(\id\otimes \phi^{\wedge}_M )\phi^{\wedge}_M \label{344}\\ (\phi^{\wedge}_M \otimes \id)\beta^{\wedge}_M =(\id\otimes \beta^{\wedge}_M )\beta^{\wedge}_M \label{345}\\ (\id\otimes \widehat{F})\beta^{\wedge}_M =(\beta^{\wedge}_M \otimes \id)\widehat{F} \label{346}\\ (\id\otimes \e^{\wedge}_M )\phi^{\wedge}_M =(\e^{\wedge}_M \otimes \id)\phi^{\wedge}_M =\id \label{347}\\ (\e^{\wedge}_M \otimes \id)\beta_M =\id \label{348}\\ m^{\wedge}_M (\k^{\wedge}_M \otimes \id)\phi^{\wedge}_M =m^{\wedge}_M (\id\otimes \k^{\wedge}_M )\phi^{\wedge}_M =j^{\wedge}_M \e^{\wedge}_M , \label{349}\end{gather} where $m^{\wedge}_M$ is the multiplication map in $\Omega(\ADP). $ \smallskip \bla{iv} We have \begin{equation}\label{350} *\k^{\wedge}_M *=(\k^{\wedge}_M )^{-1}, \end{equation} while $\bigl\{\e^{\wedge}_M,j^{\wedge}_M, \beta^{\wedge}_M,\phi^{\wedge}_M\bigr\}$ are *-preserving maps. \end{pro} \begin{pf} Using the (anti)multiplicativity, the intertwining differentials properties, and the fact that all considered differential algebras are generated by corresponding zero-th order subalgebras, it is easy to see that extensions of all maps in the game are, if exist, unique. The same properties, together with Proposition~\ref{pro:33}, imply that identities \eqref{344}--\eqref{349} hold. The statement ({\it iv\/}) follows from ({\it ii\/}) Proposition~\ref{pro:32} in a similar way. Finally, existence of maps $\e^{\wedge}_M$, $j^{\wedge}_M$, $\beta^{\wedge}_M$, $\phi^{\wedge}_M$ and $\k^{\wedge}_M$ can be established in a similar way as for maps $\e_M$, $j_M$, $\beta_M$, $\phi_M$ and $\k_M$. \end{pf} Every $\gamma\in \cal{G}$ understood as a *-homomorphism $f\colon \cal{D}\rightarrow S(M)$ is uniquely extendible to a $\Omega(M)$-linear *-homomorphism $f^{\wedge} \colon \Omega(\ADP)\rightarrow \Omega(M)$ of graded-differential algebras. The following correspondences hold \begin{align} \gamma^{-1}& \leftrightarrow f^{\wedge} \k^{\wedge}_M \label{351}\\ \gamma\gamma' &\leftrightarrow m^{\wedge}_M (f^{\wedge} \otimes {f}^{\prime\wedge} )\phi^{\wedge}_M. \label{352}\end{align} \section{Gauge Fields} In this section we shall present a generalization of the classical gauge theory, within the geometrical framework of quantum principal bundles. The base manifold $M$ will play the role of space-time. The quantum group $G$ will describe `internal symmetries'. In order to simplify considerations, we shall deal only with a `pure gauge theory'. Let us assume that $\Gamma_{\inv}$ is endowed with an $\Ad$-invariant scalar product $(,)$. It means that $$(\vartheta ,\eta )\otimes 1=\sum_{kl} (\vartheta_k ,\eta_l ) \otimes c^*_k d_l $$ for each $\vartheta ,\eta \in\Gamma_{\inv} ,$ where $\Sum_k \vartheta_k \otimes c_k =\Ad(\vartheta )$ and $\Sum_l\eta_l\otimes d_l =\Ad(\eta )$. Let us assume that M is oriented and endowed with a (pseudo)riemannian structure. Let us denote by ${\star}$ the Hodge operation on $\Omega (M)$. It can be (uniquely) extended to a linear map ${\star}\colon \hor(P)\rightarrow\hor(P)$ such that $${\star}\bigl[i^{\wedge} (\alpha )b\bigr]=i^{\wedge} \bigl[{\star}(\alpha ) \bigr]b, $$ for each $\alpha\in \Omega (M)$ and $b\in \cal{B}$. Following the classical analogy gauge fields will be geometrically represented by connection forms $\omega$ on the bundle $P$. To make possible dynamical considerations it is necessary to fix a lagrangian. Generalizing the classical situation, it is natural to consider lagrangians which are quadratic functions of the curvature $R_{\omega} $. The curvature operator $R_{\omega}$ depends, other then on the connection $\omega ,$ also on a choice of the embedded differential map $\delta \colon\Gamma_{\inv} \rightarrow\Gamma_{\inv} \otimes\Gamma_{\inv}$. As a consequence of this, dynamical properties of the gauge theory will be essentially influenced by $\delta$. In the classical case the curvature is $\delta$-independent. Let us consider a map $L\colon\con(P)\rightarrow\hor(P)$ given by \begin{equation}\label{41} L(\omega )=\sum_iR_{\omega}(e_i){\star}\left[R_{\omega}(\bar{e}_i)\right], \end{equation} where elements $e_i$ form an orthonormal system in $\Gamma_{\inv}$ and the bar denotes the conjugation in $\Gamma_{\inv}$. It is easy to see that $L(\omega )$ is independent of the choice of the mentioned orthonormal system. The map $L$ in fact takes values from the space $\Omega^n (M)$ (where $n$ is the dimension of $M$). Indeed, in terms of local trivializations we have \begin{equation}\label{42} \pi^{\wedge}_U \bigl[L(\omega )\bigr]=\sum_iF^U (e_i ){\star} \bigl[F^U (\bar{e}_i )\bigr]\otimes 1. \end{equation} This easily follows from the fact that $\Sum_i e_i\otimes \bar{e}_i$ is adjointly-invariant. We shall interpret the map $L$ as the lagrangian. In terms of the local representation, principal stationary points \cite{d-qgf} of the corresponding action functional $S(\omega )=\int_M L(\omega )$ are given by the following equations of motion \begin{equation}\label{45} d{\star} F^U (\bar{e}_k )-\frac{1}{2}\sum_{ij}(d^{jk}_i -d^{kj}_i )A^U(e_j) {\star} F^U(\bar{e}_i)=0 \end{equation} where numbers $d^{ij}_k$ are determined by \begin{equation}\label{44} \delta (e_k )=-\frac{1}{2} \sum_{ij}d^{ij}_k e_i \otimes e_j. \end{equation} The above equations correspond to the classical Yang-Mills equations of motion. The numbers $(d^{jk}_i -d^{kj}_i)/2$ play the role of the structure constants of (the Lie algebra of) $G$. If the space $\Gamma_{\inv}$ is infinite-dimensional a technical difficulty arises, related to a question of convergence of the sum in \eqref{41}--\eqref{42}. In such cases, it is necessary to restrict possible values of $\omega$ on some subspace of $\con(P),$ consisting of connections having sufficiently rapidly decreasing components, in an appropriate sense. We pass to the study of symmetry properties of the introduced lagrangian. At first, it is easy to see that $L(\omega )$ is invariant under gauge transformations of the bundle $P. $ The group $\cal{G}$ naturally acts on the left, via compositions, on the space $\psi (P)$ of pseudotensorial forms. The space $\tau (P)$ is invariant with respect to this action, because $\hor(P)$ is $\cal{G}$-invariant. The connection space is gauge invariant, too. In terms of gauge potentials the transformation of connections is \begin{equation}\label{46} A^U (\vartheta )\longrightarrow \sum_kA^U (\vartheta_k )\gamma^U(c_k) +\partial^U(\vartheta ). \end{equation} Here, $\Ad(\vartheta )=\Sum_k\vartheta_k \otimes c_k$, the map $\partial^U\colon\Gamma_{\inv} \rightarrow\Omega^1 (U)$ is given by $$\partial^U\pi (a)=\gamma^U\!\k(a^{(1)})d\gamma^U(a^{(2)}),$$ while $\gamma^U\colon \cal{A}\rightarrow S(U)$ is the map locally representing $\gamma$. Further, the transformation of the curvature is \begin{equation}\label{47} F^U (\vartheta )\longrightarrow \sum_kF^U (\vartheta_k )\gamma^U(c_k). \end{equation} The lagrangian \eqref{41} is invariant under gauge transformations of the bundle $P$. This is a simple consequence of the unitarity of the representation $\Ad$. This invariance is a manifestation of {\it classical symmetry properties} of the lagrangian. These symmetry properties are completely expressible in terms of the classical part $P_{cl}$ of $P$. On the other hand, the lagrangian $L(\omega )$ possesses symmetry properties which are not expressible in classical terms. The appearance of these `quantum symmetries' is a purely quantum phenomena caused by the quantum nature of the space $G$. Formally, they can be described as the invariance of the lagrangian under a natural action of the quantum gauge bundle $\ADP$. Let $\psi (P,\ADP)$ be the space of linear maps $f\colon\Gamma_{\inv} \rightarrow\Omega (\ADP)\grten_M \Omega (P)$ satisfying \begin{equation}\label{48} (f\otimes\id)\varpi=(\id\otimes F^\wedge)f. \end{equation} If $\varphi \in\psi (P)$ then $\beta^{\wedge}_M \varphi \in\psi (P,\ADP)$. Hence it is possible to introduce the map $\beta^{\wedge}_M \colon\psi (P) \rightarrow\psi (P,\ADP)$ (via compositions). Let us compute the element $\beta^{\wedge}_M \omega$ for $\omega \in \con(P)$. Using the definition of $\beta^{\wedge}_M$ and the local expression for $\omega$ we obtain \begin{multline}\label{49} (\pi_U^\sharp\otimes\pi_U)\bigl[\beta^{\wedge}_M (\omega )(\vartheta )\bigr] =1_U \otimes 1\otimes\vartheta+\\ {}+\sum_k\Bigl\{A^U (\vartheta _k)\otimes c^{(1)}_k\otimes c^{(2)}_k +1_U \otimes\vartheta_k \otimes c_k \Bigr\}. \end{multline} Here an identification $$\bigl[\Omega (U)\grten\Gamma^{\wedge} \bigr] \grten_U\bigl[\Omega (U)\grten\Gamma^{\wedge}\bigr] =\Omega (U)\grten\Gamma^{\wedge} \grten\Gamma^{\wedge}$$ is assumed. It is worth noticing that the transformation law \eqref{46} is contained in \eqref{49}. Indeed, understanding gauge transformations as differential algebra homomorphisms $f^\wedge\colon\Omega(\ADP) \rightarrow\Omega (M)$ we obtain \eqref{46} by composing $\beta^{\wedge}_M (\omega )$ and $f^\wedge\otimes \id$. The curvature is transformed as follows \begin{equation}\label{410} (\pi_U^\sharp\otimes\pi_U)\bigl[\beta^{\wedge}_M (R_{\omega} ) (\vartheta )\bigr]=\sum_kF^U (\vartheta_k)\otimes c^{(1)}_k \otimes c^{(2)}_k . \end{equation} That is, in local terms we have \begin{equation}\label{411} F^U \longrightarrow (F^U \otimes \id)\Ad. \end{equation} The curvature operator is gauge-covariant in the sense that \begin{equation}\label{412} \beta^{\wedge}_M (R_{\omega})=d\beta^{\wedge}_M (\omega ) -\langle \beta^{\wedge}_M(\omega),\beta^{\wedge}_M(\omega)\rangle. \end{equation} A possible interpretation of the above equation (which is a trivial consequence of the fact that $\beta^{\wedge}_M \colon\Omega (P) \rightarrow\Omega (\ADP)\grten_M\Omega (M)$ is a differential algebra homomorphism) is this. The relation between the connection $\omega$ and its curvature is, being expressible in intrinsicly geometrical terms, preserved under the action of $\ADP$. Expression \eqref{47} for the curvature of the transformed connection under a gauge transformation also directly follows from \eqref{410}. In order to find the transformation of the local expression for the lagrangian, we should insert in \eqref{42} the local expression for the transformed curvature, under the action $\beta^{\wedge}_M$ of $\ADP$ on $P$. The lagrangian transforms as follows \begin{equation}\label{413} \Bigl\{\sum_kF^U (e_k){\star} F^U(\bar{e}_k)\Bigr\} \longrightarrow \sum_{kln}F^U (e_l){\star} F^U (\bar{e}_n)\otimes c_{lk} c^*_{nk} \end{equation} where $\Ad(e_i)=\Sum_j e_j\otimes c_{ji}$. On the other hand \begin{equation}\label{414} \Bigl[\sum_kF^U(e_k){\star} F^U(\bar{e}_k)\Bigr]\otimes 1= \sum_{kln}F^U (e_l){\star} F^U (\bar{e}_n)\otimes c_{lk} c_{nk}^*, \end{equation} because of the $\ad$-invariance of $\Sum_ke_k\otimes \bar{e}_k$. Hence, the lagrangian is invariant with respect to the action $\beta^{\wedge}_M$ of the gauge bundle $\ADP$ on $P$. It is important to mention that the property of `quantum gauge invariance' of the lagrangian can not be viewed as an inherent property of the local expression \eqref{42}. Because this property essentially depends on {\it the ordering of terms} $F^U$ and ${\star} F^U$. However, in the general case, the ordering of terms $R_{\omega}$ and ${\star} R_{\omega}$ in the global representation of the lagrangian is essential, because $\cal{B}$ is a noncommutative algebra. \section{Example} We shall now illustrate the presented formalism on a concrete example, assuming that $G=SU_{\mu} (2)$ (with $\mu =(-1,1)\setminus\{0\}$). By definition \cite{w1} this compact matrix quantum group is based on the $2\times 2$ matrix \begin{equation}\label{51} u=\begin{pmatrix}\alpha & -\mu\gamma^*\\ \gamma &\phantom{-\mu}\alpha^* \end{pmatrix}\end{equation} where the elements $\alpha$ and $\gamma$ satisfy the following relations \begin{equation}\label{52}\begin{gathered} \alpha \gamma =\mu \gamma \alpha \quad\gamma\alpha^*=\mu\alpha^*\gamma \quad \gamma \gamma ^*=\gamma^*\gamma \\ \alpha^*\alpha +\gamma^*\gamma =1 \qquad\alpha \alpha^*+\mu^2 \gamma^* \gamma =1. \end{gathered}\end{equation} The classical part of $G$ is isomorphic to $U(1)$. An explicit isomorphism is given by $g\leftrightarrow g(\alpha)$. It turns out (\cite{d1}--Section 6) that the right $\cal{A}$-ideal $\widehat{\cal{R}}\subseteq \ker(\e)$ determining the minimal admissible (bicovariant $*-$) first-order calculus $\Gamma$ over $G$ is given by \begin{equation}\label{53} \widehat{\cal{R}}=\bigl(\mu^2\alpha +\alpha^*-(1+\mu^2)1\bigr)\ker(\e). \end{equation} Let $X\colon \cal{A}\rightarrow \Bbb{C}$ be a generator of $\lie(G_{cl} )$ specified by \begin{equation}\label{54}\begin{gathered} X(\alpha )=-X(\alpha^*)=1/2\\ X(\gamma )=X(\gamma^*)=0.\end{gathered}\end{equation} Let $\rho\colon \cal{A}\rightarrow\cal{A}$ be a map given by $\rho =(X\otimes \id)\ad$. Let $\nu \colon\Gamma_{\inv}\rightarrow \Bbb{C}$ and $\widetilde{\rho}\colon \Gamma_{\inv}\rightarrow\cal{A}$ be the maps defined by $\nu \pi =X$ and $\widetilde{\rho} \pi =\rho $. Then $\widetilde{\rho} =(\nu\otimes \id)\Ad,$ and $\widetilde{\rho}$ maps isomorphically the space $\Gamma_{\inv}$ onto the *-subalgebra $\cal{Q}\subseteq\cal{A}$ of left $G_{cl}$-invariant elements of $\cal{A}$. The subalgebra $\cal{Q}$ is interpretable as the algebra of polynomial functions on a quantum $2$-sphere. The adjoint action $\Ad$ is reducible. The space $\Gamma_{\inv}$ is decomposable into the orthogonal sum $$\Gamma_{\inv}=\sideset{}{^\oplus}\sum_{k\geq 0}\Gamma^k_{\inv}$$ of irreducible subspaces. The subspace $\Gamma^k_{\inv}$ is $(2k+1)$-dimensional (that is, all integer-spin irreducible multiplets are in the game). The space $\widetilde{\rho} (\Gamma^k_{\inv})=\cal{Q}_k$ is spanned by quantum spherical harmonics $\zeta_{km}$, where $m\in\{-k,\dots, k\}$. They constitute a standard basis for the action of $G$. Explicitly, these elements are given by \begin{equation}\label{55}\begin{aligned} \zeta_{km}&=(-)^m \mu^{km-m}\bigl[(k-m)_{\mu}!/(k+m)_{\mu}!\bigr]^{1/2} \partial^mp_k (\gamma \gamma^*)\gamma^m\alpha^m \\ \zeta_{k,-m}&= \mu^{km}\alpha^{*m}\gamma^{*m}\bigl[(k-m)_{\mu}!/ (k+m)_{\mu}!\bigr]^{1/2}\partial^mp_k (\gamma \gamma^*).\end{aligned}\end{equation} Here, $m\geq 0$ and $\partial\colon P(x)\rightarrow P(x)$ is a `quantum differential' (acting on the space $P(x)$ of $x$-polynoms) specified by $\partial (x^n)=n_{\mu} x^{n-1} $. Finally, $p_k (x)$ are polynomials given by \begin{equation}\label{56}\begin{aligned} p_k (x)&=(-)^k c_k \partial^k\Bigl[x^k\prod^k_{j=1}(1-\mu^{1-j}x)\Bigr]\\ p_0 (x)&=1, \end{aligned}\end{equation} while $c_k>0$ and $$k_{\mu}!=\prod^k_{j=1}j_{\mu} \quad\quad j_{\mu} =\frac{1-\mu^{2j}}{1-\mu^{2\phantom{j}}}.$$ Let us now describe a construction of the natural embedded differential map $\delta $. We shall first construct a complement $\cal{L}\subseteq \ker(\e)$ of the space $\widehat{\cal{R}}$. The elements $\gamma^k$ ($k\in\Bbb{N} )$ are primitive for the adjoint action of $G$ on $\ker(\e)$. Let $\cal{L}\subseteq \ker(\e)$ be the minimal $\Ad$-invariant subspace containing these elements, and the $\ad$-invariant element $\mu^2 \alpha +\alpha^* -(1+\mu^2)1$. It turns out that the restriction $(\pi{\restr}\cal{L})\colon\cal{L}\rightarrow \Gamma_{\inv}$ is bijective. Evidently, this restriction intertwines the adjoint actions. Let $\delta \colon\Gamma_{\inv} \rightarrow\Gamma_{\inv} \otimes\Gamma_{\inv}$ be defined by \begin{equation}\label{57} \delta (\vartheta )=-(\pi \otimes\pi )\phi\left[(\pi{\restr}\cal{L})^{-1} (\vartheta)\right]. \end{equation} It is clear, by construction, that $\delta$ is an embedded differential map. Moreover, \begin{equation}\label{58} \delta \varkappa =-(\varkappa \otimes\varkappa )\delta \end{equation} where the extension of the antipode $\varkappa \colon\Gamma_{\inv}\rightarrow \Gamma_{\inv}$ is given by \begin{equation}\label{59} \varkappa \pi =-\pi \k^2. \end{equation} Let us compute the values of $\delta$ on the singlet and the triplet subspace of $\Gamma_{\inv}$. The singlet space $\Gamma^0_{\inv}$ is spanned by \begin{equation}\label{510} \varepsilon =\pi (\mu^2 \alpha +\alpha^* ), \end{equation} while the triplet space $\Gamma^1_{\inv}$ is spanned by \begin{equation}\label{511} \eta_+=\pi(\gamma),\quad \eta =\pi (\alpha -\alpha^* ), \quad \eta_-=\pi (\gamma^* ). \end{equation} Applying the definition of $\delta$ we obtain \begin{gather*} -\delta (\varepsilon )=\bigl(\varepsilon \otimes\varepsilon +\mu^2 \eta \otimes\eta \bigr)/(1+\mu^2 )- \mu \eta_+\otimes \eta_- -\mu^3 \eta_- \otimes\eta_+ \\ -\delta (\eta_+ )=\bigl((\varepsilon -\mu^2 \eta )\otimes\eta_+ +\eta_+ \otimes(\varepsilon +\eta ) \bigr)/(1+\mu^2 )\\ -\delta (\eta_-)=\bigl(\eta_-\otimes (\varepsilon -\mu^2 \eta ) +(\varepsilon +\eta )\otimes\eta_- \bigr)/(1+\mu^2 )\\ -\delta (\eta )=\bigl(\varepsilon \otimes\eta +\eta \otimes\varepsilon +(1-\mu^2 ) \eta \otimes\eta \bigr)/(1+\mu^2 )+\mu \bigl(\eta_+\otimes \eta_- -\eta_- \otimes\eta_+ \bigr) \end{gather*} The corresponding gauge theory based on the bundle $P$, calculus $\Gamma$ group $G$ and the lagrangian $L(\omega )$ is essentially different from the classical gauge theory with $G=\SU(2). $ At first, gauge fields possess infinitely many internal degrees of freedom. In the classical limit $\mu\rightarrow 1$ the restriction $A^U {\restr}\Gamma^1_{\inv}$ on the triplet subspace can be interpreted as a classical $\SU(2)$ gauge field. Restrictions on other irreducible subspaces are classically interpretable as additional vector fields. According to the general theory, the connection $A^U$ can be decomposed into `classical' and `purely quantum' parts $$A^U=A_{cl}^{U} +A_\perp^U,$$ where $A_{cl}^U{\restr} \ker(\nu)=0$ and $A_{\perp}^U (\varepsilon )=0$. The map $A_{cl}^U$ can be interpreted as a connection on the classical $U(1)$-bundle $P_{cl}$. It is important to point out that the decomposition $\Gamma_{\inv}=\ker(\nu)+ \Bbb{C}\varepsilon$ is incompatible with the decomposition of $\Gamma_{\inv}$ into irreducible multiplets. Let us compute the singlet and the triplet components of the curvature. Applying the definition of $\delta$ and using the local expression of the curvature we find \begin{gather*} F^U(\varepsilon )=dA^U(\varepsilon )+\mu (1-\mu^2)A^U(\eta_-)A^U(\eta_+)\\ F^U (\eta_+)=dA^U (\eta_+)+A^U (\eta_+ )A^U (\eta ) \\ F^U (\eta_-)=dA^U (\eta_-)+A^U (\eta )A^U (\eta_-)\\ F^U (\eta )=dA^U (\eta )+2\mu A^U (\eta_+ )A^U (\eta_-). \end{gather*} In general, components of the restriction $F^U {\restr} \Gamma^k_{\inv}$ will be expressible through fields $A^U (\vartheta ),$ where $\vartheta \in\Gamma^l_{\inv}$ and $1\leq l\leq k$. Equations of motion are mutually {\it essentially correlated}. Indeed, the equation describing the propagation of fields $A^U{\restr} \Gamma^k_{\inv}$ will generally contain terms of the form $A^U (\vartheta ){\star} F^U (\eta)$, where $\vartheta,\eta \in\Gamma^{i,j}_{\inv}$ and $\vert i-k\vert \leq j \leq i+k$. This easily follows from the definition of $\delta$. It is interesting to observe that nonsinglet components are not explicitly influenced by the singlet component $A^U(\varepsilon )$. On the other hand, the singlet propagation is intertwined only with $A^U (\eta_\pm)$. Explicitly, $$d{\star} F^U(\varepsilon )=0.$$ \section{Concluding Remarks} In this study we have presented a gauge theory over classical space-time, in which internal symmetry groups are quantum. The basic elements of the formalism can be further naturally incorporated into the fully quantum context \cite{d-qgf}, where the space-time is quantum. However, the admissibility of the calculus $\Gamma$ is replaced by another global condition, since in the general quantum context it is not possible to speak in terms of local trivializations. We have only considered `pure' gauge theory in this study. The matter fields can be introduced via the analogs of the associated vector bundles. In the general quantum context, $M$ is represented by a noncommutative *-algebra $\cal{V}\leftrightarrow S(M)$, and it is natural to identify associated vector bundles with the intertwiner $\cal{V}$-bimodules $\cal{F}_u=\Mor(u,F)$ consisting of intertwining operators between finite-dimensional representations $u$ of $G$ and the action map $F$. It turns out that, quite generally \cite{d-tann}, the system of all these bimodules completely determines the internal structure of the bundle. In this study we have also assumed that the higher-order differential calculus on the structure group is based on the corresponding universal envelope. All constructions can be performed also in the case when the higher-order calculus is described by the corresponding bicovariant (braided) exterior algebra \cite{w3}. The admissibility assumption for $\Gamma$ ensures full local trivializability of differential structures on $P$ and $\ADP$. However, from the `local' point of view, the whole formalism works for an arbitrary bicovariant *-calculus $\Gamma$. Physical properties of the presented gauge theory are essentially influenced by two additional structural elements. At first, it is necessary to fix a bicovariant *-calculus $\Gamma$ over $G$. This determines kinematical degrees of freedom. Secondly, the curvature is determined only after fixing an embedded differential map $\delta$. In such a way the dynamics becomes $\delta$-dependent. As an example of how strongly the map $\delta$ can influence the dynamics, let us mention gauge fields based on a $4D$-calculus \cite{w3} over quantum $\SU(2)$. This (non-admissible) calculus is spanned by the triplet $\{\eta_+,\eta,\eta_-\}$ and the singlet $\{\tau\}$. As explained in \cite{d1}, changing appropriately $\delta$ we can pass from the model of noninteracting fields, to a model where the triplet fields interact similarly as the components of the classical $\SU(2)$ gauge field, modulo the presence of the singlet $\tau$. A different quantum bundle formalism \cite{bm2} was used in \cite{bm1} and \cite{haj} to construct a quantum analogue of standard gauge theory. From our point of view, geometrical formulation proposed in \cite{bm2} lacks the flexibility, because the basic entities of the formalism (covariant derivative, horizontal projection, curvature) can be constructed only in very special situations--for bundles possessing some additional properties, or using the universal calculus (where we have only trivial relations at the level of the calculus). One possibility to overcome this difficulty is to consider only universal calculi, and to restrict the values of the curvature tensor to matrix elements of a given representation of the structure group. The embedded differential is then simulated by the action of the coproduct map on these matrix elements. However, this a priori excludes all non-trivial quantum phenomenas we have considered. Another interesting possibility for developing a gauge theory in the framework of \cite{bm2} is given by quantum principal bundles possessing `strong' connections \cite{haj}, which gives an effective regularity condition. This works also for certain non-universal calculi on the bundle. Such connections are associated to the base, in the appropriate sense, and they can be taken as proper analogs of gauge fields. The construction of quantum gauge bundles proposed in this paper depends on the classicality of the base space (and possibility to locally trivialize the bundle). For general quantum principal bundles, it is necessary to use essentially different methods \cite{d-qgf}. General quantum gauge bundles can be constructed by combining the intrinsic braided structure on quantum principal bundles \cite{d-bqgb} with the natural structuralization \cite{d-tann} in terms of the intertwiner bimodules (this replaces in a certain sense local trivializations). It is important to mention that, for general quantum principal bundles, it has been proven in \cite{b} that gauge transformations (understood as vertical automorphisms of the bundle) are in a natural correspondence with the appropriate maps from the structure group algebra to the bundle algebra intertwining the adjoint action and the right action $F$. In particular, this property implies the first statement of Proposition~\ref{pro:34}. A conceptually different approach in constructing a quantum group gauge theory was proposed in \cite{sud1}, using the dual picture of quantized universal enveloping algebras. The gauge fields are based on the concept of the associated quantum Lie algebra \cite{sud2}. In particular, the formulation proposed in \cite{sud1} has a `good' classical limit, because the dimension of the quantum Lie algebra is the same as the dimension of its classical counterpart. It would be very interesting to find an invariant geometrical formulation for such a structure. From our point of view however, it is unnatural to expect the existence of such a classical limit, because of the explained inherent geometrical inhomogeneity of quantum groups. Furthermore, it is plausible to addopt the following interpretation: In a gauge theory with a quantum group $G$, the `true' local symmetries are described by the classical part $G_{cl}$. The `complement' of $G_{cl}$ in $G$ is a `purely quantum' space, describing `deformed' symmetry-like properties. This residual symmetry should be able, in principle, to unify the particle multiplets associated to $G_{cl}$. Such interpretation is very close to the supersymmetry philosophy. In accordance with this line of thinking, it is conceptually incorect to try to {\it deform} classical gauge theory. Insead of this, classical gauge theory should be {\it refined}, by considering an appropriate quantum extension of the classical internal symmetry group. The concept of symmetry is logically independent of the concept of a quantum space. For example \cite{d-spin}, it is possible to define consistenly classical differential-geometric structures on a quantum space. A natural framework for such structures is given by quantum bundles possessing classical structure groups. If the structure group is classical, then the construction of the quantum gauge bundle is simplified, and can be completed \cite{d-bqgb} using the intrinsic (in this case involutive) braiding for quantum principal bundles. \appendix \section{The Minimal Admissible Calculus} In this Appendix some properties of entities associated with the minimal admissible calculus $\Gamma$ over the quantum $\SU(2)$ group are collected. In particular, we shall analyze in more details the structure of the space $\cal{L}$ which determines the embedded differential map. For each integer $n\geq 1$ let $u_n$ be the $n\times n$ matrix over $\cal{A},$ corresponding to the irreducible representation \cite{w1,w2} of $G$, having the spin $(n-1)/2$ and acting in $\Bbb{C}^n$. Let $\cal{A}_n$ be the lineal spanned by matrix elements of $u_n$. We have $$ \cal{A}=\sideset{}{^\oplus}\sum_{n\geq 1}\cal{A}_n ,$$ according to the representation theory of $G$. The spaces $\cal{A}_n$ are invariant under the adjoint action of $G$. They are mutually orthogonal, relative to the scalar product induced by the Haar measure $h\colon\cal{A} \rightarrow\Bbb{C}$. In subspaces $\cal{A}_n$ the adjoint action decomposes (without degeneracy) into irreducible multiplets with spins from the set $\{0,1,\dots,n-1\}. $ \begin{lem}\label{lem:A1} Let $\xi \in \cal{A}_n$ be a primitive element for the $k$-spin subrepresentation of $\ad{\restr} \cal{A}_n$. Then, \begin{equation}\label{A1} \xi =p_{kn} (\lambda )\gamma^k \end{equation} where $\lambda =\mu \alpha +\mu^{-1} \alpha^*$ and $p_{kn}$ is a polynom of degree $n-k-1$ with real coefficients. \end{lem} \begin{pf} From the representation theory of $G$ it follows that $$ \cal{A}_2\cal{A}_n=\cal{A}_n\cal{A}_2=\cal{A}_{n-1}\oplus \cal{A}_{n+1}$$ for each $n\geq 2$. This implies that $\cal{A}_n\setminus\{0\}$ is consisting of certain polynoms of degree $n-1$ (over generators). Further, polynoms of degree $k\leq n-1$ form the space $\Sum_i^{*\oplus}\cal{A}_i$, where $i\leq n$. Also, from the reality of commutation relations \eqref{52} and the orthogonality of spaces $\cal{A}_n$ it follows that we can write $$ \cal{A}_n=\cal{A}_n^\Re\oplus i\cal{A}_n^\Re$$ where $\cal{A}_n^\Re$ is consisting of polynoms with real coefficients. On the other hand, every non-zero element of the form \eqref{A1} is primitive, and generates an irreducible $k$-spin multiplet relative to the adjoint representation. Having in mind the form of the decomposition of $\Ad{\restr} \cal{A}_n$ into irreducible multiplets we conclude that \eqref{A1} covers all primitive elements of the restriction $\Ad{\restr}\cal{A}_n$. \end{pf} Let us assume that polynoms $p_{kn}$ are fixed. For fixed $k$, polynoms $p_{kn}$ are orthogonal, with respect to the scalar product given by \begin{equation}\label{A2} (p,q)=h\bigl(q(\lambda )\gamma^k \gamma^{*k} p(\lambda )^* \bigr), \end{equation} Let $j\colon \cal{A}\rightarrow\cal{A}$ be the modular automorphism \cite{w2} corresponding to the Haar measure. This map is characterized by the identity \begin{equation}\label{A3} h(ba)=h\bigl(j(a)b\bigr).\end{equation} In the case of the quantum $\SU(2)$ group we have \begin{equation}\label{A4}\begin{gathered} j(\gamma )=\gamma \qquad j(\alpha )=\mu^2 \alpha \\ j(\alpha^* )=\mu^{-2}\alpha^*\qquad j(\gamma^* )=\gamma^* .\end{gathered}\end{equation} Applying \eqref{A3}--\eqref{A4} we see that the scalar product defined in \eqref{A2} can be rewritten in the form \begin{equation}\label{A5} (p,q)=h\bigl[p^* (\lambda )q(\lambda )(\gamma \gamma^* )^k \bigr]. \end{equation} Now, starting from \eqref{A5}, observing that the above scalar product is invariant under the replacement $\lambda\rightarrow \alpha +\alpha^* ,$ and using elementary properties of polynoms it can be shown that all zeroes of $p_{kn}$ are contained in the interval $[-2,2]$. We have \begin{equation}\label{A6} \cal{L}=\Bbb{C}\bigl(\mu \lambda -(1+\mu^2 )1\bigr)\oplus \Bigl\{\sideset{}{^\oplus}\sum_{k\geq 1}\cal{L}_k \Bigr\}, \end{equation} where $\cal{L}_k\subseteq \cal{A}_{k+1}$ is the $k$-spin irreducible subspace (for the adjoint action). Let $\cal{L}_*\subseteq\cal{A}$ be the lineal given by \begin{equation}\label{A7} \cal{L}_*=\Bbb{C}1\oplus\Bigl\{\sideset{}{^\oplus}\sum_{k\geq 1} \cal{L}_k\Bigr\}.\end{equation} Let $P(\lambda )\subseteq\cal{A}$ be the subalgebra generated by $\lambda$. \begin{lem}\label{lem:A2} \bla{i} We have \begin{equation}\label{A8} \pi (\cal{A}_n)=\sideset{}{^\oplus}\sum_{k\leq n-1} \Gamma^k_{\inv} \end{equation} for each $n\geq 2$. \smallskip \bla{ii} The map $\mho\colon P(\lambda )\otimes \cal{L}_* \rightarrow\cal{A}$ given by \begin{equation}\label{A9} \mho\bigl(p(\lambda )\otimes a\bigr)=p(\lambda )a \end{equation} is bijective. \end{lem} \begin{pf} Let us prove that $\mho$ is bijective. At first, let us observe that the elements from $P(\lambda )$ are $\ad$-invariant. In particular, \begin{equation}\label{A10} \ad\bigl(p(\lambda )a\bigr)=p(\lambda )\ad(a), \end{equation} for each $a\in\cal{A}$. According to Lemma~\ref{lem:A1} all primitive elements for $\ad\colon\cal{A}\rightarrow\cal{A}\otimes\cal{A}$ are contained in the image of $\mho$. Now \eqref{A10} implies that $\mho$ is surjective. We prove that $\mho$ is injective. It is sufficient to check that $\mho {\restr}\bigl( P(\lambda )\otimes \cal{L}_k\bigr)$ is injective, for each $k\in\Bbb{N}$. However, it follows again from Lemma~\ref{lem:A1} and \eqref{A10}, because $\mho\bigl(p_{kn} (\lambda )\otimes\cal{L}_k\bigr)\subseteq\cal{A}_n$ is exactly the $k$-spin irreducible subspace. The following identity holds on nontrivial $\ad$-multiplets \begin{equation}\label{A11} \rho \mho =\bigl(\e\otimes (\rho{\restr}\cal{L}_*)\bigr). \end{equation} The statement ({\it i\/}) now follows from the definition of $\Gamma_{\inv}$ and from the facts that $\e(\lambda )=\mu +\mu^{-1}$ and $p_{kn} (\mu +\mu^{-1} )\neq 0$. \end{pf} Using \eqref{A6} and the definition of $\delta$, it can be shown that \begin{equation}\label{A12} \delta (\Gamma^n_{\inv} )\subseteq \sideset{}{^\oplus_*}\sum_{ij} (\Gamma^i_{\inv}\otimes \Gamma^j_{\inv} ), \end{equation} for each $n\in\Bbb{N}$, where the sum is taken over pairs $(i,j)$ satisfying $\vert i-j\vert \leq n\leq i+j$. In particular, \begin{gather*} \delta (\vartheta )^{0,n} =d_n \varepsilon \otimes \vartheta \\ \delta (\vartheta )^{n,0} =d_n \vartheta \otimes \varepsilon , \end{gather*} for each $\vartheta \in\Gamma^n_{\inv} ,$ with $d_n \in\Re\setminus\{0\}$. 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