Covering and gluing of algebras, and quantum principal bundles (Q2711089)

Adapting scheme theoretic methods, the authors developed a theory of covering and gluing of algebras [\textit{D. Calow} and \textit{R. Matthes}, J. Geom. Phys. 32, No. 4, 364-396 (2000; Zbl 0969.46051)] and applied them to the study of quantum principal bundles and connections. In this paper, these are reviewed. As an example, it is shown that gluing of two quantum discs becomes a quantum sphere, and as \(q\)-deformings of the Hopf bundle and the Dirac monopole, a \(U(1)\) bundle and a connection are constructed on this sphere.NEWLINENEWLINENEWLINEThe outline of the paper is as follows: In section 1, covering and gluing of a unital algebra \(A\) over \(\mathbb{C}\) are defined. A covering of \(A\) is a finite family \((J_i)_{i= 1,\dots,n}\) of ideals of \(A\) such that \(\bigcap_i J_i= 0\). Let \(\pi_i: A\to A/J_i\) and \(\pi^i_j: A_i\to A_{ij}= A/(J_i+ J_j)\). Then the gluing of the \(A_i\) along the \(A_{ij}\) via the \(\pi^i_j\) is \(\bigoplus_{\pi^i_j} A_i= \{(a_i)_i\in \bigoplus_i A_i\mid\pi^i_j(a_i)= \pi^j_i(a_j)\}\). Applying these notions to differential algebras, an algebraic explanation of pullbacks of embeddings is given (Proposition 1). In section 2, these are applied to locally trivial quantum principal bundles and connections. Gluing of quantum discs and \(q\)-deformation of the Hopf bundle are explained in the final section 3.NEWLINENEWLINEFor the entire collection see [Zbl 0944.00084].