Quantum Principal Bundles on Projective Bases

Abstract

The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study noncommutative principal bundles corresponding to $$G \rightarrow G/P$$ G → G / P , where G is a semisimple group and P a parabolic subgroup.

Gauge theory models on $κ$-Minkowski space: Results and prospects

“…The idea lies in a new formalism of classical Rieamannian geometry based on frame resolution, generalising the frame bundle. This appraoch was found compatible with Drinfeld twists [85,84] and was enlarged to non-affine bases with a sheaf theoretic setting [86,87].…”

$N=2$ quantum chiral superfields and quantum super bundles

“…In particular if H is a Hopf algebra with bijective antipode over a field, the condition of equivariant projectivity of A is equivalent to that of faithful flatness of A (see [31], [32]). We now follow [22] Sec. 2, in giving the definition of quantum principal bundle, though it differs slightly from the one given in the literature, which also requires the existence of a differential structure (see e.g.…”

“…In this section we want to reinterpret our construction in the framework of quantum principal bundles, as in [22] and references therein. We shall concentrate our attention on the local picture, that is, we want to look at the quantization of a super bundle S −→ C 4|4 , with base space the chiral Minkowski superspace C 4|4 , which we interpret as the big cell into the Grassmannian supermanifold Gr (see also Sec.…”

“…This program could, in principle be proposed for general homogeneous superspaces, not necessarily superprojective. But the projectivity gives us an advantage: the algebra associated to the projective embedding (super Plücker embedding) can be seen both, as a quotient algebra of the projective superspace P 8|8 modulo some homogeneous polynomial relations (super Plücker relations) as well as a graded subring of the superring C[SL(4|2)], encoding its projective embedding (see also [20,21,22]). We will see this in detail in Section 3.…”

Noncommutative numerable principal bundles from group actions on C*-algebras

“…[55]) of the bundle is not assumed. The work of Aschieri, Fioresi, and Latini [4] introduces a notion of local triviality which is suitable for noncommutative algebraic geomtery, but it cannot be used in the C*-algebraic context. Furthermore, the setting of Hopf-Galois extensions cannot be used to obtain a generalization of principal bundles beyond the compact case.…”