Principal Pairs of Quantum Homogeneous Spaces

Abstract

We propose a simple but effective framework for producing examples of covariant faithfully flat (generalised) Hopf-Galois extensions from a nested pair of quantum homogeneous spaces. Our construction is modelled on the classical situation of a homogeneous fibration G/N → G/M , for G a group, and N ⊆ M ⊆ G subgroups. Variations on Takeuchi's equivalence and Schneider's descent theorem are presented in this context. Quantum flag manifolds and their associated quantum Poisson homogeneous spaces are taken as motivating examples. Moreover, a large collection of noncommutative fibrations (in the spirit of Brzezinski and Szymanski) are constructed.

Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

“…This grading is strong, as explained in [9, Section 4], meaning that each E k is a line module in the sense of [3,Section 3.5]. Moreover, we see that each E k admits a natural presentation as an associated module to our O(U 1 )-comodule algebra (see [9,Section 5] for a detailed discussion). By construction each E k is a left O q (G)-subcomodule of O q (G), and so, it is a relative Hopf module in the sense of Takeuchi (see Section 4.3).…”

“…the quantum flag manifold associated to S. We note that O q (G/L S ) is a left O q (G)-comodule algebra by construction. Moreover, O q (G) is faithfully flat as a right O q (G/L S )-module (see for example [9,Section 5.4]). It now follows from [32, Theorem 1] that O q (G/L S ) coincides with the space of right coinvariants of the coaction ∆ R := (id ⊗ π S ) • ∆, where…”

“…Oq(G/L S ) Mod 0 , whose objects F are those satisfying FO q (G/L S ) + = O q (G/L S ) + F. This is a monoidal category with respect to the bimodule tensor product ⊗ B . Moreover, the calculi Ω (0,1) q (G/L S ) and Ω (1,0) q (G/L S ) live in this subcategory [18,Proposition 3.3], as do the relative line modules E k , as explained for example in [9,Appendix A.3].…”

“…The relative line modules E k over each O q (G/L S ), which are indexed by integers k ∈ Z, were shown in [11] to possesses a unique left O q (G)-covariant connection ∇ with respect to Ω • q (G/L S ). Now as established in the series of papers [7,8,9], the Heckenberger-Kolb calculi admit quantum principal bundle descriptions. In this presentation, the total algebra of the quantum bundle is the quantum Poisson homogeneous space…”

Quantum Principal bundle and Non Commutative differential calculus

“…therein for a more general discussion on differential calculi on comodule algebras) there was a recent revived interest on quantum flag varieties (see e.g. [28,8]). It is important to observe that the classical framework detailed above, namely the one inspired by parabolic geometries, cannot be fully understood and treated by looking at affine objects only.…”