On synthetic interpretation of quantum principal bundles

Quantum principal bundles or principal comodule algebras are re-interpreted as principal bundles within a framework of Synthetic Noncommutative Differential Geometry. More specifically, the notion of a noncommutative principal bundle within a braided monoidal category is introduced and it is shown that a noncommutative principal bundle in the category opposite to the category of vector spaces is the same as a faithfully flat Hopf-Galois extension.Comment: 18 page

Hopf modules and the fundamental theorem for Hopf (co)quasigroups

The notion of a Hopf module over a Hopf (co)quasigroup is introduced and a version of the fundamental theorem for Hopf (co)quasigroups is proven.Comment: 11 pages; missing (co)associativity in Definition 2.1 adde

Curved differential graded algebras and corings

A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with surjective counits. Under this equivalence, comodules over a coring correspond to integrable connections or quasi-cohesive curved modules, while contramodules over a coring correspond to a specific class of curved modules introduced and termed Z-divergences in here.Comment: 25 page

Circle actions on a quantum Seifert manifold

The quotients of a (non-orientable) quantum Seifert manifold by circle actions are described. In this way quantum weighted real projective spaces that include the quantum disc and the quantum real projective space as special cases are obtained. Bounded irreducible representations of the coordinate algebras and the K-groups of the algebras of continuous functions on quantum weighted real projective spaces are presented.Comment: 9 page

Special normalised affine matrices. An example of a Lie affgebra

The affine space of traceless complex matrices in which the sum of all elements in every row and every column is equal to one is presented as an example of an affine space with a Lie bracket or a Lie affgebra.Comment: 8 pages; XL Workshop on Geometric Methods in Physics, Bia{\l}owie\.za 202