Bartosz Zieliński

Piecewise principal comodule algebras

A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P->Pi whose kernels intersect to zero, P is principal if and only if all Pis are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P->Pi and such that the comodule algebra of global sections is P.

Quantum principal bundles over quantum real projective spaces

Abstract Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U ( 1 ) as a structure group, the other has the quantum group S U q ( 2 ) as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf–Galois extension.

The K -theory of Heegaard quantum lens spaces

Representing Z/NZ as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/NZ, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/NZ to construct an associated complex loine bundle. This paper proves the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer-Vietoris sequence, and an explicit form of the Bass connecting homomorphism to prove the stable non-triviality of the bundles. On the algebraic side we prove the universality of the coordinate algebra of such a lens space for aparticular set of generators and relations. We also prove the non-existence of non-trivial invertibles in the coordinate algebra of a lens space. Finally, we prolongate the Z/NZ-fibres of the Heegaard quantum sphere to U (1), and determine the algebraic structure of such U (1) prolongation.

Hopf fibration and monopole connection over the contact quantum spheres

Abstract Non-commutative geometry of quantised contact spheres introduced by Omori et al. in [J. Math. Soc. Jpn. 50 (1998) 915; Noncommutative 3-sphere as an Example of Noncommutative Contact Algebras, Banach Center Publications, vol. 40, 1997, pp. 329–334] is studied. In particular it is proven that these spheres form a non-commutative Hopf fibration in the sense of Hopf–Galois extensions. The monopole (strong) connection is constructed, and projectors describing projective modules of all monopole charges are computed.

Modelling role hierarchy structure using the Formal Concept Analysis

We demonstrate how one can use the formal concept analysis (FCA) to obtain the role hierarchy for the role based access control from the existing access control matrix. We also discuss assesed by means of FCA the quality of security system and finding users with excess permissions.

Quantum complex projective spaces from Toeplitz cubes

From N -tensor powers of the Toeplitz algebra, we construct a multi-pullback C*-algebra that is a noncommutative deformation of the complex projective space P.C/. Using Birkhoff’s Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multi-pullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space. Mathematics Subject Classification (2010). 06B10, 06B25.

Allegories for Database Modeling

Allegories are categories modeled upon the category of sets and binary relations where sets are objects and binary relations are morphisms composed using joins. In this paper we present a new conceptual data modeling formalism based on the language of allegories. We show that allegories provide more convenient framework for modeling data than more traditional categorical approaches in which arrows are interpreted as functional dependencies and in which many to many or partial relationships have to be represented as spans. Finally, we demonstrate that by using allegories different than the allegory of sets and binary relations, for example the allegory of sets and lattice valued relations, one can model replicated data or data stored in a valid time temporal database.

Using maude rewriting system to modularize and extend SQL

One of the deficiencies of SQL is that it allows little practical support for factoring, parametrization and modularization of complex SQL statements. Using dynamic SQL or macroprocessors for this purpose is error prone and inconvenient. This paper shows how the rewriting systems such as Maude can be used to build better SQL preprocessors which understand the SQL syntax and thus reduce the possibility of error. Rewriting systems also allow creating of language extensions supporting modularity and data model abstractions. We use as an example the Maude based system currently developed at our department.

Weak n-Ary Relational Products in Allegories

Allegories are enriched categories generalizing a category of sets and binary relations. Accordingly, relational products in an allegory can be viewed as a generalization of Cartesian products. There are several definitions of relational products currently in the literature. Interestingly, definitions for binary products do not generalize easily to n-ary ones. In this paper, we provide a new definition of an n-ary relational product, and we examine its properties.

Modular Term-Rewriting Framework for Artifact-Centric Business Process Modelling.

A plethora of formalisms is used to express various dimensions of business processes such as data and organization models, orchestration of tasks, and so on. Traditionally, those dimensions are treated as largely independent. Artifact centric models, however, require tighter integration between components. Thus, it is desirable to base a formal semantics of business process constructs on a mathematical formalism which is capable of expressing all or most of the dimensions of interest and their mutual dependence for the purpose of effective simulation and verification. We propose such a specification framework based on term rewriting and implemented in Maude system. Its important feature is the ability to specify the semantics in a piecewise, extensible manner reminiscent of aspect oriented approaches to business process modelling.