arXiv subjects

Bartosz Zielinski

Publications and source records attributed to Bartosz Zielinski.

16 recordsLinked to original sources

arXiv2026

Quantum CW-complexes in a Waldhausen category for unital C*-algebras

Using the ring structure of the K-groups of finite CW-complexes, Atiyah and Todd unravelled the K-theory of complex projective spaces $CP^n$. Herein, in the realm of noncommutative topology, we develop a new framework of finite quantum CW-complexes using the language of Waldhausen categories, which allows us to enrich the class of standard morphisms between unital C*-algebras by adding inverses of *-homomorphisms that are isomorphisms in K-theory. Our concept of quantum CW-complexes subsumes earlier constructions and enjoys a plethora of examples. Moreover, the framework allows us to reduce problems concerning the multipushout quantum complex projective space $CP^n_H$ to the much more approachable setting of the Vaksman-Soibelman quantum complex projective space $CP^n_q$ enjoying the availability of graph-algebraic methods. In particular, these methods permit us to transport the ring structure from $K^0(CP^n)$ to $K^0(CP_q^n)$. Finally, we adapt the formalism of Waldhausen categories to determine a natural set of free generators of $K^0(CP_H^n)$ from a natural set of free generators of $K^0(CP_q^n)$, and to transport the ring structure from $K^0(CP_q^n)$ to $K^0(CP_H^n)$.

math.KT↗

arXiv2026

Rank-two Milnor idempotents for the multipullback quantum complex projective plane

The $K_0$-group of the C*-algebra of multipullback quantum complex projective plane is known to be $\mathbb{Z}^3$, with one generator given by the C*-algebra itself, one given by the section module of the noncommutative (dual) tautological line bundle, and one given by the Milnor module associated to a generator of the $K_1$-group of the C*-algebra of Calow-Matthes quantum 3-sphere. Herein we prove that these Milnor modules are isomorphic either to the section module of a noncommutative vector bundle associated to the $SU_q(2)$-prolongation of the Heegaard quantum 5-sphere $S^5_H$ viewed as a $U(1)$-quantum principal bundle, or to a complement of this module in the rank-four free module. Finally, we demonstrate that one of the above Milnor modules always splits into the direct sum of the rank-one free module and a rank-one non-free projective module that is \emph{not} associated with $S^5_H$.

math.KT↗

arXiv2022

A Non-Deterministic Multiset Query Language

We develop a multiset query and update language executable in a term rewriting system. Its most remarkable feature, besides non-standard approach to quantification and introduction of fresh values, is non-determinism - a query result is not uniquely determined by the database. We argue that this feature is very useful, e.g., in modelling user choices during simulation or reachability analysis of a data-centric business process - the intended application of our work. Query evaluation is implemented by converting the query into a terminating term rewriting system and normalizing the initial term which encapsulates the current database. A normal form encapsulates a query result. We prove that our language can express any relational algebra query. Finally, we present a simple business process specification framework (and an example specification). Both syntax and semantics of our query language is implemented in Maude.

cs.LO↗

arXiv2021

Reductions of piecewise-trivial principal comodule algebras

Let $G'$ be a closed subgroup of a topological group $G$. A principal $G$-bundle $X$ is reducible to a locally trivial principal $G'$-bundle $X'$ if and only if there exists a local trivialisation of $X$ such that all transition functions take values in $G'$. We prove a noncommutative-geometric counterpart of this theorem. To this end, we employ the concept of a piecewise-trivial principal comodule algebra as a replacement of a locally trivial compact principal bundle. To illustrate our theorem, first we define a new noncommutative deformation of the $\mathbb{Z}/2\mathbb{Z}$-principal bundle $S^2\rightarrow \mathbb{R}P^2$ that yields a piecewise-trivial principal comodule algebra. It is the C*-algebra of a quantum cube whose each face is given by the Toeplitz algebra. The $\mathbb{Z}/2\mathbb{Z}$-invariant subalgebra defines the C*-algebra of a quantum $\mathbb{R}P^2$. It is given as a triple-pullback of Toeplitz algebras. Next, we prolongate this noncommutative $\mathbb{Z}/2\mathbb{Z}$-principal bundle to a noncommutative $U(1)$-principal bundle, so that the former becomes a reduction of the latter thus instantiating our theorem. Moreover, using K-theory results, we prove that the prolongated noncommutative bundle is not trivial.

math.QA↗

arXiv2019

Persistence Codebooks for Topological Data Analysis

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper we introduce persistence codebooks, a novel expressive and discriminative fixed-size vectorized representation of PDs. To this end, we adapt bag-of-words (BoW), vectors of locally aggregated descriptors (VLAD) and Fischer vectors (FV) for the quantization of PDs. Persistence codebooks represent PDs in a convenient way for machine learning and statistical analysis and have a number of favorable practical and theoretical properties including 1-Wasserstein stability. We evaluate the presented representations on several heterogeneous datasets and show their (high) discriminative power. Our approach achieves state-of-the-art performance and beyond in much less time than alternative approaches.

stat.ML↗

arXiv2017

A Study on Topological Descriptors for the Analysis of 3D Surface Texture

Methods from computational topology are becoming more and more popular in computer vision and have shown to improve the state-of-the-art in several tasks. In this paper, we investigate the applicability of topological descriptors in the context of 3D surface analysis for the classification of different surface textures. We present a comprehensive study on topological descriptors, investigate their robustness and expressiveness and compare them with state-of-the-art methods including Convolutional Neural Networks (CNNs). Results show that class-specific information is reflected well in topological descriptors. The investigated descriptors can directly compete with non-topological descriptors and capture complementary information. As a consequence they improve the state-of-the-art when combined with non-topological descriptors.

cs.CV↗

arXiv2013

Semi-entwining structures and their applications

Semi-entwining structures are proposed as concepts simpler than entwining structures, yet they are shown to have interesting applications in constructing intertwining operators and braided algebras, lifting functors, finding solutions for Yang-Baxter systems, etc. While for entwining structures one can associate corings, for semi-entwining structures one can associate comodule algebra structures where the algebra involved is a bialgebra satisfying certain properties.

math.QA↗

arXiv2012

The cocycle condition for multi-pullbacks of algebras

Take finitely many topological spaces and for each pair of these spaces choose a pair of corresponding closed subspaces that are identified by a homeomorpism. We note that this gluing procedure does not guarantee that the building pieces, or the gluings of some pieces, are embedded in the space obtained by putting together all given ingredients. Dually, we show that a certain sufficient condition, called the cocycle condition, is also necessary to guarantee sheaf-like properties of surjective multi-pullbacks of algebras with distributive lattices of ideals.

math.QA↗

arXiv2012

Finite closed coverings of compact quantum spaces

We show that a projective space P^\infty(Z/2) endowed with the Alexandrov topology is a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over this projective space. In technical terms, we prove that the category of finitely supported flabby sheaves of algebras is equivalent to the category of algebras with a finite set of ideals that intersect to zero and generate a distributive lattice. In particular, the Gelfand transform allows us to view finite closed coverings of compact Hausdorff spaces as flabby sheaves of commutative C*-algebras over P^\infty(Z/2).

math.QA↗

arXiv2011

The K-theory of Heegaard quantum lens spaces

Representing Z/N as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/N, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/N to construct an associated complex line 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 a particular 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/N-fibres of the Heegaard quantum sphere to U(1), and determine the algebraic structure of such a U(1)-prolongation.

math.KT↗

arXiv2010

Quantum projective space from Toeplitz cubes

From N-tensor powers of the Toeplitz algebra, we construct a multipullback C*-algebra that is a noncommutative deformation of the complex projective space CP(N). Using Birkhoff's Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multipullback 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.

math.QA↗

arXiv2007

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 Pi's 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.

math.QA↗

arXiv2006

A generalization of Margolus-Levitin bound

The Margolus-Levitin lower bound on minimal time required for a state to be transformed into an orthogonal state is generalized. It is shown that for some initial states new bound is stronger than the Margolus-Levitin one.

quant-ph↗

arXiv2005

Locally coalgebra-Galois extensions

The paper introduces the notion of a locally coalgebra-Galois extension and, as its special case, a locally cleft extension. The necessary and sufficient conditions for a locally coalgebra-Galois extension to be a (global) coalgebra-Galois extension are stated. As an important special case, it is proven, that under not very restrictive conditions the gluing of two locally cleft extensions is a globally coalgebra-Galois extension. As an example, the quantum lens space of positive charge is constructed by gluing of two quantum solid tori.

math.QA↗

arXiv2005

Cotensor products of quantum principal bundles

A cotensor product A\Box_H P of an H-Hopf Galois extension A and a C-coalgebra Galois extension P, such that P is an (H,C)-bicomodule, is analyzed. Conditions are stated, when A\Box_H P is a C-coalgebra Galois extension and when there exists a strong connection on A\Box_H P. Two examples are given, in both, A and P are Matsumoto spheres, and H=C=C(U(1)).

math.QA↗

arXiv2003

Hopf fibration and monopole connection over the contact quantum spheres

Noncommutative geometry of quantised contact spheres introduced by Omori, Maeda, Miyazaki and Yoshioka is studied. In particular it is proven that these spheres form a noncommutative 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.

math.QA↗