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Tomasz Brzezinski

Publications and source records attributed to Tomasz Brzezinski.

At least 19 recordsLinked to original sources
arXiv2023

Heaps of modules: Categorical aspects

Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and algebraic theories. In particular, it is shown that the category of groups with a compatible action of a truss T (also called pointed T-modules) is isomorphic to the category of modules over the ring R(T) universally associated to the truss. This is widely used in the explicit description of free objects. Next, it is proven that the category of heaps of modules over T is isomorphic to the category of affine modules over R(T) and, in order to make the picture complete, that (in the unital case) these are in turn equivalent to a specific subcategory of the slice category of pointed T-modules over R(T). These correspondences and properties are then used to describe explicitly various (co)limits and to compare short exact sequences in the Barr-exact category of heaps of T-modules with short exact sequences as defined previously.

math.RA↗
arXiv2018

On twisted reality conditions

We study the twisted reality condition of Math. Phys. Anal. Geom. 19 (2016),no. 3, Art. 16, for spectral triples, in particular with respect to the product and the commutant. Motivated by this we present the procedure, which allows one to "untwist" the twisted spectral triples studied in Lett. Math. Phys. 106 (2016), 1499-1530. We also relate this construction to conformally rescaled real twisted spectral triples, and discuss the untwisting of the `minimal twist' procedure of an even spectral triple.

math.QA↗
arXiv2012

Noncommutative differential operators, Sobolev spaces and the centre of a category

We consider differential operators over a noncommutative algebra $A$ generated by vector fields. These are shown to form a unital associative algebra of differential operators, and act on $A$-modules $E$ with covariant derivative. We use the repeated differentials given in the paper to give a definition of noncommutative Sobolev space for modules with connection and Hermitian inner product. The tensor algebra of vector fields, with a modified bimodule structure and a bimodule connection, is shown to lie in the centre of the bimodule connection category ${}_A\mathcal{E}_A$, and in fact to be an algebra in the centre. The crossing natural transformation in the definition of the centre of the category is related to the action of the differential operators on bimodules with connection.

math.QA↗
arXiv2009

Non-commutative integral forms and twisted multi-derivations

Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every finitely cogenerated injective module admits a hom-connection with respect to any differential graded algebra. The bulk of the paper is devoted to describing a method of constructing hom-connections from twisted multi-derivations. The notion of a free twisted multi-derivation is introduced and the induced first order differential calculus is described. It is shown that any free twisted multi-derivation on an algebra A induces a unique hom-connection on A that vanishes on the dual basis for the module of one-forms. To any flat hom-connection \nabla on A one associates a chain complex, termed a complex of integral forms on A. The canonical cokernel morphism to the zeroth homology space is called a \nabla-integral. Examples of free twisted multi-derivations, hom-connections and corresponding integral forms are provided by covariant calculi on Hopf algebras or quantum groups and on quantum spaces. The example of a flat hom-connection within the 3D left-covariant differential calculus on the quantum group SL(2) is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf-Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podles sphere is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the \nabla-integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.

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arXiv2009

Galois-Type Extensions and Equivariant Projectivity

The theory of general Galois-type extensions is presented, including the interrelations between coalgebra extensions and algebra (co)extensions, properties of corresponding (co)translation maps, and rudiments of entwinings and factorisations. To achieve broad perspective, this theory is placed in the context of far reaching generalisations of the Galois condition to the setting of corings. At the same time, to bring together K-theory and general Galois theory, the equivariant projectivity of extensions is assumed resulting in the centrepiece concept of a principal extension. Motivated by noncommutative geometry, we employ such extensions as replacements of principal bundles. This brings about the notion of a strong connection and yields finitely generated projective associated modules, which play the role of noncommutative vector bundles. Subsequently, the theory of strong connections is developed. It is purported as a basic ingredient in the construction of the Chern character for Galois-type extensions (called the Chern-Galois character).

math.QA↗
arXiv2008

Strong connections and the relative Chern-Galois character for corings

The Chern-Galois theory is developed for corings or coalgebras over non-commutative rings. As the first step the notion of an entwined extension as an extension of algebras within a bijective entwining structure over a non-commutative ring is introduced. A strong connection for an entwined extension is defined and it is shown to be closely related to the Galois property and to the equivariant projectivity of the extension. A generalisation of the Doi theorem on total integrals in the framework of entwining structures over a non-commutative ring is obtained, and the bearing of strong connections on properties such as faithful flatness or relative injectivity is revealed. A family of morphisms between the K0-group of the category of finitely generated projective comodules of a coring and even relative cyclic homology groups of the base algebra of an entwined extension with a strong connection is constructed. This is termed a relative Chern-Galois character. Explicit examples include the computation of a Chern-Galois character of depth 2 Frobenius split (or separable) extensions over a separable algebra R. Finitely generated and projective modules are associated to an entwined extension with a strong connection, the explicit form of idempotents is derived, the corresponding (relative) Chern characters are computed, and their connection with the relative Chern-Galois character is explained.

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arXiv2008

Cleft extensions of Hopf algebroids

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid $H= (H_L,H_R)$) is cleft if and only if it is $H_R$-Galois and has a normal basis property relative to the base ring $L$ of $H_L$. Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the Chern-Galois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined.

math.QA↗
arXiv2008

A note on coring extensions

A notion of a coring extension is defined and it is related to the existence of an additive functor between comodule categories that factorises through forgetful functors. This correspondence between coring extensions and factorisable functors is illustrated by functors between categories of descent data. A category in which objects are corings and morphisms are coring extensions is also introduced.

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arXiv2008

Bimodule herds

The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a $B$-$A$ bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies untiality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definition of a bimodule herd. Every bicomodule coherd defines a pair of (non-unital) rings. It is shown that a tame $B$-$A$ bimodule herd defines a bicomodule coherd, and sufficient conditions for the derived rings to be isomorphic to $A$ and $B$ are discussed. The composition of bimodule herds via the tensor product is outlined. The notion of a bimodule herd is illustrated by the example of Galois co-objects of a commutative, faithfully flat Hopf algebra.

math.RA↗
arXiv2008

Hopf-cyclic homology with contramodule coefficients

A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions.

math.KT↗
arXiv2008

Monads and comonads in module categories

Let $A$ be a ring and $\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\otimes_A B: \M_A\to \M_A$ is a monad (or triple). Similarly, an $A $-bimodule $\C$ is an $A$-coring provided the functor $-\otimes_A\C:\M_A\to \M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\otimes_A B$ and comodules (or coalgebras) of $-\otimes_A\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\Hom_A(B,-)$ and $\Hom_A(\C,-)$ are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of $\Hom_A(B,-)$-comodules is isomorphic to the category of $B$-modules, while the category of $\Hom_A(\C,-)$-modules (called $\C$-contramodules by Eilenberg and Moore) need not be equivalent to the category of $\C$-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of $\C$-comodules and $\Hom_A(\C,-)$-modules are equivalent provided $\C$ is a coseparable coring. Furthermore, a bialgebra $H$ over a commutative ring $R$ is a Hopf algebra if and only if $\Hom_R(H-)$ is a Hopf bimonad on $\M_R$ and in this case the categories of $H$-Hopf modules and mixed $\Hom_R(H,-)$-bimodules are both equivalent to $\M_R$.

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arXiv2008

Non-commutative connections of the second kind

A connection-like objects, termed {\em hom-connections} are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.

math.QA↗
arXiv2007

Pre-torsors and equivalences

Properties of (most general) non-commutative torsors or A-B torsors are analysed. Starting with pre-torsors it is shown that they are equivalent to a certain class of Galois extensions of algebras by corings. It is shown that a class of faithfully flat pre-torsors induces equivalences between categories of comodules of associated corings. It is then proven that A-B torsors correspond to monoidal functors (and, under some additional conditions, equivalences) between categories of comodules of bialgebroids.

math.QA↗
arXiv2007

Notes on formal smoothness

The definition of an S-category is proposed by weakening the axioms of a Q-category introduced by Kontsevich and Rosenberg. Examples of Q- and S-categories and (co)smooth objects in such categories are given.

math.RA↗
arXiv2007

Separable Functors and Formal Smoothness

The natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of $ad$-(co)invariant integrals for a Hopf algebra $H$ is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double $D(H)$ over $H$. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that given a bialgebra surjection $π:E\to H$ with nilpotent kernel such that $H$ is a Hopf algebra which is formally smooth as a $K$-algebra, then $π$ has a section which is a right $H$-colinear algebra homomorphism. Moreover, if $H$ is also endowed with an $ad$-invariant integral, then this section can be chosen to be $H$-bicolinear. We also deal with the dual case.

math.QA↗
arXiv2007

Flat connections and (co)modules

The relationship between comodules of a coring and flat connections is reviewed. In particular we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in one-to-one correspondence with entwining structures, and their comodules are entwined modules. These include Yetter-Drinfeld and anti-Yetter-Drinfeld modules and their generalisations, hence all the modules of interest to Hopf-cyclic cohomology. In this way the interpretation of the latter as modules with flat connections [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential geometry, Lett. Math. Phys. 76 (2006), 77--91] is obtained as a corollary of a more general theory. We also introduce the notions of a connection in a comodule and of a bicomodule connection, and show how comodules with flat connections can be interpreted as modules of a C-ring. In this way all the above mentioned Hopf modules can be interpreted as comodules with flat connections.

math.QA↗
arXiv2006

Descent cohomology and corings

A coring approach to non-Abelian descent cohomology of [P Nuss and M Wambst, Non-Abelian Hopf cohomology, Preprint arXiv:math.KT/0511712, (2005)] is described and a definition of a Galois cohomology for partial group actions is proposed.

math.RA↗