# Tomasz BrzezinskiSwansea University | SWAN · Department of Mathematics

Tomasz Brzezinski

PhD

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173

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Introduction

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October 2015 - present

## Publications

Publications (173)

It is shown that any Lie affgebra, that is an algebraic system consisting of an affine space together with a bi-affine bracket satisfying affine versions of the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together with an element and a specific generalised derivation (in the sense of Leger and Luks, [G.F.\ Leger \& E.M.\ Luks,...

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then sho...

Let A be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra \mathcal{O}(G) of a semisimple algebraic group G . It is shown that if A admits a suitably non-degenerate comodule V and the induced G -module structure of V is non-trivial, then the third Hochschild homology group of A is non-trivial.

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine spaces) and algebraic topology (chain contractions) are presented. Relationships between heaps of modules, modules...

The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced by Carinena, Grabowski and...

It is argued that a nonsingular elliptic curve admits a natural or fundamental abelian heap structure uniquely determined by the curve itself. It is shown that the set of complex analytic or rational functions from a nonsingular elliptic curve to itself is a truss arising from endomorphisms of this heap.

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then sho...

The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced in [{\sc J.F.\ Cari\~nena,...

It is shown that generalized Rota–Baxter operators introduced in [W. A. Martinez, E. G. Reyes, M. Ronco, Int. J. Geom. Meth. Mod. Phys. 18, 2150176 (2021)] are a special case of Rota–Baxter systems [T. Brzeziński, J. Algebra 460, 1–25 (2016)]. The latter are enriched by homothetisms and then shown to give examples of Dyck\(^m\)-algebras.KeywordsRot...

Let $A$ be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra $\mathcal{O}(G)$ of a semisimple algebraic group $G$. It is shown that if $A$ admits a suitably non-degenerate comodule $V$ and the induced $G$-module structure of $V$ is non-trivial, then the third Hochschild homology group of $A$ is non-trivial.

It is argued that a nonsingular elliptic curve admits a natural or fundamental abelian heap structure uniquely determined by the curve itself. It is shown that the set of complex analytic or rational functions from a nonsingular elliptic curve to itself is a truss arising from endomorphisms of this heap.

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of...

A frame-independent formulation of Lie brackets on affine spaces or {\em Lie affebras} introduced in [K.\ Grabowska, J.\ Grabowski \& P.\ Urba\'nski, Lie brackets on affine bundles, {\em Ann.\ Global Anal.\ Geom.} {\bf 24} (2003), 101--130] is given.

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine spaces) and algebraic topology (chain contractions) are presented. Relationships between heaps of modules and modu...

Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss versions of the Eilenberg-Watts theorem and Morita equivalence are formulated. Projective and small-projective...

Categorical constructions on heaps and modules over trusses are considered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coproducts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. I...

It is shown that generalized Rota-Baxter operators introduced in [W.A. Martinez, E.G. Reyes & M. Ronco, Int. J. Geom. Meth. Mod. Phys. 18 (2021) 2150176] are a special case of Rota-Baxter systems [T. Brzezi\'nski, J. Algebra 460 (2016), 1-25]. The latter are enriched by homothetisms and then shown to give examples of Dyck$^m$-algebras.

It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated...

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of...

It is shown that the Baer-Kaplansky theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated...

The notion of a pre-truss, that is, a set that is both a heap and a semigroup is introduced. Pre-trusses themselves as well as pre-trusses in which one-sided or two-sided distributive laws hold are studied. These are termed near-trusses and skew trusses respectively. Congruences in pre-trusses are shown to correspond to paragons defined here as sub...

Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss versions of the Eilenberg-Watts theorem and Morita equivalence are formulated. Projective and small-projective...

Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring [Formula: see text] by an ideal [Formula: see text] is a paragon in the truss [Formula: see text] associated to [Formu...

Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring $R$ by an ideal $I$ is a paragon in the truss $\mathrm{T}(R)$ associated to $R$. Second, an extension of a truss by a...

An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which might be additionally equipped with a Hopf algebra symmetry. The proposed framework is supported by two examples...

The structure of the C *-algebra of functions on the quantum flag manifold SUq(3)/T2 is investigated. Building on the representation theory of C ( SUq(3) ) , we analyze irreducible representations and the primitive ideal space of C ( SUq(3)/T2) , with a view towards unearthing the “quantum sphere bundle” CP1 q → SUq(3)/T2 → CP2 q .

Trusses, defined as sets with a suitable ternary and a binary operations, connected by the distributive laws, are studied from a ring and module theory point of view. The notions of ideals and paragons in trusses are introduced and several constructions of trusses are presented. A full classification of truss structures on the Abelian group of inte...

Categorical constructions on heaps and modules over trusses are considered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coproducts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. I...

Up to Morita equivalence, every quasi-hereditary algebra is the dual algebra of a directed bocs or coring. From the bocs, an exact Borel subalgebra is obtained. In this paper a characterisation of exact Borel subalgebras arising in this way is given.

The quantum flag manifold ${SU_q(3)/\mathbb{T}^2}$ is interpreted as a noncommutative bundle over the quantum complex projective plane with the quantum or Podle\'s sphere as a fibre. A connection arising from the (associated) quantum principal $U_q(2)$-bundle is described.

The structure of the $C^*$-algebra of functions on the quantum flag manifold $SU_q(3)/\mathbb{T}^2$ is investigated. Building on the representation theory of $C(SU_q(3))$, we analyze irreducible representations and the primitive ideal space of $C(SU_q(3)/\mathbb{T}^2)$, with a view towards unearthing the `quantum sphere bundle' $\mathbb{C} P_q^1 \t...

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...

A class of skew derivations on complex Noetherian generalized down-up algebras L = L(f, r, s, γ) is constructed.

Trusses, defined as sets with a suitable ternary and a binary operations, connected by the distributive laws, are studied from a ring and module theory point of view. The notions of ideals and paragons in trusses are introduced and several construction of trusses are presented. A full classification of truss structures on the Abelian group of integ...

The $\theta$-deformed Hopf fibration $\mathbb{S}^3_\theta\to \mathbb{S}^2$ over the commutative $2$-sphere is compared with its classical counterpart. It is shown that there exists a natural isomorphism between the corresponding associated module functors and that the affine spaces of classical and deformed connections are isomorphic. The latter is...

The θ-deformed Hopf fibration S 3 θ → S 2 over the commutative 2-sphere is compared with its classical counterpart. It is shown that there exists a natural isomorphism between the corresponding associated module functors and that the affine spaces of classical and deformed connections are isomorphic. The latter isomorphism is equivariant under an a...

A non-classical differential calculus on the quantum disc and cones is constructed and the associated integral is calculated.

These lectures describe an algebraic approach to differentiation and integration that is characteristic for non-commutative geometry. The material contained in Section 2 is standard and can be found in any text on non-commutative geometry, for example [4]. Items 5 and 6, which describe concepts introduced in [3], are exceptions. The bulk of Section...

A class of skew derivations on complex Noetherian generalized down-up algebras $L=L(f,r,s,\gamma)$ is constructed.

A general or truss distributive laws between two associative operations on the same set are studied for cancellative and inverse semigroups.

In an attempt to understand the origins and the nature of the law binding together two group operations into a {\em skew brace}, introduced in [L.\ Guarnieri \& L.\ Vendramin, Math.\ Comp.\ \textbf{86} (2017), 2519--2534] as a non-Abelian version of the {\em brace distributive law} of [W.\ Rump, J.\ Algebra {\bf 307} (2007), 153--170] and [F.\ Ced\...

A non-classical differential calculus on the quantum disc and cones is constructed and the associated integral is calculated.

An extended summary of the lecture course given at the V School on Geometry and Physics, Bia\l owe\.za 2016, in which an algebraic approach to differentiation and integration that is characteristic for non-commutative geometry is described.

A wide class of skew derivations on degree-one generalized Weyl algebras $R(a,\varphi)$ over a ring $R$ is constructed. All these derivations are twisted by a degree-counting extensions of automorphisms of $R$. It is determined which of the constructed derivations are $Q$-skew derivations. The compatibility of these skew derivations with the natura...

It is shown that, under some natural assumptions, the tensor product of differentially smooth algebras and the skew-polynomial rings over differentially smooth algebras are differentially smooth.

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones we propose a new twisted reality condition for the Dirac operator.

Rota-Baxter systems are modified by the inclusion of a curvature term. It is shown that, subject to specific properties of the curvature form, curved Rota-Baxter systems $(A,R,S,\omega)$ induce associative and (left) pre-Lie products on the algebra $A$. It is also shown that if both Rota-Baxter operators coincide with each other and the curvature i...

It is shown that the algebra of continuous functions on the quantum $2n+1$-dimensional lens space $C(L^{2n+1}_q(N; m_0,\ldots, m_n))$ is a graph $C^*$-algebra, for arbitrary positive weights $ m_0,\ldots, m_n$. The form of the corresponding graph is determined from the skew product of the graph which defines the algebra of continuous functions on t...

Elements of noncommutative differential geometry of generalized Z-graded Weyl algebras A(p;q) over the ring of polynomials in two variables, and their zero-degree subalgebras B(p;q) which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of A(p;q) a...

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones we propose a new twisted reality condition for the Dirac operator.

A generalisation of the notion of a Rota-Baxter operator is proposed. This
generalisation consists of two operators acting on an associative algebra and
satisfying equations similar to the Rota-Baxter equation. Rota-Baxter operators
of any weights and twisted Rota-Baxter operators are solutions of the proposed
system. It is shown that dendriform al...

Two-dimensional integrable differential calculi for classes of Ore extensions
of the polynomial ring and the Laurent polynomial ring in one variable are
constructed. Thus it is concluded that all affine pointed Hopf domains of
Gelfand-Kirillov dimension two which are not polynomial identity rings are
differentially smooth.

It is shown that the coordinate algebra of the quantum $2n+1$-dimensional
lens space $\mathcal{O}(L^{2n+1}_q(\prod_{i=0}^n m_i; m_0,\ldots, m_n))$ is a
principal $\mathbb{Z}$-comodule algebra or the coordinate algebra of a circle
principal bundle over the weighted quantum projective space
$\mathbb{WP}^n_q(m_0,\ldots, m_n)$. Furthermore, the weighte...

Differential geometry of the deformed pillow, cones and lenses is studied.
More specifically, a new notion of smoothness of algebras is proposed. This
notion, termed differential smoothness combines the existence of a top form in
a differential calculus over the algebra together with a strong version of the
Poincar\'e duality realized as an isomorp...

The algebras obtained as fixed points of the action of the cyclic group $Z_N$
on the coordinate algebra of the quantum disc are studied. These can be
understood as coordinate algebras of quantum or non-commutative cones. The
following observations are made. First, contrary to the classical situation,
the actions of $Z_N$ are free and the resulting...

Strongly $\mathbb{Z}$-graded algebras or principal circle bundles and
associated line bundles or invertible bimodules over a class of generalized
Weyl algebras $\mathcal{B}_{p;\; q,0}$ (over a ring of polynomials in one
variable) are constructed. The Chern-Connes pairing between the cyclic
cohomology of $\mathcal{B}_{p;\; q,0}$ and the isomorphism...

It is proven that the coordinate algebra of the noncommutative pillow
orbifold and, for l=1,2,3,4, the coordinate algebras of quantum teardrops
$WP_q(1,l)$ and of quantum lens spaces $L_q(l;1,l)$ are smooth in the sense of
[U. Kraehmer, On the Hochschild (co)homology of quantum homogeneous spaces,
Israel J. Math. 189 (2012), 237-266], i.e. have fin...

Weighted circle actions on the quantum Heeqaard 3-sphere are considered. The
fixed point algebras, termed quantum weighted Heegaard spheres, and their
representations are classified and described on algebraic and topological
levels. On the algebraic side, coordinate algebras of quantum weighted Heegaard
spheres are interpreted as generalised Weyl a...

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-cohe...

The algebraic approach to bundles in non-commutative geometry and the
definition of quantum real weighted projective spaces are reviewed. Principal
U(1)-bundles over quantum real weighted projective spaces are constructed. As
the spaces in question fall into two separate classes, the {\em negative} or
{\em odd} class that generalises quantum real p...

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 contin...

A method of constructing a (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence a...

It is proven that every flat connection or covariant derivative $\nabla$ on a
left $A$-module $M$ (with respect to the universal differential calculus)
induces a right $A$-module structure on $M$ so that $\nabla$ is a bimodule
connection on $M$ or $M$ is a flat differentiable bimodule. Similarly a flat
hom-connection on a right $A$-module $M$ induc...

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 w...

Algebras of functions on quantum weighted projective spaces are introduced,
and the structure of quantum weighted projective lines or quantum teardrops are
described in detail. In particular the presentation of the coordinate algebra
of the quantum teardrop in terms of generators and relations and classification
of irreducible *-representations are...

The first steps towards linearisation of partial orders and equivalence
relations are described. The definitions of partial orders and equivalence
relations (on sets) are formulated in a way that is standard in category theory
and that makes the linearisation (almost) automatic. The linearisation is then
achieved by replacing sets by coalgebras and...

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 $SU_q(2)$ 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 o...

We consider differential operators over a noncommutative algebra $A$ generated by vector fields. This is shown to form an associative algebra of differential operators, and acts on $A$-modules $E$ with covariant derivative. For bimodule covariant derivatives on $E$, we consider a module map $U_E$ which classifies how similar to the classical case t...

The idea of a line bundle in classical geometry is transferred to
noncommutative geometry by the idea of a Morita context. From this we can
construct Z and N graded algebras, the Z graded algebra being a Hopf-Galois
extension. A non-degenerate Hermitian metric gives a star structure on this
algebra, and an additional star operation on the line bund...

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.

The theory of R-smash products for Hopf quasigroups is developed. Comment: 10 pages; to appear in AJSE D-Mathematics

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 inject...

Definitions of actions of Hopf quasigroups are discussed in the context of Long dimodules and smash products. In particular, Long dimodules are defined for Hopf quasigroups and coquasigroups, and solutions to Militaru's D-equation are constructed. A necessary compatibility condition between action and multiplication of a Hopf quasigroup acting on i...

A coring is one of the most basic algebraic structures dual to that of a ring (but also a ring itself can be viewed as a coring). The notion of a coring appeared first in the algebra literature in 1975 in Sweedler's paper on a predual version of the Jacobson–Bourbaki correspondence. This chapter discusses the basic definitions and the basic propert...

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 added

Theorem 2.2 stated a monoidal isomorphism between the comodule categories of two bialgebroids in a Hopf algebroid. The proof
of Theorem 2.2 was based on the journal version of Brzeziński (Ann Univ Ferrara Sez VII (NS) 51:15–27, 2005, Theorem 2.6), whose proof turned out to contain an unjustified step. Here we show that all other results in our pape...

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 categ...

A construction of Kleisli objects in 2-categories of noncartesian internal categories or categories internal to monoidal categories is presented. Comment: 17 pages

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 G...

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The...

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.

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 unitality 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....

Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor −A⊗B:MA→MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor −A⊗C:MA→MA is a comonad (or cotriple). The related categories of modules (or algebras) of −A⊗B and c...

An explicit formula for a strong connection form in a principal extension by a coseparable coalgebra is given.

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...

The following incorrect claim occurred: An A-coringC˜is a left, equivalently, right extension of an A-coringCCif and only if there exists a homomorphism of A-coringsC→C˜. Here we present a corrected statement and claim that the error does not influence other results in the paper.

The 23 articles in this volume encompass the proceedings of the International Conference on Modules and Comodules held in Porto (Portugal) in 2006 and dedicated to Robert Wisbauer on the occasion of his 65th birthday. These articles reflect Professor Wisbauer's wide interests and give an overview of different fields related to module theory, some o...