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  • 1.
    Back, Per
    et al.
    Mälardalens universitet, SWE.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    The hom-associative Weyl algebras in prime characteristic2022In: International Electronic Journal of Algebra, E-ISSN 1306-6048, Vol. 31, no 31, p. 203-229Article in journal (Refereed)
    Abstract [en]

    We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general twisting'' procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket. © 2022, Hacettepe University. All rights reserved.

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  • 2.
    Baghdari, Samaneh
    et al.
    Isfahan University of Technology, Iran.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Pure semisimple and Kothe group rings2023In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 51, no 7, p. 2779-2790Article in journal (Refereed)
    Abstract [en]

    In this article, we provide a complete characterization of abelian group rings which are Kothe rings. We also provide characterizations of (possibly non-abelian) group rings over division rings which are Kothe rings, both in characteristic zero and in prime characteristic, and prove a Maschke type result for pure semisimplicity of group rings. Furthermore, we illustrate our results by several examples.Communicated by Eric Jespers

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  • 3.
    Bagio, Dirceu
    et al.
    Universidade Federal de Santa Catarina, Brazil.
    Gonçalves, Daniel
    Universidade Federal de Santa Catarina, Brazil.
    Moreira, Paula Savana Estácio
    Universidade Federal de Santa Catarina, Brazil.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras2024In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337Article in journal (Refereed)
    Abstract [en]

    Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring R ⋊ α G {R\rtimes_{\alpha}G}, which carries a natural G-grading. We show that there is a one-to-one correspondence between the G-invariant ideals of R and the graded ideals of the G-graded ring R ⋊ α G {R\rtimes_{\alpha}G}. We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of R ⋊ α G {R\rtimes_{\alpha}G}. We show that every ideal of R ⋊ α G {R\rtimes_{\alpha}G} is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G-simplicity of R, topological transitivity of θ is equivalent to G-primeness of R, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra. © 2024 Walter de Gruyter GmbH, Berlin/Boston 2024.

  • 4.
    Beuter, Viviane
    et al.
    Universidade Federal de Santa Catarina, BRA.
    Gonçalves, Daniel
    Universidade Federal de Santa Catarina, BRA.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Royer, Danilo
    Universidade Federal de Santa Catarina, BRA.
    Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics2019In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 31, no 3, p. 543-562Article in journal (Refereed)
    Abstract [en]

    Given a partial action π of an inverse semigroup S on a ring A {\mathcal{A}}, one may construct its associated skew inverse semigroup ring A π S {\mathcal{A}\rtimes-{\pi}S}. Our main result asserts that, when A {\mathcal{A}} is commutative, the ring A π S {\mathcal{A}\rtimes-{\pi}S} is simple if, and only if, A {\mathcal{A}} is a maximal commutative subring of A π S {\mathcal{A}\rtimes-{\pi}S} and A {\mathcal{A}} is S-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra A R (g) {A-{R}(\mathcal{G})} associated with a Hausdorff and ample groupoid g {\mathcal{G}}. © 2018 Walter de Gruyter GmbH, Berlin/Boston.

  • 5.
    Bäck, Per
    et al.
    Mälardalen University, SWE.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Hilbert’s Basis Theorem for Non-associative and Hom-associative Ore Extensions2023In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 26, no 4, p. 1051-1065Article in journal (Refereed)
    Abstract [en]

    We prove a hom-associative version of Hilbert’s basis theorem, which includes as special cases both a non-associative version and the classical Hilbert’s basis theorem for associative Ore extensions. Along the way, we develop hom-module theory. We conclude with some examples of both non-associative and hom-associative Ore extensions which are all noetherian by our theorem.

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  • 6.
    Bäck, Per
    et al.
    Mälardalen University, SWE.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    On the hom-associative Weyl algebras2020In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 224, no 9, article id 106368Article in journal (Refereed)
    Abstract [en]

    The first (associative) Weyl algebra is formally rigid in the classical sense. In this paper, we show that it can however be formally deformed in a nontrivial way when considered as a so-called hom-associative algebra, and that this deformation preserves properties such as the commuter, while deforming others, such as the center, power associativity, the set of derivations, and some commutation relations. We then show that this deformation induces a formal deformation of the corresponding Lie algebra into what is known as a hom-Lie algebra, when using the commutator as bracket. We also prove that all homomorphisms between any two purely hom-associative Weyl algebras are in fact isomorphisms. In particular, all endomorphisms are automorphisms in this case, hence proving a hom-associative analogue of the Dixmier conjecture to hold true. © 2020 Elsevier B.V.

  • 7.
    Ibragimov, Nail
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Integration of dynamical systems admitting nonlinear superposition2016In: JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS, ISSN 2330-152X, Vol. 4, no 2, p. 91-106Article, review/survey (Refereed)
    Abstract [en]

    A method of integration of non-stationary dynamical systems admitting nonlinear superpositions is presented. The method does not require knowledge of symmetries of the differential equations under consideration. The integration procedure is based on classification of Vessiot-Guldberg-Lie algebras associated with nonlinear superpositions. It is shown that the systems associated with one-and two-dimensional Lie algebras can be integrated by quadrature upon introducing Lie's canonical variables. It is not necessary to know symmetries of a system in question in this approach. Two-dimensional non-stationary dynamical systems with three-dimensional Vessiot-Guldberg-Lie algebras are classified into thirteen standard forms. Ten of them are integrable by quadrature. The remaining three standard forms lead to the Riccati equations. Integration of perturbed dynamical systems possessing approximate nonlinear superposition is discussed.

  • 8.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Gainetdinova, A. A.
    Ufa State Aviat Tech Univ, Rus.
    Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras2016In: Applied Mathematics Letters, ISSN 0893-9659, E-ISSN 1873-5452, Vol. 52, p. 126-131Article in journal (Refereed)
    Abstract [en]

    The recent method of integration of non-stationary dynamical systems admitting nonlinear superpositions is applied to the three-dimensional dynamical systems associated with three-dimensional Vessiot-Guldberg-Lie algebras L-3. The investigation is based on Bianchi's classification of real three-dimensional Lie algebras and realizations of these algebras in the three-dimensional space. Enumeration of the Vessiot-Guldberg-Lie algebras L-3 allows to classify three-dimensional dynamical systems admitting nonlinear superpositions into thirty one standard types by introducing canonical variables. Twenty four of them are associated with solvable Vessiot-Guldberg-Lie algebras and can be reduced to systems of first-order linear equations. The remaining seven standard types are nonlinear. Integration of the latter types is an open problem. (C) 2015 Elsevier Ltd. All rights reserved.

  • 9.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Gainetdinova, Aliya
    Ufimskij Gosudarstvennyj Aviacionnyj Tehniceskij Universitet, RUS.
    Three-dimensional dynamical systems with four-dimensional vessiot-guldberg-lie algebras2017In: The Journal of Applied Analysis and Computation, ISSN 2156-907X, E-ISSN 2158-5644, Vol. 7, no 3, p. 872-883Article in journal (Refereed)
    Abstract [en]

    - Dynamical systems attract much attention due to their wide applications. Many significant results have been obtained in this field from various points of view. The present paper is devoted to an algebraic method of integration of three-dimensional nonlinear time dependent dynamical systems admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras L4. The invariance of the relation between a dynamical system admitting nonlinear superposition and its Vessiot-Guldberg-Lie algebra is the core of the integration method. It allows to simplify the dynamical systems in question by reducing them to standard forms. We reduce the three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras to 98 standard types and show that 86 of them are integrable by quadratures.

  • 10.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Gainetdinova, Aliya. A.
    Ufa State Aviation Technical University, RUS.
    Classification and integration of four-dimensional dynamical systems admitting non-linear superposition2017In: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 90, p. 50-71Article in journal (Refereed)
    Abstract [en]

    The method of integration of dynamical systems admitting non-linear superpositions is applied to four-dimensional non-linear dynamical systems. All four-dimensional dynamical systems admitting non-linear superpositions with four-dimensional Vessiot-Guldberg-Lie algebras are classified into 160 standard forms. The integration method is described and illustrated.

  • 11.
    Lorensen, Karl
    et al.
    Pennsylvania State University, United States.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Generating numbers of rings graded by amenable and supramenable groups2024In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 109, no 1, article id e12826Article in journal (Refereed)
    Abstract [en]

    A ring 𝑅 has unbounded generating number (UGN) if,for every positive integer 𝑛, there is no 𝑅-module epimorphism 𝑅𝑛 → 𝑅𝑛+1. For a ring 𝑅 = ⨁g∈𝐺 𝑅g gradedby a group 𝐺 such that the base ring 𝑅1 has UGN, weidentify several sets of conditions under which 𝑅 mustalso have UGN. The most important of these are: (1)𝐺 is amenable, and there is a positive integer 𝑟 suchthat, for every g ∈ 𝐺, 𝑅g ≅ (𝑅1)𝑖 as 𝑅1-modules for some𝑖 = 1, … , 𝑟; (2) 𝐺 is supramenable, and there is a positive integer 𝑟 such that, for every g ∈ 𝐺, 𝑅g ≅ (𝑅1)𝑖 as𝑅1-modules for some 𝑖 = 0, … , 𝑟. The pair of conditions(1) leads to three different ring-theoretic characterizations of the property of amenability for groups. We alsoconsider rings that do not have UGN; for such a ring𝑅, the smallest positive integer 𝑛 such that there is an𝑅-module epimorphism 𝑅𝑛 → 𝑅𝑛+1 is called the generating number of 𝑅, denoted gn(𝑅). If 𝑅 has UGN, then wedefine gn(𝑅) ∶= ℵ0. We describe several classes of examples of a ring 𝑅 graded by an amenable group 𝐺 such thatgn(𝑅) ≠ gn(𝑅1).

    MSC 2020

    16P99, 16S35, 16W50, 20F65, 43A07 (primary), 16D90 (secondary)

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  • 12.
    Lundstrom, Patrik
    et al.
    University West.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    SIMPLICITY OF LEAVITT PATH ALGEBRAS VIA GRADED RING THEORY2023In: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 108, no 3, p. 428-437Article in journal (Refereed)
    Abstract [en]

    Suppose that R is an associative unital ring and that E= (E-0, E-1, r, s) is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra L-R(E) is simple if and only if R is simple, E-0 has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

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  • 13.
    Lundstrom, Patrik
    et al.
    University West.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    NON-UNITAL ORE EXTENSIONS2023In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 172, no 2, p. 217-229Article in journal (Refereed)
    Abstract [en]

    We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings R[x; delta], under the hy-pothesis that R is s-unital and ker(delta) contains a non-zero idempotent. This result gener-alizes a result by oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.

  • 14.
    Lundström, Patrik
    et al.
    University West, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Strongly graded leavitt path algebras2022In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 21, no 07, article id 2250141Article in journal (Refereed)
    Abstract [en]

    Let R be a unital ring, let E be a directed graph and recall that the Leavitt path algebra LR(E) carries a natural-gradation. We show that LR(E) is strongly-graded if and only if E is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained. © 2022 World Scientific Publishing Company

  • 15.
    Lännström, Daniel
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    A characterization of graded von Neumann regular rings with applications to Leavitt path algebras2021In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 567, p. 91-113Article in journal (Refereed)
    Abstract [en]

    We prove a new characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra LR(E) with coefficients in a unital ring R is graded von Neumann regular if and only if R is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path algebras over fields. © 2020 The Author(s)

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  • 16.
    Lännström, Daniel
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Chain conditions for epsilon-strongly graded rings with applications to Leavitt path algebras2020In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 23, no 4, p. 1707-1726Article in journal (Refereed)
    Abstract [en]

    Let G be a group with neutral element e and let S=⊕g∈GSg be a G-graded ring. A necessary condition for S to be noetherian is that the principal component Se is noetherian. The following partial converse is well-known: If S is strongly-graded and G is a polycyclic-by-finite group, then Se being noetherian implies that S is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain characterizations of noetherian and artinian Leavitt path algebras with coefficients in a general unital ring. This extends a recent characterization by Steinberg for Leavitt path algebras with coefficients in a commutative unital ring and previous characterizations by Abrams, Aranda Pino and Siles Molina for Leavitt path algebras with coefficients in a field. Secondly, we obtain characterizations of noetherian and artinian unital partial crossed products. © 2019, The Author(s).

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  • 17.
    Lännström, Daniel
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Induced quotient group gradings of epsilon-strongly graded rings2020In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 9, no 9, article id 2050162Article in journal (Refereed)
    Abstract [en]

    Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well-known that if $S$ is strongly $G$-graded, then the induced $G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings was recently introduced by Nystedt, Öinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and sufficient conditions for the induced $G/N$-grading of an epsilon-strongly $G$-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units (s-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings. 

  • 18.
    Lännström, Daniel
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    The graded structure of algebraic Cuntz-Pimsner rings2020In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 224, no 9, article id UNSP 106369Article in journal (Refereed)
    Abstract [en]

    The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We  classify strongly, epsilon-strongly and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. As an application, we characterize noetherian and artinian fractional skew monoid rings by a single corner automorphism.

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    The graded structure of algebraic Cuntz-Pimsner rings
  • 19.
    Lännström, Daniel
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    The structure of epsilon-strongly graded rings with applications to Leavitt path algebras and Cuntz-Pimsner rings2019Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings.

     In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras.  In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.

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  • 20.
    Lännström, Daniel
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    The structure of epsilon-strongly group graded rings2021Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The development of a general theory of strongly group graded rings was initiated by Dade, Năstăsescu and Van Oystaeyen in the 1980s, and since then numerous structural results have been established.  In this thesis we develop a general theory of so-called (nearly) epsilon-strongly group graded rings which were recently introduced by Nystedt, Öinert and Pinedo and which generalize strongly group graded rings. Moreover, we obtain applications to  Leavitt path algebras, unital partial crossed products and algebraic Cuntz-Pimsner rings. 

    This thesis is based on five scientific papers (A, B, C, D, E). 

    Papers A and B are concerned with structural properties of epsilon-strongly graded rings. In Paper A, we consider an important construction called the induced quotient group grading. In Paper B, using results from Paper A, we obtain a Hilbert Basis Theorem for epsilon-strongly graded rings.  In Paper C, we study the graded structure of algebraic  Cuntz-Pimsner rings. In particular, we obtain a partial characterization of unital strongly graded, epsilon-strongly graded and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. 

    In Paper D, we give a complete characterization of group graded rings that are graded von Neumann regular.

    Finally, in Paper E, written in collaboration with Lundström, Öinert and Wagner, we consider prime nearly epsilon-strongly graded rings. Generalizing Passman's work from the 1980s, we give  necessary and sufficient conditions for a nearly epsilon-strongly graded ring to be prime. 

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  • 21.
    Lännström, Daniel
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Graded von Neumann regularity of rings graded by semigroups2022In: Beitraege zur Algebra und Geometrie, ISSN 0138-4821, E-ISSN 2191-0383Article in journal (Refereed)
    Abstract [en]

    In this article, we give a complete characterization of semigroup graded rings which are graded von Neumann regular. We also demonstrate our results by applying them to several classes of examples, including matrix rings and groupoid graded rings. © 2022, The Author(s).

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  • 22.
    Lännström, Daniel
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Lundström, Patrik
    University West, SWE.
    Prime group graded rings with applications to partial crossed products and Leavitt path algebrasManuscript (preprint) (Other academic)
    Abstract [en]

    In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime s-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over s-unital rings; (ii) characterizations of prime s-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterization of prime Leavitt path algebras, by Abrams, Bell and Rangaswamy.

  • 23.
    Mai Hoang, Bien
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Quasi-duo differential polynomial rings2018In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 17, no 4, article id 1850072Article in journal (Refereed)
    Abstract [en]

    In this article we give a characterization of left (right) quasi-duo differential polynomial rings. We provide non-trivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

  • 24.
    Musonda, John
    et al.
    The University of Zambia, ZMB.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Silvestrov, Sergei
    Mälardalens Högskola, SWE.
    Reordering in noncommutative algebras associated with iterated function systems2020In: Springer Proceedings in Mathematics and Statistics / [ed] Silvestrov S.,Malyarenko A.,Rancic M., Springer , 2020, Vol. 317, p. 509-552Conference paper (Refereed)
    Abstract [en]

    A general class of multi-parametric families of unital associative complex algebras, defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems, is considered. A generalization of these commutation relations in three generators is also considered, modifying Lie algebra type commutation relations, typical for usual differential or difference operators, to relations satisfied by more general twisted difference operators associated with general twisting maps. General reordering and nested commutator formulas for arbitrary elements in these algebras are presented, and some special cases are considered, generalizing some well-known results in mathematics and physics. © Springer Nature Switzerland AG 2020.

  • 25.
    Musonda, John
    et al.
    The University of Zambia, ZMB.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Silvestrov, Sergei
    Mälardalens Högskola, SWE.
    Twisted difference operator representations of deformed lie type commutation relations2020In: Springer Proceedings in Mathematics and Statistics / [ed] Silvestrov S.,Malyarenko A.,Rancic M., Springer , 2020, Vol. 317, p. 553-573Conference paper (Refereed)
    Abstract [en]

    Operator representations of deformed Lie type commutation relations, associated with group or semigroup actions of dynamical systems and iterated function systems are considered. In particular, it is shown that some multi-parameter deformed symmetric difference and multiplication operators satisfy these commutation relations. The operator representations are considered also in the context of twisted derivations. © Springer Nature Switzerland AG 2020.

  • 26.
    Nystedt, Patrik
    et al.
    Högskolan Väst, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Group gradations on Leavitt path algebras2020In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 19, no 9, article id 2050165Article in journal (Refereed)
    Abstract [en]

    Given a directed graph E and an associative unital ring R one may define the Leavitt path algebra with coefficients in R, denoted by LR(E). For an arbitrary group G, LR(E) can be viewed as a G-graded ring. In this paper, we show that LR(E) is always nearly epsilon-strongly G-graded. We also show that if E is finite, then LR(E) is epsilon-strongly G-graded. We present a new proof of Hazrat's characterization of strongly g-graded Leavitt path algebras, when E is finite. Moreover, if E is row-finite and has no source, then we show that LR(E) is strongly-graded if and only if E has no sink. We also use a result concerning Frobenius epsilon-strongly G-graded rings, where G is finite, to obtain criteria which ensure that LR(E) is Frobenius over its identity component. © 2020 World Scientific Publishing Company.

  • 27.
    Nystedt, Patrik
    et al.
    University West, Sweden.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Simple graded rings, non-associative crossed products and Cayley-Dickson doublings2020In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 19, no 12, article id 2050231Article in journal (Refereed)
    Abstract [en]

    We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers from the associative case to the non-associative situation. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply this result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.

  • 28.
    Nystedt, Patrik
    et al.
    Univ West, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Pinedo, Hector
    Univ Ind Santander, COL.
    EPSILON-STRONGLY GROUPOID-GRADED RINGS, THE PICARD INVERSE CATEGORY AND COHOMOLOGY2020In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 62, no 1, p. 233-259, article id PII S0017089519000065Article in journal (Refereed)
    Abstract [en]

    We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid-graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group-graded situation to the groupoid-graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.

  • 29.
    Nystedt, Patrik
    et al.
    University West, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Pinedo, Héctor
    Industrial University of Santander, COL.
    Artinian and noetherian partial skew groupoid rings2018In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 503, p. 433-452Article in journal (Refereed)
    Abstract [en]

    Let α={α_g : R_{g^{−1}}→R_g}_{g∈mor(G)} be a partial action of a groupoid G on a (not necessarily associative) ring R and let S=R⋆G be the associated partial skew groupoid ring. We show that if α is global and unital, then S is left (right) artinian if and only if R is left (right) artinian and R_g={0}, for all but finitely many g∈mor(G). We use this result to prove that if α is unital and R is alternative, then S is left (right) artinian if and only if R is left (right) artinian and R_g={0}, for all but finitely many g∈mor(G). This result applies to partial skew group rings, in particular. Both of the above results generalize a theorem by J. K. Park for classical skew group rings, i.e. the case when R is unital and associative, and G is a group which acts globally on R. We provide two additional applications of our main results. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (not necessarily associative) groupoid rings. This result is in turn applied to partial group algebras. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we relate noetherian and artinian properties of partial skew groupoid rings to those of global skew groupoid rings, as well as establish two Maschke-type results, thereby generalizing results by M. Ferrero and J. Lazzarin for partial skew group rings to the case of partial skew groupoid rings.

  • 30.
    Nystedt, Patrik
    et al.
    University West, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Pinedo, Héctor
    Industrial University of Santander, COL.
    Epsilon-strongly graded rings, separability and semisimplicity2018In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 514, no Nov., p. 1-24Article in journal (Refereed)
    Abstract [en]

    We introduce the class of epsilon-strongly graded rings and show that it properly contains both the collection of strongly graded rings and the family of unital partial crossed products. We determine when epsilon-strongly graded rings are separable over their principal components. Thereby, we simultaneously generalize a result for strongly group-graded rings by Nastasescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the family of unital partial crossed products appear in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the family of strongly graded rings. Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Simon concerning when graded rings can be presented as partial crossed products.

  • 31.
    Nystedt, Patrik
    et al.
    Högskolan Väst, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences. Blekinge Inst Technol, Dept Math & Nat Sci, SE-37179 Karlskrona, Sweden..
    Richter, Johan
    Mälardalens högskola, SWE.
    NON-ASSOCIATIVE ORE EXTENSIONS2018In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 224, no 1, p. 263-292Article in journal (Refereed)
    Abstract [en]

    We introduce non-associative Ore extensions, S = R[X; sigma, delta], for any non-ssociative unital ring R and any additive maps sigma, delta : R -> R satisfying sigma(1) = 1 and delta(1) = 0. In the special case when delta is either left or right R-delta-linear, where R-delta = ker(delta), and R is delta-simple, i.e. {0} and R are the only delta-invariant ideals of R, we determine the ideal structure of the non-associative differential polynomial ring D = R[X; id(R),delta]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z(D) = R-delta[p] for a monic p is an element of R-delta [X], unique up to addition of elements from Z(R)(delta) . Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is delta-simple and Z(D) equals the field R-delta boolean AND Z(R). This provides us with a non-associative generalization of a result by Oinert, Richter and Silve-strov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field R-delta boolean AND Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.

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  • 32.
    Nystedt, Patrik
    et al.
    Högskolan Väst, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Richter, Johan
    Mälardalens högskola, SWE.
    Simplicity of Ore monoid rings2019In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 530, p. 69-85Article in journal (Refereed)
    Abstract [en]

    Given a non-associative unital ring R, a monoid G and a set π of additive maps R→R, we introduce the Ore monoid ring R[π;G], and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D-structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids. © 2019 Elsevier Inc.

  • 33.
    Ongong'a, Elvice
    et al.
    Mälardalens högskola, SWE.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Silvestrov, Sergei
    Mälardalens högskola, SWE.
    Hom-Lie structures on 3-dimensional skew symmetric algebras2019In: Journal of Physics: Conference Series, Institute of Physics Publishing (IOPP), 2019, Vol. 1416, article id 012025Conference paper (Refereed)
    Abstract [en]

    We describe the dimension of the space of possible linear endomorphisms that turn skew-symmetric three-dimensional algebras into Hom-Lie algebras. We find a correspondence between the rank of a matrix containing the structure constants of the bilinear product and the dimension of the space of Hom-Lie structures. Examples from classical complex Lie algebras are given to demonstrate this correspondence.

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    Hom-Lie structures on 3-dimensional skew symmetric algebras
  • 34.
    Ongong’a, Elvice
    et al.
    University of Nairobi, KEN ; Mälardalens högskola, SWE.
    Silvestrov, Sergei
    Mälardalens Högskola, SWE.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Classification of low-dimensional hom-lie algebras2020In: Springer Proceedings in Mathematics and Statistics / [ed] Silvestrov S.,Malyarenko A.,Rancic M., Springer , 2020, Vol. 317, p. 223-256Conference paper (Refereed)
    Abstract [en]

    We derive conditions for an arbitrary n-dimensional algebra to be a Hom-Lie algebra, in the form of a system of polynomial equations, containing both structure constants of the skew-symmetric bilinear map and constants describing the twisting linear endomorphism. The equations are linear in the constants representing the endomorphism and non-linear in the structure constants. When the algebra is 3 or 4-dimensional we describe the space of possible endomorphisms with minimum dimension. For the 3-dimensional case we give families of 3-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphism constructed upto isomorphism together with non-isomorphic canonical representatives for all the families in that case. We further give a list of 4-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphisms. © Springer Nature Switzerland AG 2020.

  • 35.
    Schwieger, Kay
    et al.
    iteratec GmbH, DEU.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    An Atiyah Sequence for Noncommutative Principal Bundles2022In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 18, p. 1-22Article in journal (Refereed)
    Abstract [en]

    We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given ∗∗-automorphism on the quantum base space lifts to a ∗∗-automorphism on the quantum total space that commutes with the underlying structure group.

  • 36.
    Schwieger, Kay
    et al.
    iteratec GmbH, DEU.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Lifting spectral triples to noncommutative principal bundles2022In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 396, article id 108160Article in journal (Refereed)
    Abstract [en]

    Given a free action of a compact Lie group G on a unital C*-algebra A and a spectral triple on the corresponding fixed point algebra A^G, we present a systematic and in-depth construction of a spectral triple on A that is build upon the geometry of A^G and G. We compare our construction with a selection of established examples. 

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  • 37.
    Schwieger, Kay
    et al.
    iteratec GmbH, GER.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Noncommutative coverings of quantum tori2020In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 126, no 1, p. 99-116Article in journal (Refereed)
    Abstract [en]

    We investigate a framework for coverings of noncommutative spaces. Furthermore, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

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    Noncommutative coverings of quantum tori
  • 38.
    Schwieger, Kay
    et al.
    Iteratec GmbH, GER.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Part III, Free Actions of Compact Quantum Groups on C*-Algebras2017In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 13, article id 62Article in journal (Refereed)
    Abstract [en]

    We study and classify free actions of compact quantum groups on unital C*-algebras in terms of generalized factor systems. Moreover, we use these factor systems to show that all finite coverings of irrational rotation C*-algebras are cleft.

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  • 39.
    Tarizadeh, Abolfazl
    et al.
    University of Maragheh, IRN.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Homogeneity in commutative graded ringsManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper, we establish several new results on commutative G-graded rings where G is a totally ordered abelian group. McCoy’s theorem and Armendariz’ theorem are classical results in the theory of polynomial rings. We generalize both of these celebrated theorems to the more general setting of G-graded rings and simultaneously to the setting of ideals rather than to that of elements. Next, we give a complete characterization of invertible elements in G-graded rings. We generalize Bergman’s famous theorem (which asserts that the Jacobson radical of a Z-graded ring is a graded ideal) to the setting of G-graded rings and then proceed to give a natural and quite elementary proof of it. This generalization allows us to show that an abelian group is a totally ordered group if and only if the Jacobson radical of every ring graded by that group is a graded ideal, or equivalently, nonzero idempotents of every ring graded by that group are homogeneous of degree zero. Finally, some topological aspects of graded prime ideals are investigated.

  • 40.
    Tumwesigye, Alex Behakanira
    et al.
    Makerere University, UGA.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Silvestrov, Sergei
    Mälardalens Högskola, SWE.
    Centralizers in pbw extensions2020In: Springer Proceedings in Mathematics and Statistics / [ed] Silvestrov S.,Malyarenko A.,Rancic M., Springer , 2020, Vol. 317, p. 469-490Conference paper (Refereed)
    Abstract [en]

    In this article we give a description for the centralizer of the coefficient ring R in the skew PBW extension σ (R)<x1, x2, ..., xn>. We give an explicit description in the quasi-commutative case and state a necessary condition in the general case. We also consider the PBW extension σ (A)<x1, x2, ..., xn> of the algebra of functions with finite support on a countable set, describing the centralizer of A and the center of the skew PBW extension. © Springer Nature Switzerland AG 2020.

  • 41.
    Tumwesigye, Alex Behakanira
    et al.
    Makerere University, UGA.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Silvestrov, Sergei
    Mälardalens Högskola, SWE.
    Commutants in crossed product algebras for piecewise constant functions on the real line2020In: Springer Proceedings in Mathematics and Statistics / [ed] Silvestrov S.,Malyarenko A.,Rancic M., Springer , 2020, Vol. 317, p. 427-444Conference paper (Refereed)
    Abstract [en]

    In this paper we consider commutants in crossed product algebras, for algebras of piece-wise constant functions on the real line acted on by the group of integers Z. The algebra of piece-wise constant functions does not separate points of the real line, and interplay of the action with separation properties of the points or subsets of the real line by the function algebra become essential for many properties of the crossed product algebras and their subalgebras. In this article, we deepen investigation of properties of this class of crossed product algebras and interplay with dynamics of the actions. We describe the commutants and changes in the commutants in the crossed products for the canonical generating commutative function subalgebras of the algebra of piece-wise constant functions with common jump points when arbitrary number of jump points are added or removed in general positions, that is when corresponding constant value set partitions of the real line change, and we give complete characterization of the set difference between commutants for the increasing sequence of subalgebras in crossed product algebras for algebras of functions that are constant on sets of a partition when partition is refined. © Springer Nature Switzerland AG 2020.

  • 42.
    Tumwesigye, Alex Behakanira
    et al.
    Makerere University, UGA.
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Silvestrov, Sergei
    Mälardalens Högskola, SWE.
    Ore extensions of function algebras2020In: Springer Proceedings in Mathematics and Statistics / [ed] Silvestrov S.,Malyarenko A.,Rancic M., Springer , 2020, Vol. 317, p. 445-467Conference paper (Refereed)
    Abstract [en]

    In this article we consider the Ore extension algebra for the algebra A of functions with finite support on a countable set. We derive explicit formulas for twisted derivations on A, give a description for the centralizer of A, and the center of the Ore extension algebra under specific conditions. © Springer Nature Switzerland AG 2020.

  • 43.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Extending characters of fixed point algebras2018In: Axioms, ISSN 2075-1680, Vol. 7, no 4, article id 79Article in journal (Refereed)
    Abstract [en]

    A dynamical system is a triple (A, G, α) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism α: G → Aut(A) that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units A× is open in A and the inversion map i: A× → A×, a → a-1 is continuous at 1A. Given a dynamical system (A, G, α) with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A. © 2018 by the authors.

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  • 44.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences. Blekinge Tekniska Hgsk, Karlskrona, Sweden..
    SECONDARY CHARACTERISTIC CLASSES OF LIE ALGEBRA EXTENSIONS2019In: Bulletin de la Société Mathématique de France, ISSN 0037-9484, E-ISSN 2102-622X, Vol. 147, no 3, p. 443-453Article in journal (Refereed)
    Abstract [en]

    We introduce the notion of secondary characteristic classes of Lie algebra extensions. As an application of our construction we obtain a new proof of Lecomte's generalization of the Chern-Weil homomorphism.

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  • 45.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Bimodules in Group Graded Rings2017In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, no 6, p. 1483-1494Article in journal (Refereed)
    Abstract [en]

    In this article we introduce the notion of a controlled group graded ring. Let G be a group, with identity element e, and let R = aS center dot (gaG) R (g) be a unital G-graded ring. We say that R is G-controlled if there is a one-to-one correspondence between subsets of the group G and (mutually non-isomorphic) R (e) -sub-bimodules of R, given by G aSc Ha dagger broken vertical bar aS center dot (haH) R (h) . For strongly G-graded rings, the property of being G-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general G-graded ring to be G-controlled. We also give a characterization of strongly G-graded rings which are G-controlled. As an application of our main results we give a description of all intermediate subrings T with R (e) aS dagger T aS dagger R of a G-controlled strongly G-graded ring R. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras.

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  • 46.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Units, zero-divisors and idempotents in rings graded by torsion-free groups2024In: Journal of group theroy, ISSN 1433-5883, E-ISSN 1435-4446Article in journal (Refereed)
    Abstract [en]

    The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.

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  • 47.
    Öinert, Johan
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Complex group rings and group C ∗ -algebras of group extensions2023In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 58, no 2, p. 387-397Article in journal (Refereed)
    Abstract [en]

    Let N and H be groups, and let G be an extension of H by N. In this article, we describe the structure of the complex group ring of G in terms of data associated with N and H. In particular, we present conditions on the building blocks N and H guaranteeing that G satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison–Kaplansky conjecture holds for the group C∗-algebra of G. © 2022, The Author(s).

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