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  • 1.
    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 dynamics2018In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337Article 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.

  • 2.
    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.

  • 3.
    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.

  • 4.
    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.

  • 5.
    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, 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.

  • 6.
    Mai Hoang, Bien
    et al.
    University of Science, VNM.
    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.

  • 7.
    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 doublingsManuscript (preprint) (Other academic)
    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.

  • 8.
    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.

  • 9.
    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.

  • 10.
    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.

  • 11.
    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.

  • 12.
    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.

  • 13.
    Wagner, Stefan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Secondary Characteristic Classes of Lie Algebra ExtensionsIn: Article in journal (Other academic)
    Abstract [en]

    We introduce a notion of secondary characteristic classes of Lie algebra extensions. As a spin-off of our construction we obtain a new proof of Lecomte’s generalization of the Chern–Weil homomorphism.

  • 14.
    Wagner, Stefan
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Schwieger, Kay
    iteratec GmbH, GER.
    Noncommutative Coverings of Quantum ToriIn: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807Article in journal (Other academic)
    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.

  • 15.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences. Blekinge Inst Technol, Dept Math & Nat Sci, S-37179 Karlskrona, Sweden..
    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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