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  • 51.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Valued Custom Skew Fields with Generalised PBW Property from Power Series Construction2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016, p. 33-55Chapter in book (Refereed)
    Abstract [en]

    This chapter describes a construction of associative algebras that, despite starting from a commutation relation that the user may customize quite considerably, still manages to produce algebras with a number of useful properties: they have a Poincaré–Birkhoff–Witt type basis, they are equipped with a norm (actually an ultranorm) that is trivial to compute for basis elements, they are topologically complete, and they satisfy their given commutation relation. In addition, parameters can be chosen so that the algebras will in fact turn out to be skew fields and the norms become valuations. The construction is basically that of a power series algebra with given commutation relation, stated to be effective enough that the other properties can be derived. What is worked out in detail here is the case of algebras with two generators, but only the analysis of the commutation relation is specific for that case.

  • 52.
    Hellström, Lars
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Makhlouf, Abdenacer
    University of Haute Alsace, LMIA.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Universal Algebra Applied to Hom-Associative Algebras, and More2014In: Algebra, Geometry, and Mathematical Physics: AGMP, Mulhouse, France, October 2011, Springer Berlin/Heidelberg, 2014, p. 157-199Conference paper (Refereed)
    Abstract [en]

    The purpose of this paper is to discuss the universal algebra theory of hom-algebras. This kind of algebra involves a linear map which twists the usual identities. We focus on hom-associative algebras and hom-Lie algebras for which we review the main results. We discuss the envelopment problem, operads, and the Diamond Lemma; the usual tools have to be adapted to this new situation. Moreover we study Hilbert series for the hom-associative operad and free algebra, and describe them up to total degree equal 8 and 9 respectively.

  • 53.
    Hellström, Lars
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei D.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Commuting Elements in Q-Deformed Heisenberg Algebras2000Book (Refereed)
    Abstract [en]

    Noncommutative algebras, rings and other noncommutative objects, along with their more classical commutative counterparts, have become a key part of modern mathematics, physics and many other fields. The q-deformed Heisenberg algebras defined by deformed Heisenberg canonical commutation relations of quantum mechanics play a distinguished role as important objects in pure mathematics and in many applications in physics. The structure of commuting elements in an algebra is of fundamental importance for its structure and representation theory as well as for its applications. The main objects studied in this monograph are q-deformed Heisenberg algebras — more specifically, commuting elements in q-deformed Heisenberg algebras.

    In this book the structure of commuting elements in q-deformed Heisenberg algebras is studied in a systematic way. Many new results are presented with complete proofs. Several appendices with some general theory used in other parts of the book include material on the Diamond lemma for ring theory, a theory of degree functions in arbitrary associative algebras, and some basic facts about q-combinatorial functions over an arbitrary field. The bibliography contains, in addition to references on q-deformed Heisenberg algebras, some selected references on related subjects and on existing and potential applications.

    The book is self-contained, as far as proofs and the background material are concerned. In addition to research and reference purposes, it can be used in a special course or a series of lectures on the subject or as complementary material to a general course on algebra. Specialists as well as doctoral and advanced undergraduate students in mathematics and physics will find this book useful in their research and study.

    Contents:

    • Immediate Consequences of the Commutation Relations
    • Bases and Normal Form in H(q) and H(q,J)
    • Degree in and Gradation of H(q,J)
    • Centralisers of Elements in H(q,J)
    • Centralisers of Elements in H(q)
    • Algebraic Dependence of Commuting Elements in H(q) and H(q,J)
    • Representations of H(q,J) by q-Difference Operators
    • The Diamond Lemma
    • Degree Functions and Gradations
    • q-Special Combinatorics

    Readership: Researchers, graduate students and undergraduates in algebra and physics.

  • 54.
    Kitouni, Abdenour
    et al.
    Univ Haute Alsace, France.
    Makhlouf, Abdenacer
    Univ Haute Alsace, France.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    On (n+1)-Hom-Lie algebras induced by n-Hom-Lie algebras2016In: Georgian Mathematical Journal, ISSN 1072-947X, E-ISSN 1572-9176, Vol. 23, no 1, p. 75-95Article in journal (Refereed)
    Abstract [en]

    The purpose of this paper is to study the relationships between an n-Hom-Lie algebra and its induced (n + 1)-Hom-Lie algebra. We provide an overview of the theory and explore structure properties such as ideals, centers, derived series, solvability, nilpotency, central extensions, and the cohomology.

  • 55.
    Larsson, Daniel
    et al.
    Uppsala University.
    Sigurdsson, Gunnar
    Royal Institute of Technology (KTH).
    Silvestrov, Sergei D.
    Mälardalen University, School of Education, Culture and Communication.
    Quasi-Lie deformations on the algebra F[t]/(tn)2008In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, no 3, p. 201-205Article in journal (Refereed)
    Abstract [en]

    This paper explores the quasi-deformation scheme devised by Hartwig, Larsson and Silvestrov as applied to the simple Lie algebra sl2(F). One of the main points of this method is that the quasi-deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when the quasi-deformation method is applied to sl2(F) via representations by twisted derivations on the algebra F[t]/(tN) one obtains interesting new multi-parameter families of almost quadratic algebras.

  • 56.
    Larsson, Daniel
    et al.
    Uppsala University.
    Silvestrov, Sergei D.
    Lund University.
    On generalized N-complexes coming from twisted derivations2009In: Generalized Lie theory in mathematics, physics and beyond / [ed] Silvestrov, S.D.; Paal, E.; Abramov, V.; Stolin, A., Berlin, Heidelberg: Springer , 2009, p. 81-88Chapter in book (Refereed)
    Abstract [en]

    Inspired by a result of V. Abramov on q-differential graded algebras, we prove a theorem, analogous to Abramov's result but in a slightly different set-up, using a σ- (twisted) derivation as the differential-like map. As an application, we construct a generalized N-complex based on the ring of Laurent polynomials.

  • 57. Laustsen, Niels Jakob
    et al.
    Silvestrov, S. D.
    Lund University.
    Heisenberg-Lie commutation relations in Banach algebras2009In: Mathematical Proceedings of the Royal Irish Academy, ISSN 1393-7197, E-ISSN 2009-0021, Vol. 109, no 2, p. 163-186Article in journal (Refereed)
  • 58.
    Lundengård, Karl
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Generalized Vandermonde matrices and determinants in electromagnetic compatibility2017Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility.

    In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.

    The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.

    The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.

  • 59.
    Lundengård, Karl
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Österberg, Jonas
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Optimization of the Determinant of the Vandermonde Matrix and Related Matrices2018In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 20, no 4, p. 1417-1428Article in journal (Refereed)
    Abstract [en]

    The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Grobner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.

  • 60.
    Makhlouf, Abdenacer
    et al.
    Université de Haute Alsace, Mulhouse, France.
    Paal, EugeneTallinn University of Technology, Estonia.Silvestrov, SergeiMälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.Stolin, AlexanderGöteborg Universitet, Sweden.
    Algebra, Geometry and Mathematical Physics: Proceedings of the AGMP, Mulhouse, France, October 20112014Collection (editor) (Refereed)
    Abstract [en]

    This book collects the proceedings of the Algebra, Geometry and Mathematical Physics Conference, held at the University of Haute Alsace, France, October 2011. Organized in the four areas of algebra, geometry, dynamical symmetries and conservation laws and mathematical physics and applications, the book covers deformation theory and quantization; Hom-algebras and n-ary algebraic structures; Hopf algebra, integrable systems and related math structures; jet theory and Weil bundles; Lie theory and applications; non-commutative and Lie algebra and more.

    The papers explore the interplay between research in contemporary mathematics and physics concerned with generalizations of the main structures of Lie theory aimed at quantization, and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, non-commutative geometry and applications in physics and beyond.

  • 61.
    Makhlouf, Abdenacer
    et al.
    Université de Haute Alsace, France.
    Silvestrov, Sergei
    Lund University.
    Hom-algebras and Hom-coalgebras2010In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 9, no 4, p. 553-589Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating HomLie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and HomHopf algebras, and prove some fundamental properties and give examples. Finally, we define the concept of HomLie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.

  • 62. Makhlouf, Abdenacer
    et al.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras2009In: Generalized Lie theory in mathematics, physics and beyond / [ed] Silvestrov, S.D.; Paal, E.; Abramov, V.; Stolin, A, Berlin Heidelberg: Springer , 2009, p. 189-206Chapter in book (Refereed)
    Abstract [en]

    The aim of this paper is to develop the coalgebra counterpart of the notions introduced by the authors in a previous paper, we introduce the notions of Hom-coalgebra, Hom-coassociative coalgebra and G-Hom-coalgebra for any subgroup G of permutation group S3. Also we extend the concept of Lie-admissible coalgebra by Goze and Remm to Hom-coalgebras and show that G-Hom-coalgebras are Hom-Lie admissible Hom-coalgebras, and also establish duality correspondence between classes of G-Hom-coalgebras and G-Hom-algebras. In another hand, we provide relevant definitions and basic properties of Hom-Hopf algebras generalizing the classical Hopf algebras and define the module and comodule structure over Hom-associative algebra or Hom-coassociative coalgebra.

  • 63.
    Makhlouf, Abdenacer
    et al.
    Université de Haute Alsace, France.
    Silvestrov, Sergei
    Lund University.
    Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras2010In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 22, no 4, p. 715-739Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to extend to Hom-algebra structures the theory of 1-parameter formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis and Richardson. In this paper, formal deformations of Hom-associative and Hom-Lie algebras are studied. The first groups of a deformation cohomology are constructed and several examples of deformations are given. We also provide families of Hom-Lie algebras deforming Lie algebra 2(K) and describe as formal deformations the q-deformed Witt algebra and Jackson 2(K).

  • 64. Makhlouf, Abdenacer
    et al.
    Silvestrov, Sergei D.
    Lund university.
    Hom-algebra structures2008In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, no 2, p. 51-64Article in journal (Refereed)
    Abstract [en]

    A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov in [4] and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras in [5, 6]. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic structures which generalize the well known associative, Leibniz and Lie admissible algebras. Also, we characterize the flexible Hom-algebras in this case. We also explain some connections between Hom-Lie algebras and Santilli’s isotopies of associative and Lie algebras.

  • 65.
    Mitrovic, Melanija
    et al.
    University of Nis, Serbia.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Regular subsets and semilattice decompositions of semigroups. Hereditarness and periodicity2019In: Logic and Applications LAP 2019 / [ed] Zvonimir Sikic, Andre Scedrov, Silvia Ghilezan, Zogan Ognjanovic, Thomas Studer, Inter University Centre Dubrovnic , 2019, p. 31-33Conference paper (Refereed)
  • 66.
    Mitrovic, Melanija
    et al.
    Faculty of Mechanical Engineering, University of Nis, Serbia.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Crvenkovic, S
    Department of Mathematics and Informatics, University of Novi Sad, Serbia.
    Romano, D. A.
    International Mathematical Virtual Institute, Banja Luka, Republic of Srpska, Bosnia and Herzegovina.
    Constructive Semigroups with Apartness: Towards a New Algebraic Theory2019In: Journal of Physics: Conference Series, Vol. 1194, no 1, p. 1-8, article id 012076Article in journal (Refereed)
    Abstract [en]

    The theory of constructive semigroups with apartness is a new approach tosemigroup theory, and not a new class of semigroups. Of course, our work is partly inspired byclassical semigroup theory, but, on the other hand, it is distinguished from it by two signicantaspects: rst, we use intuitionistic logic rather than classical, secondly, our work is based onthe notion of apartness (between elements, elements and sets). Here, the focus is on E. Bishop'sapproach to constructive mathematics (BISH)

  • 67.
    Musonda, John
    Mälardalen University, School of Education, Culture and Communication. University of Zambia.
    Reordering in Noncommutative Algebras, Orthogonal Polynomials and Operators2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The main object studied in this thesis is the multi-parametric family of unital associative complex algebras generated by the element $Q$ and the finite or infinite set $\{S_j\}_{j\in J}$ of elements satisfying the commutation relations $S_jQ=\sigma_j(Q)S_j$, where $\sigma_j$ is a polynomial for all $j\in J$. A concrete representation is given by the operators $Q_x(f)(x)=xf(x)$ and $\alpha_{\sigma_j}(f)(x)=f(\sigma_j(x))$ acting on polynomials or other suitable functions. The main goal is to reorder arbitrary elements in this family and some of its generalizations, and to study properties of operators in some representing operator algebras, including their connections to orthogonal polynomials. For $J=\{1\}$ and $\sigma(x)=x+1$, the above commutation relations reduce to the famous classical Heisenberg--Lie commutation relation $SQ-QS=S$. Reordering an element in $S$ and $Q$ means to bring it, using the commutation relation, into a form where all elements $Q$ stand either to the left or to the right. For example, $SQ^2=Q^2S+2QS+S$. In general, one can use the commutation relation $SQ-QS=S$ successively and transform for any positive integer $n$ the element $SQ^n$ into a form where all elements $Q$ stand to the left. The coefficients which appear upon reordering in this case are the binomial coefficients. General reordering formulas for arbitrary elements in noncommutative algebras defined by commutation relations are important in many research directions, open problems and applications of the algebras and their operator representations. In investigation of the structure, representation theory and applications of noncommutative algebras, an important role is played by the explicit description of suitable normal forms for noncommutative expressions or functions of generators. Further investigation of the operator representations of the commutation relations by difference type operators on Hilbert function spaces leads to interesting connections to functional analysis and orthogonal polynomials.

     

    This thesis consists of two main parts. The first part is devoted to the multi-parametric family of algebras introduced above. General reordering formulas for arbitrary elements in this family are derived, generalizing some well-known results. As an example of an application of the formulas, centralizers and centers are computed. Some operator representations of the above algebras are also described, including considering them in the context of twisted derivations. The second part of this thesis is devoted to a special representation of these algebras by difference operators associated with action by shifts on the complex plane. It is shown that there are three systems of orthogonal polynomials of the class of Meixner--Pollaczek polynomials that are connected by these operators. Boundedness properties of two singular integral operators of convolution type connected to these difference operators are investigated in the Hilbert spaces related to these systems of orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on the $L^2$-spaces and estimates of the norms are obtained. This investigation is also extended to $L^p$-spaces on the real line where it is proved again that the two operators are bounded.

  • 68.
    Musonda, John
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics and Statistics, University of Zambia, Zambia.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Reordering in a multi-parametric family of algebras2019In: Journal of Physics: Conference Series, Institute of Physics Publishing , 2019, no 1Conference paper (Refereed)
    Abstract [en]

    This article is devoted to a multi-parametric family of associative complex algebras defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems. General reordering and nested commutator formulas for arbitrary elements in these families are presented, generalizing some well-known results in mathematics and physics. A generalization of this family in three generators is also considered.

  • 69.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Sweden.
    Öinert, Johan
    Blekinge Tekniska Högskola, Sweden.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    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; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right Rδ-linear, where Rδ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the nonassociative differential polynomial ring D = R[X; idR, δ]. 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) = Rδ[p] for a monic p ∈ Rδ[X], unique up to addition of elements from Z(R)δ. 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 δ-simple and Z(D) equals the field Rδ ∩ Z(R). This provides us with a non-associative generalization of a result by Öinert, Richter and Silvestrov. 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 Rδ ∩ 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. 

  • 70.
    Nystedt, Patrik
    et al.
    Department of Engineering Science, University West, Trollhättan, Sweden.
    Öinert, Johan
    Blekinge Tekniska Högskola.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    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.

  • 71.
    Ongong'a, Elvice
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. University of Nairobi, Nairobi, Kenya .
    Ongong'a, Elvice
    University of Nairobi, Nairobi, Kenya .
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Hom-Lie structures on 3-dimensional skew symmetric algebras2019In: Journal of Physics, Conference Series, ISSN 1742-6588, E-ISSN 1742-6596, Vol. 1416, no 1, p. 1-8, article id 012025Article in journal (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.

  • 72.
    Ongong'A, Elvice
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. University of Nairobi, Nairobi, Kenya .
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Classification of 3-dimensional Hom-Lie algebras2019In: Journal of Physics: Conference Series, Institute of Physics Publishing , 2019, Vol. 1194, no 1, article id 012084Conference paper (Refereed)
    Abstract [en]

    For any n-dimensional Hom-Lie algebra, a system of polynomial equations is obtained from the Hom-Jacobi identity, 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. The space of possible endomorphisms has minimum dimension 6, and we describe the possible endomorphisms in that case. We further give families of 3-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphism constructed upto isomorphism.

  • 73.
    Ostrovski\u\i, V. L.
    et al.
    Mathematics Institute, Academy of Sciences of Ukraine, Kiev, USSR; Kiev University, USSR.
    Silvestrov, S. D.
    Mathematics Institute, Academy of Sciences of Ukraine, Kiev, USSR; Kiev University, USSR.
    Representations of real forms of the graded analogue of a Lie algebra1992In: Ukraïn. Mat. Zh., ISSN 0041-6053, Vol. 44, no 11, p. 1518-1524Article in journal (Refereed)
    Abstract [en]

    Real-valued forms of the ℤ2n -graded analogue of the Lie algebra s(2,C) are described and their irreducible representations are studied.

  • 74.
    Persson, T.
    et al.
    Lund University.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Commuting Operators for Representations of Commutation Relations Defined by Dynamical Systems2012In: Numerical Functional Analysis and Optimization, ISSN 0163-0563, Vol. 33, no 7-9, p. 1126-1165Article in journal (Refereed)
    Abstract [en]

    In this article, using orbits of the dynamical system generated by the function F, operator representations of commutation relations XX*=F(X*X) and AB=BF(A) are studied and used to investigate commuting operators expressed using polynomials in A and B. Various conditions on the function F, defining the commutation relations, are derived for monomials and polynomials in operators A and B to commute. These conditions are further studied for dynamical systems generated by affine and q-deformed power functions, and for the -shift dynamical system.

  • 75.
    Radeschnig, David
    Mälardalen University, School of Education, Culture and Communication.
    Modelling Implied Volatility of American-Asian Options: A Simple Multivariate Regression Approach2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This report focus upon implied volatility for American styled Asian options, and a least squares approximation method as a way of estimating its magnitude. Asian option prices are calculated/approximated based on Quasi-Monte Carlo simulations and least squares regression, where a known volatility is being used as input. A regression tree then empirically builds a database of regression vectors for the implied volatility based on the simulated output of option prices. The mean squared errors between imputed and estimated volatilities are then compared using a five-folded cross-validation test as well as the non-parametric Kruskal-Wallis hypothesis test of equal distributions. The study results in a proposed semi-parametric model for estimating implied volatilities from options. The user must however be aware of that this model may suffer from bias in estimation, and should thereby be used with caution.

  • 76. Richard, Lionel
    et al.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    A note on quasi-Lie and Hom-Lie structures of σ-derivations of C[z1±1, \dots, zn±1]2009In: Generalized Lie theory in mathematics, physics and beyond / [ed] Silvestrov, S.D.; Paal, E.; Abramov, V.; Stolin, A., Springer, 2009, p. 257-262Chapter in book (Refereed)
    Abstract [en]

    In a previous paper we studied the properties of the bracket defined by Hartwig, Larsson and the second author in (J. Algebra 295, 2006) on σ-derivations of Laurent polynomials in one variable. Here we consider the case of several variables, and emphasize on the question of when this bracket defines a hom-Lie structure rather than a quasi-Lie one.

  • 77.
    Richard, Lionel
    et al.
    University of Edinburgh.
    Silvestrov, Sergei D.
    Lund University.
    Quasi-Lie structure of σ-derivations of C[t±1]2008In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 319, no 3, p. 1285-1304Article in journal (Refereed)
    Abstract [en]

    Hartwig, Larsson and Silvestrov in [J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2) (2006) 314–361] defined a bracket on σ-derivations of a commutative algebra. We show that this bracket preserves inner derivations, and based on this obtain structural results providing new insights into σ-derivations on Laurent polynomials in one variable.

  • 78.
    Richter, J.
    et al.
    Lunds universitet.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Burchnall-Chaundy annihilating polynomials for commuting elements in Ore extension rings2012In: Journal of Physics: Conference Series, ISSN 1742-6588, Vol. 346, no 1Article in journal (Refereed)
    Abstract [en]

    In this article further progress is made in extending the Burchnall-Chaundy type determinant construction of annihilating polynomial for commuting elements to broader classes of rings and algebras by deducing an explicit general formula for the coefficients of the annihilating polynomial obtained by the Burchnall-Chaundy type determinant construction in Ore extension rings. It is also demonstrated how this formula can be used to compute the annihilating polynomials in several examples of commuting elements in Ore extensions. Also it is demonstrated that additional properties which may be possessed by the endomorphism, such as for example injectivity, may influence strongly the annihilating polynomial.

  • 79.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Lund University, Sweden.
    Burchnall-Chaundy Theory for Ore Extensions2014In: Algebra, Geometry and Mathematical Physics / [ed] Makhlouf, Abdenacer; Paal, Eugen; Silvestrov, Sergei D.; Stolin, Alexander, Springer Berlin Heidelberg , 2014, Vol. 85, p. 61-70Chapter in book (Refereed)
    Abstract [en]

    We begin by reviewing a classical result on the algebraic dependence of commuting elements in the Weyl algebra. We proceed by describing generalizations of this result to various classes of Ore extensions, including both results that are already known and one new result.

  • 80.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Centralizers and Pseudo-Degree Functions2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Silvestrov, Sergei and Rančić, Milica, Springer, 2016, p. 65-73Chapter in book (Refereed)
    Abstract [en]

    This paper generalizes a proof of certain results by Hellström and Silvestrov on centralizers in graded algebras. We study centralizers in certain algebras with valuations. We prove that the centralizer of an element in these algebras is a free module over a certain ring. Under further assumptions we obtain that the centralizer is also commutative.

  • 81.
    Richter, Johan
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Centralizers in Ore extensions over polynomial rings2014In: International Electronic Journal of Algebra, ISSN 1306-6048, E-ISSN 1306-6048, Vol. 15, p. 15p. 196-207Article in journal (Refereed)
    Abstract [en]

    In this paper we consider centralizers of single elements in Ore extensions of the ring of polynomials in one variable over a field. We show that they are commutative and finitely generated as an algebra. We also show that for certain classes of elements their centralizer is singly generated as an algebra.

  • 82.
    Richter, Johan
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Computing Burchnall–Chaundy Polynomials with Determinants2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov, Milica Rančić, Springer, 2016, p. 57-63Chapter in book (Refereed)
    Abstract [en]

    In this expository paper we discuss a way of computing the Burchnall-Chaundy polynomial of two commuting differential operators using a determinant.We describe how the algorithm can be generalized to general Ore extensions, andwhich properties of the algorithm that are preserved.

  • 83.
    Richter, Johan
    et al.
    Lund University.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication. Lund University.
    On algebraic curves for commuting elements in $q$-Heisenberg algebras2009In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 3, no 4, p. 321-328Article in journal (Refereed)
    Abstract [en]

    In the present article we continue investigating the algebraic dependence of commutingelements in q-deformed Heisenberg algebras. We provide a simple proof that the0-chain subalgebra is a maximal commutative subalgebra when q is of free type and thatit coincides with the centralizer (commutant) of any one of its elements dierent fromthe scalar multiples of the unity. We review the Burchnall-Chaundy-type construction forproving algebraic dependence and obtaining corresponding algebraic curves for commutingelements in the q-deformed Heisenberg algebra by computing a certain determinantwith entries depending on two commuting variables and one of the generators. The coecients in front of the powers of the generator in the expansion of the determinant arepolynomials in the two variables dening some algebraic curves and annihilating the twocommuting elements. We show that for the elements from the 0-chain subalgebra exactlyone algebraic curve arises in the expansion of the determinant. Finally, we present severalexamples of computation of such algebraic curves and also make some observations onthe properties of these curves.

  • 84.
    Richter, Johan
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ssembatya, Vincent
    Makerere University.
    Tumwesigye, Alex
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Crossed Product Algebras for Piece-Wise Constant Functions2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov; Milica Rančić, Springer, 2016Chapter in book (Refereed)
    Abstract [en]

    In this paper we consider algebras of functions that are constant on the sets of a partition. We describe the crossed product algebras of the mentioned algebras with Z. We show that the function algebra is isomorphic to the algebra of all functions on some set. We also describe the commutant of the function algebra and finish by giving an example of piece-wise constant functions on a real line.

  • 85.
    Richter, Johan
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Tumwesigye, Alex
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Commutants in Crossed Product Algebras for Piece-Wise Constant Functions2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and OptimizationEditors / [ed] Sergei Silvestrov; Milica Rančić, Springer, 2016, p. 95-108Chapter in book (Refereed)
    Abstract [en]

    In this paper we consider crossed product algebras of algebras of piecewiseconstant functions on the real line with Z. For an increasing sequence of algebras (in which case the commutants form a decreasing sequence), we describe the set difference between the corresponding commutants.

  • 86.
    Sigurdsson, Gunnar
    et al.
    KTH.
    Silvestrov, S. D.
    Mälardalen University, School of Education, Culture and Communication.
    Matrix bosonic realizations of a Lie colour algebra with three generators and five relations of Heisenberg Lie type2009In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 3, no 4, p. 329-340Article in journal (Refereed)
    Abstract [en]

    We describe realizations of a Lie colour algebra with three generators and five relations by matrices of power series in noncommuting indeterminates satisfying Heisenberg's canonical commutation relation of quantum mechanics. The obtained formulas are used to construct new operator representations of this Lie colour algebra using canonical representation of the Heisenberg commutation relation and creation and annihilation operators of the quantum mechanical harmonic oscillator.

  • 87.
    Sigurdsson, Gunnar
    et al.
    KTH.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Lie color and Hom-Lie algebras of Witt type and their central extensions2009In: Generalized Lie theory in mathematics, physics and beyond, Berlin, Heidelberg: Springer Berlin/Heidelberg, 2009, p. 247-255Chapter in book (Refereed)
    Abstract [en]

    In this article, two classes of Г-graded Witt-type algebras, Lie color and hom-Lie algebras of Witt type, are considered. These algebras can be seen as generalizations of Lie algebras of Witt type. One-dimensional central extensions of Lie color and hom-Lie algebras of Witt type are investigated.

  • 88.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Foundations of Santilli's Lie-Isotopic Theory2010In: Numerical Analysis and Applied Mathematics (AIP Conference Proceedings), American Institute of Physics (AIP), 2010, p. 860-863Conference paper (Refereed)
    Abstract [en]

    The need and benefits of new universal mathematical and especially algebraic and operator structures and methods in all branches and levels of modem physics from experiments to fundamentals is nowdays widely recognized. The far reaching pioneering discoveries of R. M. Santilli extend significantly the Lie analysis and operator methods and models of modem physics opening new areas of applications with possible essential improvements and revisions in models in particle physics and cosmology. In this article, some fundamental aspects of Lie-Santilli isotopic theory and relevant references are presented.

  • 89.
    Silvestrov, Sergei
    et al.
    Lund University.
    Paal, EugenTallinn University of Technology, Estonia.Abramov, ViktorUniversity of Tartu, Estonia.Stolin, AlexanderChalmers.
    Generalized Lie theory in mathematics, physics and beyond2009Collection (editor) (Refereed)
  • 90.
    Silvestrov, Sergei
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Rancic, MilicaMälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization2016Collection (editor) (Refereed)
    Abstract [en]

    This book highlights the latest advances in engineering mathematics with a main focus on the mathematical models, structures, concepts, problems and computational methods and algorithms most relevant for applications in modern technologies and engineering. It addresses mathematical methods of algebra, applied matrix analysis, operator analysis, probability theory and stochastic processes, geometry and computational methods in network analysis, data classification, ranking and optimisation.

    The individual chapters cover both theory and applications, and include a wealth of figures, schemes, algorithms, tables and results of data analysis and simulation. Presenting new methods and results, reviews of cutting-edge research, and open problems for future research, they equip readers to develop new mathematical methods and concepts of their own, and to further compare and analyse the methods and results discussed.

    The book consists of contributed chapters covering research developed as a result of a focused international seminar series on mathematics and applied mathematics and a series of three focused international research workshops on engineering mathematics organised by the Research Environment in Mathematics and Applied Mathematics at Mälardalen University from autumn 2014 to autumn 2015: the International Workshop on Engineering Mathematics for Electromagnetics and Health Technology; the International Workshop on Engineering Mathematics, Algebra, Analysis and Electromagnetics; and the 1st Swedish-Estonian International Workshop on Engineering Mathematics, Algebra, Analysis and Applications.

    It serves as a source of inspiration for a broad spectrum of researchers and research students in applied mathematics, as well as in the areas of applications of mathematics considered in the book.

  • 91.
    Silvestrov, Sergei
    et al.
    Mälardalen University, School of Education, Culture and Communication.
    Svensson, Christian
    de Jeu, Marcel
    Algebraic dependence of commuting elements in algebras2009In: Generalized Lie theory in mathematics, physics and beyond / [ed] Silvestrov, S.D.; Paal, E.; Abramov, V.; Stolin, A., Springer Berlin/Heidelberg, 2009, p. 265-280Chapter in book (Refereed)
    Abstract [en]

    The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall—Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall–Chaundy approach from differential operators and the Heisenberg algebra to the q-deformed Heisenberg algebra, showing that the Burchnall—Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the q-deformed Heisenberg algebras for q not a root of unity.

  • 92.
    Svensson, Christian
    et al.
    Leiden University, Netherlands .
    Silvestrov, Sergei
    Lund university.
    de Jeu, Marcel
    Leiden University, Netherlands .
    Dynamical systems associated with crossed products2009In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, ISSN 0167-8019, Vol. 108, no 3, p. 547-559Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A'. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.

  • 93.
    Tumwesigye, Alex Behakanira
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ore extensions of function algebras2018Conference paper (Other (popular science, discussion, etc.))
  • 94.
    Tumwesigye, Alex
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Commutants in crossed products for algebras of piecewise constant functions on the real lineManuscript (preprint) (Other (popular science, discussion, etc.))
  • 95.
    Tumwesigye, Alex
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    On monomial commutativity of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems2014In: 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014 Conference date: 15–18 July 2014 Location: Narvik, Norway ISBN: 978-0-7354-1276-7 Editor: Seenith Sivasundaram Volume number: 1637 Published: 10 december 2014 / [ed] Seenith Sivasundaram, American Institute of Physics (AIP), 2014, p. 1110-1119Conference paper (Refereed)
    Abstract [en]

    T. Persson and S. D. Sivestrov investigated representations of operators satisfying the relation XX* = F(X*X) in connection with periodic points and orbits of the map F. In particular they derived commutativity conditions for two monomials in operators A and B on a Hilbert space satisfying the relation AB = BF(A). In this article we shall apply their results to special one-dimensional dynamical systems and and give an explicit description of the interplay between periodic orbits of one-dimensional piecewise polynomial maps and commutativity of monomials for special operators A and B. Furthermore we shall apply our results to derive conditions on β for the special case when F β is the β–shift dynamical system.

  • 96.
    Öinert, Johan
    et al.
    Lund University.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Commutativity and ideals in algebraic crossed products2008In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, no 4, p. 287-302Article in journal (Refereed)
    Abstract [en]

    We investigate properties of commutative subrings and ideals in non-commutative algebraic crossed products for actions by arbitrary groups. A description of the commutant of the coefficient subring in the crossed product ring is given. Conditions for commutativity and maximal commutativity of the commutant of the coefficient subring are provided in terms of the action as well as in terms of the intersection of ideals in the crossed product ring with the coefficient subring, specially taking into account both the case of coefficient rings without non-trivial zero-divisors and the case of coefficient rings with non-trivial zero-divisors.

  • 97. Öinert, Johan
    et al.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Crossed product-like and pre-crystalline graded rings2009In: Generalized Lie theory in mathematics, physics and beyond / [ed] Silvestrov, S.D.; Paal, E.; Abramov, V.; Stolin, A., Berlin, Heidelberg: Springer Berlin/Heidelberg, 2009, p. 281-296Chapter in book (Refereed)
    Abstract [en]

    We introduce crossed product-like rings, as a natural generalization of crystalline graded rings, and describe their basic properties. Furthermore, we prove that for certain pre-crystalline graded rings and every crystalline graded ring A, for which the base subring A0 is commutative, each non-zero two-sided ideal has a nonzero intersection with CA(A0), i.e. the commutant of A0 in A. We also show that in general this property need not hold for crossed product-like rings.

  • 98.
    Öinert, Johan
    et al.
    Lund University.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    On a correspondence between ideals and commutativity in algebraic crossed products2008In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, no 3, p. 216-220Article in journal (Refereed)
    Abstract [en]

    In this paper we will give an overview of some recent results which display a connection between commutativity and the ideal structure in algebraic crossed products.

  • 99.
    Öinert, Johan
    et al.
    Lund University.
    Silvestrov, Sergei D.
    Lund University.
    Commutativity and ideals in pre-crystalline graded rings2009In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 108, no 3, p. 603-615Article in journal (Refereed)
    Abstract [en]

    Pre-crystalline graded rings constitute a class of rings which share many properties with classical crossed products. Given a pre-crystalline graded ring A, we describe its center, the commutant CA(A0) of the degree zero grading part, and investigate the connection between maximal commutativity of A0 in A and the way in which two-sided ideals intersect A0.

  • 100.
    Öinert, Johan
    et al.
    Lund University.
    Silvestrov, Sergei
    Lund University.
    Theohari-Apostolidi, Theodora
    Aristotle University of Thessaloniki, Greece .
    Vavatsoulas, Harilaos
    Aristotle University of Thessaloniki, Greece .
    Commutativity and ideals in strongly graded rings2009In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 108, no 3, p. 585-602Article in journal (Refereed)
    Abstract [en]

    In some recent papers by the first two authors it was shown that for any algebraic crossed product A, where A0, the subring in the degree zero component of the grading, is a commutative ring, each non-zero two-sided ideal in A has a non-zero intersection with the commutant CA(A0) of A0 in A. This result has also been generalized to crystalline graded rings; a more general class of graded rings to which algebraic crossed products belong. In this paper we generalize this result in another direction, namely to strongly graded rings (in some literature referred to as generalized crossed products) where the subring A0, the degree zero component of the grading, is a commutative ring. We also give a description of the intersection between arbitrary ideals and commutants to bigger subrings than A0, and this is done both for strongly graded rings and crystalline graded rings.

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