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  • 1.
    Bäck, Per
    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.
    Hom-associative Ore extensions2018In: Journal of Physics, Conference Series, ISSN 1742-6588, E-ISSN 1742-6596, Vol. 965, no 1, article id 012006Article in journal (Refereed)
    Abstract [en]

    We introduce hom-associative Ore extensions as non-associative, non-unital Ore extensions with a hom-associative multiplication, as well as give some necessary and sufficient conditions when such exist. Within this framework, we also construct a family of hom-associative Weyl algebras as generalizations of the classical analogue, and prove that they are simple.

  • 2.
    Bäck, Per
    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.
    Hom-associative Ore extensions and weak unitalizations2018In: International Electronic Journal of Algebra, ISSN 1306-6048, E-ISSN 1306-6048, Vol. 24, p. 174-194Article in journal (Refereed)
    Abstract [en]

    We introduce hom-associative Ore extensions as non-unital, nonassociative Ore extensions with a hom-associative multiplication, and give some necessary and sucient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebra, which we call a weak unitalization.

  • 3.
    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. 

  • 4.
    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.

  • 5.
    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.

  • 6.
    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.

  • 7.
    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.

  • 8.
    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.

  • 9.
    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.

  • 10.
    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.

  • 11.
    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 (Refereed)
  • 12.
    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.))
1 - 12 of 12
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