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  • 1. Frégier, Yaël
    et al.
    Gohr, Aron
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Unital algebras of Hom-associative type and surjective or injective twistings2009In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 3, no 4, p. 285-295Article in journal (Refereed)
  • 2.
    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.

  • 3. 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.

  • 4.
    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.

  • 5.
    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.

  • 6.
    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.

  • 7.
    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.

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