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  • 1.
    Darpö, Erik
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mathematical Institute, St Giles', Oxford, United Kingdom.
    Pérez Izquierdo, José María
    Universidad de la Rioja, Spain.
    Autotopies and quasigroup identities: new aspects of non-associative division algebras2015In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 27, no 5, p. 2691-2745Article in journal (Refereed)
    Abstract [en]

    In this article, we explore new aspects in the classification of non-associative division algebras. By a detailed study of the representations of the Lie group of autotopies of real division algebras we show that, if the group of autotopies has a sufficiently rich structure, then the algebra is isotopic to one of the classical real division algebras. This turns out to be the case for large classes of real division algebras, including many that are defined by identities. In several cases, a classification up to isomorphism can be worked out from this information.

  • 2.
    Elchinger, O.
    et al.
    Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, Mulhouse, France.
    Lundengård, Karl
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Makhlouf, A.
    Laboratoire de Mathématiques Informatique et Applications, Université de Haute Alsace, Mulhouse, France.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras2016In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 28, no 4, p. 657-673Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to study some brackets defined on (τ,σ)-derivations satisfying quasi-Lie identities. Moreover, we provide examples of (p, q)-deformations of Witt and Virasoro algebras as well as sl(2) algebra. These constructions generalize the results obtained by Hartwig, Larsson and Silvestrov on σ-derivations, arising in connection with discretizations and deformations of algebras of vector fields.

  • 3.
    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).

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