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  • 1. Abramov, V.
    et al.
    Paal, E.Tallinn University of Technology.Silvestrov, Sergei D.Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.Stolin, A.Chalmers University of Techology.
    Proceedings of the 3rd Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics”2008Conference proceedings (editor) (Refereed)
  • 2.
    Ammar, F.
    et al.
    Faculté des Sciences, Université de Sfax, Tunisia .
    Makhlouf, A.
    Université de Haute Alsace, France .
    Silvestrov, S. D.
    Lund University.
    Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras2010In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 43, no 26, p. 265204-Article in journal (Refereed)
    Abstract [en]

    In this paper we construct ternary q-Virasoro-Witt algebras which q-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using su(1, 1) enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary q-Virasoro-Witt algebras constructed in this paper are also Hom-Nambu-Lie because they are obtained from the ternary Nambu-Lie algebras using the composition method. For other parameter values this composition method does not yield a Hom-Nambu-Lie algebra structure for q-Virasoro-Witt algebras. We show however, using a different construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary q-Virasoro-Witt algebras we construct, carry a structure of the ternary Hom-Nambu-Lie algebra for all values of the involved parameters.

  • 3.
    Andren, Daniel
    et al.
    Umeå University.
    Hellström, Lars
    Umeå University.
    Markström, Klas
    Umeå University.
    Fast multiplication of matrices over a finitely generated semiring2008In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, no 6, p. 230-234Article in journal (Refereed)
  • 4.
    Arnlind, Joakim
    et al.
    Linköping University, Sweden.
    Kitouni, Abdennour
    Université de Haute-Alsace, Mulhouse, France.
    Makhlouf, Abdenacer
    Université de Haute-Alsace, Mulhouse, France .
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras2014In: Springer Proceedings in Mathematics and Statistics, Berlin, Heidelberg: Springer, 2014, Vol. 85, p. 123-144Conference paper (Refereed)
    Abstract [en]

    The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced 3-Lie algebras

  • 5.
    Arnlind, Joakim
    et al.
    Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
    Makhlouf, Abdenacer
    Université de Haute Alsace, Lab. de Mathématiques Informatique et Applications, 4, rue des Frères Lumière, F-68093 Mulhouse, France.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication.
    Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, no 12, p. 123502-Article in journal (Refereed)
    Abstract [en]

    As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3653197]

  • 6.
    Arnlind, Joakim
    et al.
    Max Planck Institute for Gravitational Physics (AEI), Germany.
    Makhlouf, Abdenacer
    Université de Haute Alsace, France .
    Silvestrov, Sergei
    Lund University, Sweden.
    Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, no 4, p. 043515-11Article in journal (Refereed)
    Abstract [en]

    The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom-Nambu-Lie algebras obtained using this construction.

  • 7.
    Arnlind, Joakim
    et al.
    Albert Einstein Institute, Golm, Germany.
    Silvestrov, Sergei
    Lund University.
    Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems: July 21-25 2008, Lorentz Center, Leiden, The Netherlands, satellite conference of the fifth European Congress of Mathematics, Amer. Math. Soc. , 2009, Vol. 503, p. 1-25Chapter in book (Refereed)
  • 8.
    Ataguema, H.
    et al.
    Universit́e de Haute Alsace, France .
    Makhlouf, A.
    Universit́e de Haute Alsace, France .
    Silvestrov, S. D.
    Lund University.
    Generalization of n-ary Nambu algebras and beyond2009In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 50, no 8, p. Article number 083501-Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.

  • 9.
    Bergander, Philip
    Mälardalen University, School of Education, Culture and Communication.
    Twisted derivations, quasi-hom-Lie algebras and their quasi-deformations2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Quasi-hom-Lie algebras (qhl-algebras) were introduced by Larsson and Silvestrov (2004) as a generalisation of hom-Lie algebras, which are a deformation of Lie algebras. Lie algebras are defined by an operation called bracket, [·,·], and a three-term Jacobi identity. By the theorem from Hartwig, Larsson, and Silvestrov (2003), this bracket and the three-term Jacobi identity are deformed into a new bracket operation, <·,·>, and a six-term Jacobi identity, making it a quasi-hom-Lie algebra.

    Throughout this thesis we deform the Lie algebra sl2(F), where F is a field of characteristic 0. We examine the quasi-deformed relations and six-term Jacobi identities of the following polynomial algebras: F[t], F[t]/(t2), F[t]/(t3), F[t]/(t4), F[t]/(t5), F[t]/(tn), where n is a positive integer ≥2, and F[t]/((t-t0)3). Larsson and Silvestrov (2005) and Larsson, Sigurdsson, and Silvestrov (2008) have already examined some of these cases, which we repeat for the reader's convenience.

    We further investigate the following σ-twisted derivations, and how they act in the different cases of mentioned polynomial algebras: the ordinary differential operator, the shifted difference operator, the Jackson q-derivation operator, the continuous q-difference operator, the Eulerian operator, the divided difference operator, and the nilpotent imaginary derivative operator. We also introduce a new, general, σ-twisted derivation operator, which is σ(t) as a polynomial of degree k.

  • 10.
    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.

  • 11.
    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.

  • 12.
    Carlsen, Toke M.
    et al.
    Norwegian University of Science and Technology.
    Eilers, SorenUniversity of Copenhagen.Restorff, GunnarUniversity of the Faroe Islands.Silvestrov, SergeiMälardalen University, School of Education, Culture and Communication.
    Operator Algebra and Dynamics: Nordforsk Network Closing Conference, Faroe Islands, May 20122013Collection (editor) (Refereed)
  • 13.
    Carlsen, Toke Meier
    et al.
    University of Southern Denmark.
    Silvestrov, Sergei
    Lund University.
    On the Exel crossed product of topological covering maps2009In: 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. 573-583Article in journal (Refereed)
    Abstract [en]

    For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C *-algebras C(X)⋊ α,ℒℕintroduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ α,ℒℕis a maximal abelian C *-subalgebra of C(X)⋊ α,ℒN; any nontrivial two sided ideal of C(X)⋊ α,ℒℕhas non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ α,ℒℕis faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C *-algebras of homeomorphism dynamical systems. 

  • 14.
    Cuenca Mira, José Antonio
    et al.
    Universidad de Málaga.
    Darpö, Erik
    Uppsala Universitet.
    Dieterich, Ernst
    Uppsala Universitet.
    Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity2010In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 134, no 3, p. 247-277Article in journal (Refereed)
    Abstract [en]

    An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramirez Alvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent. 

  • 15.
    Darpö, Erik
    Uppsala universitet.
    Classification of pairs of rotations in finite-dimensional Euclidean space2009In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 12, p. 333-342Article in journal (Refereed)
  • 16.
    Darpö, Erik
    Uppsala universitet.
    Normal forms for the G2-action on the real symmetric 7 × 7-matrices by conjugation2007In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 312, p. 668-688Article in journal (Refereed)
    Abstract [en]

    The exceptional Lie group G2 ⊂ O7 (R) acts on the set of real symmetric 7 × 7-matrices by conjugation. We solve the normal form problem for this group action. In view of the earlier results [G.M. Benkart, D.J. Britten, J.M. Osborn, Real flexible division algebras, Canad. J. Math. 34 (1982) 550-588; J.A. Cuenca Mira, R. De Los Santos Villodres, A. Kaidi, A. Rochdi, Real quadratic flexible division algebras, Linear Algebra Appl. 290 (1999) 1-22; E. Darpö, On the classification of the real flexible division algebras, Colloq. Math. 105 (1) (2006) 1-17], this gives rise to a classification of all finite-dimensional real flexible division algebras. By a classification is meant a list of pairwise non-isomorphic algebras, exhausting all isomorphism classes. We also give a parametrisation of the set of all real symmetric matrices, based on eigenvalues.

  • 17.
    Darpö, Erik
    Uppsala Universitet.
    On the classification of the real flexible division algebras2006In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 105, p. 1-17Article in journal (Refereed)
    Abstract [en]

    The exceptional Lie group G2 ⊂ O7 (R) acts on the set of real symmetric 7 × 7-matrices by conjugation. We solve the normal form problem for this group action. In view of the earlier results [G.M. Benkart, D.J. Britten, J.M. Osborn, Real flexible division algebras, Canad. J. Math. 34 (1982) 550-588; J.A. Cuenca Mira, R. De Los Santos Villodres, A. Kaidi, A. Rochdi, Real quadratic flexible division algebras, Linear Algebra Appl. 290 (1999) 1-22; E. Darpö, On the classification of the real flexible division algebras, Colloq. Math. 105 (1) (2006) 1-17], this gives rise to a classification of all finite-dimensional real flexible division algebras. By aclassification is meant a list of pairwise non-isomorphic algebras, exhausting all isomorphism classes. We also give a parametrisation of the set of all realsymmetric matrices, based on eigenvalues. 

  • 18.
    Darpö, Erik
    Mälardalen University, School of Education, Culture and Communication.
    Reella vektorprodukter2007In: Normat, ISSN 0801-3500, Vol. 55, p. 157-165Article in journal (Other (popular science, discussion, etc.))
  • 19.
    Darpö, Erik
    Mälardalen University, School of Education, Culture and Communication. Uppsala University.
    Some modern developments in the theory of real division algebras2008Report (Other academic)
  • 20.
    Darpö, Erik
    Mathematical Institute, St Giles, Oxford.
    Some modern developments in the theory of real division algebras2010In: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, E-ISSN 1736-7530, Vol. 59, no 1, p. 53-59Article in journal (Refereed)
    Abstract [en]

    The study of real division algebras was initiated by the construction of the quaternion and the octonion algebras in the mid-19th century. In spite of its long history, the problem, of classifying all finite-dimensional real division algebras is still unsolved. We review the theory of this problem, with focus on recent contributions.

  • 21.
    Darpö, Erik
    Uppsala universitet.
    Vector product algebras2009In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 41, p. 898-902Article in journal (Refereed)
    Abstract [en]

    Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.

  • 22.
    Darpö, Erik
    et al.
    Uppsala Universitet.
    Dieterich, Ernst
    Uppsala Universitet.
    Real commutative division algebras2007In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 10, no 2, p. 179-196Article in journal (Refereed)
    Abstract [en]

    The category of all two-dimensional real commutative division algebras is shown to split into two full subcategories. One of them is equivalent to the category of the natural action of the cyclic group of order 2 on the open right half plane ℝ>0 × ℝ. The other one is equivalent to the category of the natural action of the dihedral group of order 6 on the set of all ellipses in ℝ2 which are centered at the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are easily described. Together with ℝ they classify all real commutative division algebras up to isomorphism. Moreover we describe all morphisms between the objects in this classifying set, thus obtaining a complete picture of the category of all real commutative division algebras, up to equivalence. This supplements earlier contributions of Kantor and Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Nauka, Moscow, 1973; Benkart et al., Hadronic J., 4: 497-529, 1981; and Althoen and Kugler, Amer. Math. Monthly, 90: 625-635, 1983, who achieved partial results on the classification of the realcommutative division algebras.

  • 23.
    Darpö, Erik
    et al.
    Mathematical Institute, St Giles', Oxford,.
    Dieterich, Ernst
    Uppsala universitet.
    The double sign of a real division algebra of finite dimension greater than one2012In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 285, no 13, p. 1635-1642Article in journal (Refereed)
    Abstract [en]

    For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a∈ A{set minus}{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category of all n-dimensional realdivision algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of.

  • 24.
    Darpö, Erik
    et al.
    Mälardalen University, School of Education, Culture and Communication.
    Dieterich, Ernst
    Herschend, Martin
    In which dimensions does a division algebra over a given ground field exist?2005In: L'Enseignement mathématique, ISSN 0013-8584, E-ISSN 2309-4672, Vol. 51, p. 255-263Article in journal (Refereed)
  • 25.
    Darpö, Erik
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Gill, Christopher C.
    Mathematical Institute, St Giles, Oxford, United Kingdom.
    Decomposing tensor products for cyclic and dihedral groups2013In: Proceedings of the 45th Symposium on Ring Theory and Representation Theory, 2013, p. 24-28Conference paper (Refereed)
    Abstract [en]

    We give a new formula for the decomposition of a tensor product of indecomposable modules of cyclic two-groups. This formula is also shown to describe thedecomposition of tensor products of an important class of modules of dihedral two-groups

  • 26.
    Darpö, Erik
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Nagoya University, Japan.
    Gill, Christopher C
    Charles University, Praha, Czech Republic.
    The Loewy length of a tensor product of modules of a dihedral two-group2014In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 218, no 4, p. 760-776Article in journal (Refereed)
    Abstract [en]

    While the finite-dimensional modules of the dihedral 2-groups over fields of characteristic 2 were classified over 30 years ago, very little is known about the tensorproducts of such modules. In this article, we compute the Loewy length of the tensor product of two modules of a dihedral 2-group in characteristic 2. As an immediate consequence, we determine when such a tensor product has a projective direct summand.

  • 27.
    Darpö, Erik
    et al.
    Nagoya University.
    Gill, Christopher C.
    Charles University, Praha.
    The Loewy length of tensor products for dihedral two-groups2012In: Proceedings of the 44th Symposium on Ring Theory and Representation Theory, 2012, p. 23-29Conference paper (Refereed)
  • 28.
    Darpö, Erik
    et al.
    Uppsala universitet.
    Herschend, Martin
    Nagoya University.
    On the representation ring of the polynomial algebra over a perfect field2010In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 265, no 3, p. 601-615Article in journal (Refereed)
    Abstract [en]

    We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type à is provided.

  • 29.
    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.

  • 30.
    Darpö, Erik
    et al.
    University of Oxford.
    Rochdi, Abdellatif
    Université Hassan II Mohammedia, Morocco.
    Classification of the four-dimensional power-commutative real division algebras2011In: Proceedings of the Royal Society of Edinburgh. Section A Mathematics, ISSN 0308-2105, E-ISSN 1473-7124, Vol. 141, no 6, p. 1207-1223Article in journal (Refereed)
    Abstract [en]

    A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative realdivision algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras. In dimension four, this leads to a complete and irredundant classification. As a special case, the finite-dimensionalpower-commutative real division algebras that have a unique non-zero idempotent are characterized.

  • 31.
    de Jeu, Marcel
    et al.
    Leiden University.
    Silvestrov, SergeiMälardalen University, School of Education, Culture and Communication.Skau, ChristianNorwegian University of Science and Technology (NTNU), Norway.Tomiyama, JunUniversity of Tokyo, Japan.
    Operator structures and dynamical systems2009Conference proceedings (editor) (Refereed)
  • 32.
    de Jeu, Marcel
    et al.
    Leiden University, Netherlands .
    Svensson, Christian
    Lund University.
    Silvestrov, Sergei
    Lund University.
    Algebraic curves for commuting elements in the $q$-deformed Heisenberg algebra2009In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 321, no 4, p. 1239-1255Article in journal (Refereed)
    Abstract [en]

    In this paper we extend the eliminant construction of Burchnall and Chaundy for commuting differential operators in the Heisenberg algebra to the q-deformed Heisenberg algebra and show that it again provides annihilating curves for commuting elements, provided q satisfies a natural condition. As a side result we obtain estimates on the dimensions of the eigenspaces of elements of this algebra in its faithful module of Laurent series.

  • 33.
    Ekström, Fredrik
    et al.
    Lund University.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Commutants and Centers in a 6-Parameter Family of Quadratically Linked Quantum Plane Algebras2014In: Springer Proceedings in Mathematics and Statistics, Springer, 2014, Vol. 85, p. 37-59Conference paper (Refereed)
    Abstract [en]

    We consider a family of associative algebras, defined as the quotient of a free algebra with the ideal generated by a set of multi-parameter deformed commutation relations between four generators consisting of five quantum plane relations between pairs of generators and one sub-quadratic relation inter-linking all four generators. For generic parameter vectors, the center and the commutants of the two of the generators are described and conditions on the parameters for these commutants to be itself commutative or non-commutative are obtained.

  • 34.
    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.

  • 35. 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)
  • 36.
    Gavel, Hillevi
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Eriksson, Kimmo
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Discrete Mathematics and Discrete Models2015 (ed. 1)Book (Other academic)
  • 37.
    Hamon, Thierry
    et al.
    LIM&BIO (EA3969), Université Paris 13, France.
    Engström, Christopher
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Manser, Mounira
    Université Paris 13, France.
    Badji, Zina
    Universiteá Lille, Villeneuve d'Ascq, France.
    Grabar, Natalia
    Universiteá Lille, Villeneuve d'Ascq, France.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Combining Compositionality and Pagerank for the Identification of Semantic Relations between Biomedical Words2012In: BioNLP: Proceedings of the 2012 Workshop on Biomedical Natural Language Processing, 2012, p. 109-117Conference paper (Refereed)
    Abstract [en]

    The acquisition of semantic resources and relations is an important task for several applications, such as query expansion, information retrieval and extraction, machine translation. However, their validity should also be computed and indicated, especially for automatic systems and applications. We exploit the compositionality based methods for the acquisition of synonymy relations and of indicators of these synonyms. We then apply pagerank-derived algorithm to the obtained semantic graph in order to filter out the acquired synonyms. Evaluation performed with two independent experts indicates that the quality of synonyms is systematically improved by 10 to 15% after their filtering.

  • 38.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication. Umeå University.
    A Rewriting Approach to Graph Invariants2009In: Generalized Lie Theory in Mathematics, Physics and Beyond / [ed] Sergei Silvestrov, Eugen Paal, Viktor Abramov, Alexander Stolin, Berlin, Heidelberg: Springer, 2009, p. 47-67Chapter in book (Refereed)
    Abstract [en]

    Diagrammatic calculation is a powerful tool that gets near indispensable when one tries to manage some of the newer algebraic structures that have been popping up in the last couple of decades. Concretely, it generalises the underlying structure of expressions to being general graphs, where traditional algebraic notation only supports path- or treelike expressions. This paper demonstrates how to apply the author's Generic Diamond Lemma in diagrammatic calculations, by solving through elementary rewriting techniques the problem of classifying all multigraph invariants satisfying a linear contract—delete recursion. (As expected, this leads one to rediscover the Tutte polynomial, along with some more degenerate invariants.) In addition, a concept of “semigraph” is defined which formalises the concept of a graph-theoretical “gadget”.

  • 39.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Critical Pairs in Network Rewriting2014In: IWC 2014: 3rd International Workshop on Confluence / [ed] Takahito Aoto & Delia Kesner, 2014, p. 9-13Conference paper (Refereed)
    Abstract [en]

    This extended abstract breifly introduces rewriting of networks (directed acyclic graphs with the extra structure needed to serve as expressions for PROducts and Permutations categories) and describes the critical pairs aspects of this theory. The author's interest in these comes from wanting to do equational reasoning in algebraic theories (such as Hopf algebras) that mix ordinary operations with co-operations; networks then serve as a formalism for expressions.

    The main message is to point out two phenomena that arise in network rewriting. The first is that of non-convexity of rules, wherein the left hand side of a rule need not be syntactically similar to a symbol in any extension of the underlying signature. The second is one of critical pairs potentially arising where two redexes wrap around each other even when they do not intersect.

  • 40.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Mathematical Formulae Beyond the Tree Paradigm2016Conference paper (Other academic)
    Abstract [en]

    A restriction in the traditional language of mathematics is that every formula has an underlying tree structure, wherein an expression decomposes into subexpressions that each contribute one result to the whole. Some modern mathematics (especially in higher algebra and differential geometry) is however not so well catered for by this tree paradigm, and would be much better off if this restriction was lifted; it is not that the mathematics requires a generalised formula language, but it becomes a whole lot easier to state and do when you use one.

    This poster shows the formula language of networks that arises when one relaxes the tree restriction to instead allow an underlying directed acyclic graph (DAG) structure. Evaluation of networks is non-obvious, but can be carried out in any PROP/symocat; conversely, evaluation of networks characterises PROPs in much the same way as evaluation of words characterises monoids. Networks are isomorphic to the abstract index interpretation of Einstein convention tensor expressions, and demonstrate that these can in fact be given a coordinate-free interpretation. Special cases of the network notation that have been invented independently include the Penrose graphical notation and quantum gate arrays.

  • 41.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication.
    Network Rewriting I: The Foundation2012Report (Other academic)
    Abstract [en]

    A theory is developed which uses "networks" (directed acyclic graphs with some extra structure) as a formalism for expressions in multilinear algebra. It is shown that this formalism is valid for arbitrary PROPs (short for 'PROducts and Permutations category'), and conversely that the PROP axioms are implicit in the concept of evaluating a network. Ordinary terms and operads constitute the special case that the graph underlying the network is a rooted tree. Furthermore a rewriting theory for networks is developed. Included in this is a subexpression concept for which is given both algebraic and effective graph-theoretical characterisations, a construction of reduction maps from rewriting systems, and an analysis of the obstructions to confluence that can occur. Several Diamond Lemmas for this rewriting theory are given. In addition there is much supporting material on various related subjects. In particular there is a "toolbox" for the construction of custom orders on the free PROP, so that an order can be tailored to suit a specific rewriting system. Other subjects treated are the abstract index notation in a general PROP context and the use of feedbacks (sometimes called traces) in PROPs.

  • 42.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Network Rewriting II: Bi- and Hopf Algebras2015In: Leibniz International Proceedings in Informatics, LIPIcs, Volume 36 / [ed] Maribel Fernández, Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik , 2015, Vol. 36, p. 194-208Conference paper (Refereed)
    Abstract [en]

    Bialgebras and their specialisation Hopf algebras are algebraic structures that challenge traditional mathematical notation, in that they sport two core operations that defy the basic functional paradigm of taking zero or more operands as input and producing one result as output. On the other hand, these peculiarities do not prevent studying them using rewriting techniques, if one works within an appropriate network formalism rather than the traditional term formalism. This paper restates the traditional axioms as rewriting systems, demonstrating confluence in the case of bialgebras and finding the (infinite) completion in the case of Hopf algebras. A noteworthy minor problem solved along the way is that of constructing a quasi-order with respect to which the rules are compatible.

  • 43.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication.
    Network Rewriting II: Bi- and Hopf AlgebrasManuscript (preprint) (Other academic)
    Abstract [en]

    Bialgebras and their specialisation Hopf algebras are algebraic structures that challenge traditional mathematical notation, in that they sport two core operations that defy the basic functional paradigm of taking zero or more operands as input and producing one result as output. On the other hand, these peculiarities do not prevent studying them using rewriting techniques, if one works within an appropriate network formalism. This paper restates the traditional axioms as rewriting systems, demonstrating confluence in the case of bialgebras and finding the (infinite) completion in the case of Hopf algebras. A noteworthy minor problem solved along the way is that of constructing a quasi-order with respect to which the rules are compatible.

  • 44.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ordered and Combinatorial Structures for Higher-Dimensional Rewriting2016In: / [ed] Samuel Mimram, 2016Conference paper (Refereed)
    Abstract [en]

    In principle, rewriting is the logically pure activity of transforming abstract expressions according to fully formalised rules, but in practice there is a significant interplay between abstract rules, more-or-less concrete interpretations, and a variety of book-keeping devices that all need to fit together if the rewriting process is to yield results. This paper presents elementary realisations of book-keeping and other formalising devices that are useful in higher-dimensional rewriting, with a focus on the "2-dimensional" case (PROPs and other types of monoidal category). In particular, it explains how one may construct a variety of ordering relations on these object that are sensitive to differences in the underlying graph structure of the objects being rewritten. It also shows how the formal feedback operation can be used to handle nonconvex redexes, which is a phenomenon of higher-dimensional rewriting that lacks a counterpart in word or term rewriting.

  • 45.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ordering Networks2014Conference paper (Refereed)
    Abstract [en]

    This extended abstract discusses the problem of defining quasi-orders that are suitable for use with network rewriting. The author’s primary interest is in using network rewriting as a tool for equational reasoning in algebraic theories with both operations and co-operations.

  • 46.
    Hellström, Lars
    Umeå universitet.
    Quantifiers and n-ary binders: an OpenMath standard enhancement proposal2013In: Proceedings of the MathUI, OpenMath, PLMMS and ThEdu Workshops and Work in Progress at CICMco-located with Conferences on Intelligent Computer Mathematics (CICM 2013), Aachen: Redaktion Sun SITE, Informatik V, RWTH Aachen , 2013Conference paper (Refereed)
    Abstract [en]

    It is proposed that the restriction in the OpenMath standard that an OMBIND element must have exactly three children should be lifted, to support more general binder symbols. The case of logics with generalised quantifiers is described in some detail, since these turn out to not have a natural encoding within OpenMath 2.0, because of precisely this restriction. That restricting quantifiers to a single body should have such consequences is not trivial, but follows from a theorem in the Logic branch of Philosophy.

  • 47.
    Hellström, Lars
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    The eval symbol for axiomatising variadic functions2014In: CICM-WS-WiP 2014: Workshop and Work in Progress Papers at CICM 2014 / [ed] Matthew England et al., Aachen: CEUR-WS.org , 2014Conference paper (Refereed)
    Abstract [en]

    This paper describes (and constitutes the source for!) the proposed list4 OpenMath content dictionary. The main feature in this content dictionary is the eval symbol, which treats a list of values as the list of children of an application element. This may, among other things, be employed to state properties of variadic functions.

  • 48.
    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.

  • 49.
    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.

  • 50.
    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.

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