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  • 1.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    A Categorical Study of Composition Algebras via Group Actions and Triality2015Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    A composition algebra is a non-zero algebra endowed with a strictly non-degenerate, multiplicative quadratic form. Finite-dimensional composition algebras exist only in dimension 1, 2, 4 and 8 and are in general not associative or unital. Over the real numbers, such algebras are division algebras if and only if they are absolute valued, i.e. equipped with a multiplicative norm. The problem of classifying all absolute valued algebras and, more generally, all composition algebras of finite dimension remains unsolved. In dimension eight, this is related to the triality phenomenon. We approach this problem using a categorical language and tools from representation theory and the theory of algebraic groups.

    We begin by considering the category of absolute valued algebras of dimension at most four. In Paper I we determine the morphisms of this category completely, and describe their irreducibility and behaviour under the actions of the automorphism groups of the algebras.

    We then consider the category of eight-dimensional absolute valued algebras, for which we provide a description in Paper II in terms of a group action involving triality. Then we establish general criteria for subcategories of group action groupoids to be full, and applying this to the present setting, we obtain hitherto unstudied subcategories determined by reflections. The reflection approach is further systematized in Paper III, where we obtain a coproduct decomposition of the category of finite-dimensional absolute valued algebras into blocks, for several of which the classification problem does not involve triality. We study these in detail, reducing the problem to that of certain group actions, which we express geometrically.

    In Paper IV, we use representation theory of Lie algebras to completely classify all finite-dimensional absolute valued algebras having a non-abelian derivation algebra. Introducing the notion of quasi-descriptions, we reduce the problem to the study of actions of rotation groups on products of spheres.

    We conclude by considering composition algebras over arbitrary fields of characteristic not two in Paper V. We establish an equivalence of categories between the category of eight-dimensional composition algebras with a given quadratic form and a groupoid arising from a group action on certain pairs of outer automorphisms of affine group schemes

    List of papers
    1. Morphisms in the Category of Finite-Dimensional Absolute Valued Algebras
    Open this publication in new window or tab >>Morphisms in the Category of Finite-Dimensional Absolute Valued Algebras
    2011 (English)In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 125, no 2, 147-174 p.Article in journal (Refereed) Published
    Abstract [en]

    This is a study of morphisms in the category of finite-dimensional absolute valued algebras whose codomains have dimension four. We begin by citing and transferring a classification of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for the morphisms from two-dimensional algebras we describe the orbits under the actions of the automorphism groups involved. Parts of these descriptions rely on a suitable choice of a cross-section of four-dimensional absolute valued algebras, and we thus end by providing an explicit means of transferring these results to algebras outside this cross-section.

    Keyword
    absolute valued algebra, division algebra, homomorphism, irreducibility, composition
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-172251 (URN)10.4064/cm125-2-2 (DOI)000299619900002 ()
    Available from: 2012-04-04 Created: 2012-04-03 Last updated: 2017-12-07
    2. Corestricted Group Actions and Eight-Dimensional Absolute Valued Algebras
    Open this publication in new window or tab >>Corestricted Group Actions and Eight-Dimensional Absolute Valued Algebras
    2015 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 5, 1519-1547 p.Article in journal (Refereed) Published
    Abstract [en]

    We define and study the class of left reflection algebras, which is a subclass of eight-dimensional absolute valued algebras. We reduce its classification problem to the problem of finding a transversal for the action of a subgroup of O-7 on O-7 by conjugation. As a basis for this study, we give a general criterion for finding full subcategories of group action categories, which themselves arise from group actions.

    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-247368 (URN)10.1016/j.jpaa.2014.06.014 (DOI)000349427000009 ()
    Available from: 2015-03-20 Created: 2015-03-18 Last updated: 2017-12-04
    3. An Approach to Finite-Dimensional Real Division Composition Algebras through Reflections
    Open this publication in new window or tab >>An Approach to Finite-Dimensional Real Division Composition Algebras through Reflections
    2015 (English)In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 139, no 4, 357-399 p.Article in journal (Refereed) Published
    Abstract [en]

    We consider the category of all finite-dimensional real composition algebras which are division algebras. These are precisely the finite-dimensional absolute valued algebras, and exist only in dimension 1, 2, 4 and 8. We construct three decompositions of this category, each determined by the number of reflections composing left and right multiplication by idempotents. As a consequence, we obtain new full subcategories in dimension 8, in which all morphisms are automorphisms of the octonions. This reduces considerable parts of the still open classification problem in dimension 8 to the normal form problem of an action of the automorphism group of the octonions, which is a compact Lie group of type  , on pairs of orthogonal maps. We describe these subcategories further in terms of subgroups of  and their cosets, which we express geometrically. This extends the study of finite-dimensional real division composition algebras with a one-sided unity.

    Keyword
    Composition algebra, division algebra, absolute valued algebra, reflection, G2-subgroup, G2-set
    National Category
    Mathematics
    Research subject
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-224094 (URN)10.1016/j.bulsci.2014.10.001 (DOI)000356200200001 ()
    Available from: 2014-05-03 Created: 2014-05-03 Last updated: 2017-12-05Bibliographically approved
    4. Classification of the Finite-Dimensional Real Division Composition Algebras having a Non-Abelian Derivation Algebra
    Open this publication in new window or tab >>Classification of the Finite-Dimensional Real Division Composition Algebras having a Non-Abelian Derivation Algebra
    2016 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 445, 35-77 p.Article in journal (Other academic) Published
    Abstract [en]

    We classify the category of finite-dimensional real division composition algebras having a non-abelian Lie algebra of derivations. Our complete and explicit classification is largely achieved by introducing the concept of a quasi-description of a category, and using it to express the problem in terms of normal form problems for certain group actions on products of 3-spheres.

    Keyword
    Composition algebras, division algebras, absolute valued algebras, derivation algebras, quasi-descriptions
    National Category
    Mathematics
    Research subject
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-224093 (URN)10.1016/j.jalgebra.2015.07.025 (DOI)000365826900002 ()
    Available from: 2014-05-03 Created: 2014-05-03 Last updated: 2017-12-05Bibliographically approved
    5. Composition Algebras and Outer Automorphisms of Algebraic Groups
    Open this publication in new window or tab >>Composition Algebras and Outer Automorphisms of Algebraic Groups
    (English)Article in journal (Other academic) Submitted
    Abstract [en]

    In this note, we establish an equivalence of categories between the category of all eight-dimensionalcomposition algebras with any given quadratic form  over a field of characteristic not two, and a category arising from anaction of the projective similarity group of on certain pairs of automorphisms of the group scheme  defined ove . This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known resultson composition algebras from our equivalence.

    Keyword
    Composition algebras, algebraic groups, outer automorphisms, triality
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-248499 (URN)
    Available from: 2015-03-30 Created: 2015-03-30 Last updated: 2015-04-07
  • 2.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    An Approach to Finite-Dimensional Real Division Composition Algebras through Reflections2015In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 139, no 4, 357-399 p.Article in journal (Refereed)
    Abstract [en]

    We consider the category of all finite-dimensional real composition algebras which are division algebras. These are precisely the finite-dimensional absolute valued algebras, and exist only in dimension 1, 2, 4 and 8. We construct three decompositions of this category, each determined by the number of reflections composing left and right multiplication by idempotents. As a consequence, we obtain new full subcategories in dimension 8, in which all morphisms are automorphisms of the octonions. This reduces considerable parts of the still open classification problem in dimension 8 to the normal form problem of an action of the automorphism group of the octonions, which is a compact Lie group of type  , on pairs of orthogonal maps. We describe these subcategories further in terms of subgroups of  and their cosets, which we express geometrically. This extends the study of finite-dimensional real division composition algebras with a one-sided unity.

  • 3.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Classification of the Finite-Dimensional Real Division Composition Algebras having a Non-Abelian Derivation Algebra2016In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 445, 35-77 p.Article in journal (Other academic)
    Abstract [en]

    We classify the category of finite-dimensional real division composition algebras having a non-abelian Lie algebra of derivations. Our complete and explicit classification is largely achieved by introducing the concept of a quasi-description of a category, and using it to express the problem in terms of normal form problems for certain group actions on products of 3-spheres.

  • 4.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Composition Algebras and Outer Automorphisms of Algebraic GroupsArticle in journal (Other academic)
    Abstract [en]

    In this note, we establish an equivalence of categories between the category of all eight-dimensionalcomposition algebras with any given quadratic form  over a field of characteristic not two, and a category arising from anaction of the projective similarity group of on certain pairs of automorphisms of the group scheme  defined ove . This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known resultson composition algebras from our equivalence.

  • 5.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Corestricted Group Actions and Eight-Dimensional Absolute Valued Algebras2012Report (Other academic)
    Abstract [en]

    A condition for when two eight-dimensional absolute valued algebras are isomorphic was given in [4]. We use this condition to deduce a description (in the sense of Dieterich, [9]) of the category of such algebras, and show how previous descriptions of some full subcategories fit in this description. Led by the structure of these examples, we aim at systematically constructing new subcategories whose classification is manageable. To this end we propose, in greater generality, the definition of sharp stabilizers for group actions, and use these to obtain conditions for when certain subcategories of groupoids are full. This we apply to the category of eight-dimensional absolute valued algebras and obtain a class of subcategories, for which we simplify, and partially solve, the classification problem.

  • 6.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Corestricted Group Actions and Eight-Dimensional Absolute Valued Algebras2015In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 5, 1519-1547 p.Article in journal (Refereed)
    Abstract [en]

    We define and study the class of left reflection algebras, which is a subclass of eight-dimensional absolute valued algebras. We reduce its classification problem to the problem of finding a transversal for the action of a subgroup of O-7 on O-7 by conjugation. As a basis for this study, we give a general criterion for finding full subcategories of group action categories, which themselves arise from group actions.

  • 7.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Determining the elements of a semigroup2007Report (Other academic)
  • 8.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Morphisms in the Category of Finite Dimensional Absolute Valued Algebras2011Report (Other academic)
    Abstract [en]

    This is a study of morphisms in the category of nite dimensional absolute valued algebras, whose codomains have dimension four. We begin by citing and transferring a classication of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for the morphisms from two-dimensional algebras we describe the orbits under the actions of the automorphism groups involved. Parts of these descriptions rely on a suitable choice of a cross-section of four-dimensional absolute valued algebras, and we thus end by providing an explicit means of transferring these results to algebras outside this crosssection.

  • 9.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Morphisms in the Category of Finite-Dimensional Absolute Valued Algebras2011In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 125, no 2, 147-174 p.Article in journal (Refereed)
    Abstract [en]

    This is a study of morphisms in the category of finite-dimensional absolute valued algebras whose codomains have dimension four. We begin by citing and transferring a classification of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for the morphisms from two-dimensional algebras we describe the orbits under the actions of the automorphism groups involved. Parts of these descriptions rely on a suitable choice of a cross-section of four-dimensional absolute valued algebras, and we thus end by providing an explicit means of transferring these results to algebras outside this cross-section.

  • 10. Alsaody, Seidon
    et al.
    Serra, Jean
    Connective segmentation generalized to arbitrary complete lattices.2011In: Mathematical Morphology and Its Applications to Image and Signal Processing: 10th international symposium, ISMM 2011, Verbania-Intra, Italy, July 6–8, 2011. Proceedings, Berlin: Springer , 2011, 61-72 p.Chapter in book (Refereed)
    Abstract [en]

    Summary: We begin by defining the setup and the framework of connective segmentation. Then we start from a theorem based on connective criteria, established for the power set of an arbitrary set. As the power set is an example of a complete lattice, we formulate and prove an analogue of the theorem for general complete lattices. Secondly, we consider partial partitions and partial connections. We recall the definitions, and quote a result that gives a characterization of (partial) connections. As a continuation of the work in the first part, we generalize this characterization to complete lattices as well. Finally we link these two approaches by means of a commutative diagram, in two manners.

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