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  • 1.
    Cuenca Mira, José Antonio
    et al.
    Departamento de Algebra, Geometria y Topologia, Facultad de Ciencias, Universidad de Málaga.
    Darpö, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    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 Ramírez Álvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.

  • 2.
    Darpö, Erik
    et al.
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Dieterich, Ernst
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Classification of the Real Commutative Division Algebras2003Report (Other scientific)
  • 3.
    Darpö, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    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 x 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 R-2 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 real commutative division algebras.

  • 4. Darpö, Erik
    et al.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    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 is an element of A\{0} are shown to form an invariant of A, called its double sign. For each n is an element of {2, 4, 8}, the double sign causes the category D-n of all n-dimensional real division 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 D-n.

  • 5.
    Darpö, Erik
    et al.
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Dieterich, Ernst
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Herschend, Martin
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    In which dimensions does a division algebra over a given ground field exists?2005Report (Other scientific)
  • 6.
    Darpö, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Herschend, Martin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    In which dimensions does a division algebra over a given ground field exists?2005In: L’Enseignement Mathématique, Vol. 51, no 3-4, p. 255-263Article in journal (Other (popular science, discussion, etc.))
  • 7.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    A general approach to finite dimensional division algebras2012In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 126, no 1, p. 73-86Article in journal (Refereed)
    Abstract [en]

    We present a short and rather self-contained introduction to the theory of finite-dimensional division algebras, setting out from the basic definitions and leading up to recent results and current directions of research. In Sections 2-3 we develop the general theory over an arbitrary ground field k, with emphasis on the trichotomy of fields imposed by the dimensions in which a division algebra exists, the groupoid structure of the level subcategories D-n(k), and the role played by the irreducible morphisms. Sections 4-5 deal with the classical case of real division algebras, emphasizing the double sign decomposition of the level subcategories D-n(R) for n is an element of {2, 4, 8} and the problem of describing their blocks, along with an account of known partial solutions to this problem. 

  • 8.
    Dieterich, Ernst
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Classification, automorphism groups and categorical structure of the two-dimensional real division algebras.2005In: Journal of Algebra and Its Applications, Vol. 4, no 5, p. 517-538Article in journal (Refereed)
    Abstract [en]

    The Faculty of Science and Technology of Uppsala University is for all practical purposes a bilingual institution, using both Swedish and English in education and research. Extensive use of English in teaching, and also in intrauniversity communication, permits recruitment of nonSwedishspeaking students, researchers and professors, and also prepares our students for

    international careers. However, the introduction of English has been somewhat haphazard, not taking into account possible negative effects on communication in Swedish, nor on students'

    learning.

    In order to improve students' and professors' language skills, and achieve a good balance between Swedish and English, the faculty board appointed a language committee in 2003 whose task was to propose a language policy for the faculty. A first part, stating as a main goal that all communication from and within the faculty should have the highest quality possible, has been adopted by the board. A second part including language planning with respect to status, corpus, and acquisition for both Swedish and English to accomplish this goal was sent to the board of the faculty in May, 2005. Implementation of this policy will affect all faculty activities, especially education. Suggested

    annual reports on language status will raise our present minimal knowledge about possible domain losses and allow for relevant countermeasures.

  • 9.
    Dieterich, Ernst
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Classification, Automorphism Groups and Categorical Structure of the Two-Dimensional Real Division Algebras2004Report (Other scientific)
  • 10.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Construction of Auslander-Reiten quivers for a class of group rings1983In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 183, no 1, p. 43-60Article in journal (Refereed)
  • 11.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Dissident algebras1999In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 82, no 1, p. 13-23Article in journal (Refereed)
  • 12.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Existence and construction of two-dimensional invariant subspaces for pairs of rotations2009In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 114, no 2, p. 203-211Article in journal (Refereed)
  • 13.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Fraleighs misstag: en varning2000In: Normat, Vol. 48, p. 153-158Article in journal (Refereed)
  • 14.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Group rings of wild representation type1983In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 266, no 1, p. 1-22Article in journal (Refereed)
  • 15.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Lattices over curve singularities with large conductor1993In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 114, no 1, p. 399-433Article in journal (Refereed)
  • 16.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Lattices over group rings of cyclic p-groups and generalized factorspace categories1985In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. s2-31, no 3, p. 407-424Article in journal (Refereed)
  • 17.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Power-associative real division algebras1998In: Canadian Mathematical Society Conference Proceedings, Vol. 24, p. 139-144Article in journal (Refereed)
  • 18.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Quadratic division algebras revisited. (Remarks on an article by J. M. Osborn.)2000In: Proceedings of the American Mathematical Society, Vol. 128, p. 3159-3166Article in journal (Refereed)
  • 19.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Real quadratic division algebras2000In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 28, no 2, p. 941-947Article in journal (Refereed)
  • 20.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Reduction of isolated singularities1987In: Commentarii Mathematici Helvetici, ISSN 0010-2571, E-ISSN 1420-8946, Vol. 62, no 1, p. 654-676Article in journal (Refereed)
  • 21.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Representation types of group rings over complete discrete valuation rings1981In: Lecture notes in mathematics, ISSN 0075-8434, E-ISSN 1617-9692, Vol. 882, p. 369-389Article in journal (Refereed)
  • 22.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Representation types of group rings over complete discrete valuation rings II1985In: Lecture notes in mathematics, ISSN 0075-8434, E-ISSN 1617-9692, Vol. 1142, p. 112-125Article in journal (Refereed)
  • 23.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Solution of a non-domestic tame classification problem from integral representation theory of finite groups1991In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 92, no 450, p. 1-140Article in journal (Refereed)
  • 24.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Tame curve singularities with large conductor1991In: Progress in Mathematics, Vol. 95, p. 327-341Article in journal (Refereed)
  • 25.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Tame orders1990In: Banach Center Publications, ISSN 0137-6934, E-ISSN 1730-6299, Vol. 26, no 1, p. 233-261Article in journal (Refereed)
  • 26.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    The Auslander-Reiten quiver of an isolated singularity1987In: Lecture notes in mathematics, ISSN 0075-8434, E-ISSN 1617-9692, Vol. 1273, p. 244-264Article in journal (Refereed)
  • 27.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Trace Invariants of Finite-Dimensional Algebras2016In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, no 5, p. 1852-1881Article in journal (Refereed)
  • 28.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Zur Klassifikation vierdimensionaler reeller Divisionsalgebren1998In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 194, p. 13-22Article in journal (Refereed)
  • 29.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Fieseler, Karl-Heinz
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Lindberg, Lars
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Liftings of Dissident Maps2003Report (Other academic)
  • 30.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Fieseler, Karl-Heinz
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Lindberg, Lars
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Liftings of dissident maps2006In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 204, no 1, p. 133-154Article in journal (Refereed)
    Abstract [en]

    We study dissident maps ηη on RmRm for m∈{3,7}m∈{3,7} by investigating liftings Φ:Rm→RmΦ:Rm→Rm of the selfbijection ηP:P(Rm)→P(Rm),ηP[v]=(η(v∧Rm))⊥ induced by ηη. Our main result (Theorem 2.4) asserts the existence and uniqueness, up to a non-zero scalar multiple, of a lifting ΦΦ whose component functions are homogeneous polynomials of degree dd, relatively prime and without non-trivial common zero. We prove that 1⩽d⩽m-21⩽d⩽m-2.

    We achieve a complete description of all dissident maps of degree one and we solve their isomorphism problem (Theorems 4.8 and 4.13). As a consequence, we achieve a complete description of all real quadratic division algebras of degree one and we solve their isomorphism problem (Theorems 5.1 and 5.3). Moreover we present examples of eight-dimensional real quadratic division algebras of degree 3 and 5 (Proposition 6.3). This extends earlier results of Osborn [Trans. Amer. Math. Soc. 105 (1962) 202–221], Hefendehl [Geometriae Dedicata 9 (1980) 129–152], Hefendehl-Hebeker [Arch. Math. 40 (1983) 50–60], Cuenca Mira et al. [Lin. Alg. Appl. 290 (1999) 1–22], Dieterich [Proc. Amer. Math. Soc. 128 (2000) 3159–3166] and Dieterich and Lindberg [Colloq. Math. 97 (2003) 251–276] on the classification of real quadratic division algebras.

  • 31.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Lindberg, Lars
    Dissident Maps on the Seven-Dimensional Euclidean Space2001Report (Other academic)
  • 32.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Lindberg, Lars
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Dissident maps on the seven-dimensional Euclidean space2003In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 97, p. 251-276Article in journal (Refereed)
    Abstract [en]

    Our article contributes to the classification of dissident maps on R7, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on R7. The first class is formed by all composed dissident maps, obtained from a vector product on R7 by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional real quadratic division algebras which arise from a 4-dimensional real quadratic division algebra by doubling. For each of these two classes we exhibit a complete (but redundant) classification, given by a 49-parameter family of composed dissident maps and a 9-parameter family of doubled dissident maps respectively. The intersection of these two classes forms one isoclass of dissident maps only, namely the isoclass consisting of all vector products on R7.

  • 33.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Rubinsztein, Ryszard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    The Degree of an Eight-Dimensional Real Quadratic Division Algebra is 1, 3, or 52009Report (Other academic)
  • 34.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Rubinsztein, Ryszard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    The Degree of an Eight-Dimensional Real Quadratic Division Algebra is 1, 3, or 52010In: Bulletin des Sciences Mathematiques, ISSN 0007-4497, Vol. 134, no 5, p. 447-453Article in journal (Refereed)
    Abstract [en]

    A celebrated theorem of Hopf, Bott, Milnor, and Kervaire states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified, the problem of classifying all 8-dimensional real quadratic division algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic division algebra has degree 1, 3, or 5. This statement is sharp.

  • 35.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Wiedemann, Alfred
    The Auslander-Reiten quiver of a simple curve singularity1986In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 294, no 2, p. 455-475Article in journal (Refereed)
  • 36.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Öhman, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    On the classification of four-dimensional quadratic division algebras over square-ordered fields2002In: Journal of the London Mathematical Society, Vol. 65, p. 285-302Article in journal (Refereed)
1 - 36 of 36
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