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Publications (10 of 17) Show all publications
Darpö, E. & Pérez Izquierdo, J. M. (2015). Autotopies and quasigroup identities: new aspects of non-associative division algebras. Forum mathematicum, 27(5), 2691-2745
Open this publication in new window or tab >>Autotopies and quasigroup identities: new aspects of non-associative division algebras
2015 (English)In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 27, no 5, p. 2691-2745Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
De Gruyter, 2015
National Category
Algebra and Logic Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21740 (URN)10.1515/forum-2012-0087 (DOI)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-29Bibliographically approved
Darpö, E. & Gill, C. C. (2014). The Loewy length of a tensor product of modules of a dihedral two-group. Journal of Pure and Applied Algebra, 218(4), 760-776
Open this publication in new window or tab >>The Loewy length of a tensor product of modules of a dihedral two-group
2014 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 218, no 4, p. 760-776Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2014
National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21739 (URN)10.1016/j.jpaa.2013.08.015 (DOI)000329475800017 ()2-s2.0-84883126885 (Scopus ID)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-06Bibliographically approved
Darpö, E. & Gill, C. C. (2013). Decomposing tensor products for cyclic and dihedral groups. In: Proceedings of the 45th Symposium on Ring Theory and Representation Theory: . Paper presented at Proceedings of the 45th Symposium on Ring Theory and Representation Theory, September 7-9, 2012, Shinshu University, Japan (pp. 24-28).
Open this publication in new window or tab >>Decomposing tensor products for cyclic and dihedral groups
2013 (English)In: Proceedings of the 45th Symposium on Ring Theory and Representation Theory, 2013, p. 24-28Conference paper, Published 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

National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21716 (URN)
Conference
Proceedings of the 45th Symposium on Ring Theory and Representation Theory, September 7-9, 2012, Shinshu University, Japan
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2016-03-10
Darpö, E. & Dieterich, E. (2012). The double sign of a real division algebra of finite dimension greater than one. Mathematische Nachrichten, 285(13), 1635-1642
Open this publication in new window or tab >>The double sign of a real division algebra of finite dimension greater than one
2012 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 285, no 13, p. 1635-1642Article in journal (Refereed) Published
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.

National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21717 (URN)10.1002/mana.201100189 (DOI)000308643300008 ()2-s2.0-84866156912 (Scopus ID)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-06Bibliographically approved
Darpö, E. & Gill, C. C. (2012). The Loewy length of tensor products for dihedral two-groups. In: Proceedings of the 44th Symposium on Ring Theory and Representation Theory: . Paper presented at 44th Symposium of Ring Theory and Representation Theory, Okayama (pp. 23-29).
Open this publication in new window or tab >>The Loewy length of tensor products for dihedral two-groups
2012 (English)In: Proceedings of the 44th Symposium on Ring Theory and Representation Theory, 2012, p. 23-29Conference paper, Published paper (Refereed)
National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21718 (URN)
Conference
44th Symposium of Ring Theory and Representation Theory, Okayama
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2014-12-08Bibliographically approved
Darpö, E. & Rochdi, A. (2011). Classification of the four-dimensional power-commutative real division algebras. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 141(6), 1207-1223
Open this publication in new window or tab >>Classification of the four-dimensional power-commutative real division algebras
2011 (English)In: 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) Published
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.

National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21719 (URN)10.1017/S0308210510000259 (DOI)000298454000004 ()2-s2.0-84857293837 (Scopus ID)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-06Bibliographically approved
Cuenca Mira, J. A., Darpö, E. & Dieterich, E. (2010). Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity. Bulletin des Sciences Mathématiques, 134(3), 247-277
Open this publication in new window or tab >>Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity
2010 (English)In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 134, no 3, p. 247-277Article in journal (Refereed) Published
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. 

National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21722 (URN)10.1016/j.bulsci.2009.03.001 (DOI)000276938900003 ()2-s2.0-77949567068 (Scopus ID)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-06Bibliographically approved
Darpö, E. & Herschend, M. (2010). On the representation ring of the polynomial algebra over a perfect field. Mathematische Zeitschrift, 265(3), 601-615
Open this publication in new window or tab >>On the representation ring of the polynomial algebra over a perfect field
2010 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 265, no 3, p. 601-615Article in journal (Refereed) Published
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.

National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21720 (URN)10.1007/s00209-009-0532-9 (DOI)000277603500007 ()2-s2.0-77952419836 (Scopus ID)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-06Bibliographically approved
Darpö, E. (2010). Some modern developments in the theory of real division algebras. Proceedings of the Estonian Academy of Sciences, 59(1), 53-59
Open this publication in new window or tab >>Some modern developments in the theory of real division algebras
2010 (English)In: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, E-ISSN 1736-7530, Vol. 59, no 1, p. 53-59Article in journal (Refereed) Published
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.

National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21721 (URN)10.3176/proc.2010.1.09 (DOI)000276989900009 ()2-s2.0-77956588514 (Scopus ID)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-06Bibliographically approved
Darpö, E. (2009). Classification of pairs of rotations in finite-dimensional Euclidean space. Algebras and Representation Theory, 12, 333-342
Open this publication in new window or tab >>Classification of pairs of rotations in finite-dimensional Euclidean space
2009 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 12, p. 333-342Article in journal (Refereed) Published
National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-21724 (URN)10.1007/s10468-009-9156-3 (DOI)000265682900013 ()2-s2.0-67349201047 (Scopus ID)
Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2017-12-06Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-9177-9774

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