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A Categorical Study of Composition Algebras via Group Actions and Triality
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.ORCID iD: 0000-0002-0182-6205
2015 (English)Doctoral 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

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2015. , 45 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 88
Keyword [en]
Composition algebra, division algebra, absolute valued algebra, triality, groupoid, group action, algebraic group, Lie algebra of derivations, classification.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-248519ISBN: 978-91-506-2454-0 (print)OAI: oai:DiVA.org:uu-248519DiVA: diva2:799477
Public defence
2015-05-21, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2015-04-27 Created: 2015-03-31 Last updated: 2015-04-27
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

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