Journal of the Ramanujan Mathematical Society

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Conjugacy Classes of Trialitarian Automorphisms and Symmetric Compositions


Affiliations
1 Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, AB T6G 2G1, Canada
2 Department Mathematik, ETH Zentrum, CH-8092 Ziirich, Switzerland
3 ICTEAM Institute, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium, and Zukunftskolleg, Universitat Konstanz, D-78457 Konstanz, Germany
     

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The trialitarian automorphisms considered in this paper are the outer automorphisms of order 3 of adjoint classical groups of type D4 over arbitrary fields. A one-to-one correspondence is established between their conjugacy classes and similarity classes of symmetric compositions on 8-dimensional quadratic spaces. Using the known classification of symmetric compositions, we distinguish two conjugacy classes of trialitarian automorphisms over algebraically closed fields. For type I, the group of fixed points is connected and simple of type G2, whereas, if the characteristic of the field is different from 3, it is adjoint of type A2 for trialitarian automorphisms of type II.
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  • Conjugacy Classes of Trialitarian Automorphisms and Symmetric Compositions

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Authors

V. Chernousov
Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, AB T6G 2G1, Canada
M. A. Knus
Department Mathematik, ETH Zentrum, CH-8092 Ziirich, Switzerland
J. P. Tignol
ICTEAM Institute, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium, and Zukunftskolleg, Universitat Konstanz, D-78457 Konstanz, Germany

Abstract


The trialitarian automorphisms considered in this paper are the outer automorphisms of order 3 of adjoint classical groups of type D4 over arbitrary fields. A one-to-one correspondence is established between their conjugacy classes and similarity classes of symmetric compositions on 8-dimensional quadratic spaces. Using the known classification of symmetric compositions, we distinguish two conjugacy classes of trialitarian automorphisms over algebraically closed fields. For type I, the group of fixed points is connected and simple of type G2, whereas, if the characteristic of the field is different from 3, it is adjoint of type A2 for trialitarian automorphisms of type II.