Heisenberg groups over composition algebras
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry - Tập 57 - Trang 667-677 - 2015
Tóm tắt
We solve the isomorphism problem for Heisenberg groups constructed over composition algebras, including the split case and characteristic two. We prove that two such groups are isomorphic if, and only if, the corresponding composition algebras are isomorphic as
$$\mathbb Z$$
-algebras.
Tài liệu tham khảo
Albert, A.A.: Quasigroups I. Trans. Am. Math. Soc. 54, 507–519 (1943). doi:10.2307/1990259
Dembowski, P.: Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 44. Springer, Berlin (1968)
Grundhöfer, T., Stroppel, M.J.: Automorphisms of Verardi groups: small upper triangular matrices over rings. Beitr. Algebra Geom. 49(1), 1–31 (2008). http://www.emis.de/journals/BAG/vol.49/no.1/1.html
Gulde, M., Stroppel, M.J. Stabilizers of subspaces under similitudes of the Klein quadric, and automorphisms of Heisenberg algebras. Linear Algebra Appl. 437(4), 1132–1161 (2012). doi:10.1016/j.laa.2012.03.018. arXiv:1012.0502
Knarr, N., Stroppel, M.J.: Polarities and planar collineations of Moufang planes. Monatsh Math. 169(3–4), 383–395 (2013). doi:10.1007/s00605-012-0409-6
Knarr, N., Stroppel, M.J.: Heisenberg groups, semifields, and translation planes. Beitr. Algebra Geom. 56(1), 115–127 (2015). doi:10.1007/s13366-014-0193-7
Schafer, R.D.: An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, vol. 22. Academic Press, New York (1966)
Springer, T.A., Veldkamp, F.D.: Octonions, Jordan Algebras and Exceptional Groups. Springer Monographs in Mathematics, Springer, Berlin (2000)
Stroppel, M.J.: The Klein quadric and the classification of nilpotent Lie algebras of class two. J. Lie Theory 18(2), 391–411 (2008). http://www.heldermann-verlag.de/jlt/jlt18/strola2e
Zorn, M.: Theorie der alternativen Ringe. Abh. Math. Sem. Univ. Hamburg 8, 123–147 (1930). doi:10.1007/BF02940993