Heisenberg groups over composition algebras. (English) Zbl 1368.17006
Summary: 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.
MSC:
| 17A75 | Composition algebras |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20F28 | Automorphism groups of groups |
Keywords:
Heisenberg group; nilpotent group; automorphism; isomorphism; isotopism; composition algebra; quaternion; octonion; Cayley algebraReferences:
| [1] | Albert, A.A.: Quasigroups I. Trans. Am. Math. Soc. 54, 507-519 (1943). doi:10.2307/1990259 · Zbl 0063.00039 |
| [2] | Dembowski, P.: Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 44. Springer, Berlin (1968) · Zbl 0159.50001 |
| [3] | 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 · Zbl 1200.20026 |
| [4] | 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 · Zbl 1300.17009 |
| [5] | 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 · Zbl 1281.51003 · doi:10.1007/s00605-012-0409-6 |
| [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 · Zbl 1359.12005 |
| [7] | Schafer, R.D.: An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, vol. 22. Academic Press, New York (1966) · Zbl 0145.25601 |
| [8] | Springer, T.A., Veldkamp, F.D.: Octonions, Jordan Algebras and Exceptional Groups. Springer Monographs in Mathematics, Springer, Berlin (2000) · Zbl 1087.17001 · doi:10.1007/978-3-662-12622-6 |
| [9] | 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 · Zbl 1179.17013 |
| [10] | Zorn, M.: Theorie der alternativen Ringe. Abh. Math. Sem. Univ. Hamburg 8, 123-147 (1930). doi:10.1007/BF02940993 · JFM 56.0140.01 · doi:10.1007/BF02940993 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
