A generalization of Dickson’s commutative division algebras. (English) Zbl 1460.17007
In this paper the author generalizes the Dickson’s doubling process over any base field by doubling not just finite field extensions, as in the classical process, but also by doubling a central simple algebra \(B\) over a field \(F\). There are obtained division algebras which are no longer commutative nor associative. The structure of the automorphism group, the nuclei and center of these algebras are determined.
Reviewer: Cristina Flaut (Constanta)
MSC:
| 17A35 | Nonassociative division algebras |
| 17A36 | Automorphisms, derivations, other operators (nonassociative rings and algebras) |
| 17A60 | Structure theory for nonassociative algebras |
References:
| [1] | Benkart, G. M.; Osborn, J. M., The derivation algebra of a real division algebra, Am. J. Math., 103, 6, 1135-1150 (1981) · Zbl 0474.17002 · doi:10.2307/2374227 |
| [2] | Burmester, M. V. D., On the commutative non-associative division algebras of even order of LE Dickson, Rend. Mat. Appl, 21, 143-166 (1962) · Zbl 0146.26101 |
| [3] | Burmester, M. V. D., On the non-unicity of translation planes over division algebras, Arch. Math, 15, 1, 364-370 (1964) · Zbl 0131.36803 · doi:10.1007/BF01589214 |
| [4] | Combarro, E. F.; Ranilla, J.; Rúa, I. F., Classification of semifields of order 64, J. Algebra, 322, 11, 4011-4029 (2009) · Zbl 1202.12003 · doi:10.1016/j.jalgebra.2009.02.020 |
| [5] | Combarro, E. F.; Ranilla, J.; Rúa, I. F., Finite semifields with 74 elements, Int. J. Comput. Math., 89, 13-14, 1865-1878 (2012) · Zbl 1271.12005 · doi:10.1080/00207160.2012.688113 |
| [6] | Deajim, A.; Grant, D., Space time codes and non-associative division algebras arising from elliptic curves, Contemp. Math, 463, 29-44 (2008) · Zbl 1163.17005 |
| [7] | Dickson, L. E., On commutative linear algebras in which division is always uniquely possible, Trans. Amer. Math. Soc, 7, 4, 514-522 (1906) · JFM 37.0112.01 · doi:10.1090/S0002-9947-1906-1500764-6 |
| [8] | Grant, D.; Varanasi, M. K., The equivalence of space-time codes and codes defined over finite fields and Galois rings, Adv. Math Commun, 2, 2, 131-145 (2008) · Zbl 1234.94089 · doi:10.3934/amc.2008.2.131 |
| [9] | Hui, A.; Tai, Y. K.; Wong, P. P. W., On the autotopism group of the commutative Dickson semifield K and the stabilizer of the Ganley unital embedded in the semifield plane, Innov. Incidence Geom, 14, 1, 27-42 (2015) · Zbl 1404.51002 · doi:10.2140/iig.2015.14.27 |
| [10] | Jones, J. W.; Roberts, D. P., A database of local fields, J. Symbol. Comput, 41, 1, 80-97 (2006) · Zbl 1140.11350 · doi:10.1016/j.jsc.2005.09.003 |
| [11] | Kervaire, M., Non-parallelizability of the n-sphere for n > 7, Proc. Nat. Acad. Sci, 44, 3, 280-283 (1958) · Zbl 0093.37303 · doi:10.1073/pnas.44.3.280 |
| [12] | Knus, M.; Merkurjev, A.; Rost, M.; Tignol, J., The Book of Involutions (1998), Providence: American Mathematical Soc, Providence · Zbl 0955.16001 |
| [13] | Knuth, D. E., Finite semifields and projective planes (1963), California Institute of Technology · Zbl 0128.25604 |
| [14] | Lavrauw, M., Finite semifields and nonsingular tensors, Des Codes Cryptogr, 68, 1-3, 205-227 (2013) · Zbl 1304.12004 · doi:10.1007/s10623-012-9710-6 |
| [15] | Lavrauw, M.; Polverino, O., Current Research Topics in Galois Geometry, Finite semifields, 131-160 (2011), Hauppauge, NY: Nova Science Publishers, Hauppauge, NY |
| [16] | Milnor, J.; Bott, R., On the parallelizability of the spheres, Bull. Amer. Math. Soc, 64, 3, 87-89 (1958) · Zbl 0082.16602 · doi:10.1090/S0002-9904-1958-10166-4 |
| [17] | Steele, A.; Pumplün, S., The nonassociative algebras used to build fast-decodable space-time block codes, AMC, 9, 4, 449-469 (2015) · Zbl 1366.17002 · doi:10.3934/amc.2015.9.449 |
| [18] | Pumplün, S.; Unger, T., Space-time block codes from nonassociative division algebras, AMC, 5, 3, 449-471 (2011) · Zbl 1246.94038 · doi:10.3934/amc.2011.5.449 |
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.
