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Article

MSC: 17A35, 17A36, 17A60
Keywords:
Nonassociative algebras; division algebras; automorphisms
Summary:
We generalize Knuth's construction of Case I semifields quadratic over a weak nucleus, also known as generalized Dickson semifields, by doubling of central simple algebras. We thus obtain division algebras of dimension $2s^2$ by doubling central division algebras of degree $s$. Results on isomorphisms and automorphisms of these algebras are obtained in certain cases.
References:
[1] Brown, C., Pumplün, S.: The automorphisms of Petit's algebras. Communications in Algebra, 46, 2, 2018, 834-849, Taylor & Francis, DOI 10.1080/00927872.2017.1327598 | MR 3764900
[2] Burmester, M.V.D.: On the commutative non-associative division algebras of even order of LE Dickson. Rendiconti di Matematica e delle sue Applicazioni. Serie V, 21, 1962, 143-166, MR 0141692
[3] Burmester, M.V.D.: On the non-unicity of translation planes over division algebras. Archiv der Mathematik, 15, 1, 1964, 364-370, Springer, DOI 10.1007/BF01589214 | MR 0173971
[4] Cohen, S.D., Ganley, M.J.: Commutative semifields, two dimensional over their middle nuclei. Journal of Algebra, 75, 2, 1982, 373-385, Academic Press, DOI 10.1016/0021-8693(82)90045-X | MR 0653897
[5] Cordero, M., Wene, G.P.: A survey of finite semifields. Discrete Mathematics, 208, 1999, 125-137, Elsevier, DOI 10.1016/S0012-365X(99)00068-0 | MR 1725526
[6] Dickson, L.E.: On commutative linear algebras in which division is always uniquely possible. Transactions of the American Mathematical Society, 7, 4, 1906, 514-522, JSTOR, DOI 10.1090/S0002-9947-1906-1500764-6 | MR 1500764
[7] Ganley, M.J.: Central weak nucleus semifields. European Journal of Combinatorics, 2, 4, 1981, 339-347, Elsevier, DOI 10.1016/S0195-6698(81)80041-8 | MR 0638409
[8] 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 $\Pi (K)$. Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial, 14, 1, 2015, 27-42, Mathematical Sciences Publishers, MR 3450950
[9] Knus, M., Merkurjev, A., Rost, M., Tignol, J.: The book of involutions. 1998, American Mathematical Soc., MR 1632779
[10] Knuth, D.E.: Finite semifields and projective planes. 1963, Dissertation (Ph.D.), California Institute of Technology. MR 2939390
[11] Schafer, R.D.: An introduction to nonassociative algebras. 1995, Dover Publications, MR 1375235
[12] Thompson, D.: A generalisation of Dickson's commutative division algebras. Communications in Algebra, 48, 9, 2020, 3922-3932, DOI 10.1080/00927872.2020.1751849 | MR 4124670
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