Solvable crossed product algebras revisited

Brown, Christian; Pumpluen, Susanne

Abstract

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF (M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit, and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F .

Citation

Brown, C., & Pumpluen, S. (2019). Solvable crossed product algebras revisited. Glasgow Mathematical Journal, 1-21. https://doi.org/10.1017/S0017089519000089

• Journal Article Type: Article
• Conference Name: Workshop on Nonassociative Algebra and Applications
• Start Date: Jul 9, 2018
• End Date: Jul 13, 2018
• Acceptance Date: Feb 13, 2019
• Online Publication Date: Apr 8, 2019
• Publication Date: Apr 8, 2019
• Deposit Date: Apr 8, 2019
• Publicly Available Date: Oct 9, 2019
• Journal: Glasgow Mathematical Journal
• Print ISSN: 0017-0895
• Electronic ISSN: 1469-509X
• Publisher: Cambridge University Press (CUP)
• Peer Reviewed: Peer Reviewed
• Pages: 1-21
• DOI: https://doi.org/10.1017/S0017089519000089
• Keywords: General Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/1562901
• Publisher URL: https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/solvable-crossed-product-algebras-revisited/C3E1991F0D9B3738FA311C274CC3EF5B
• Related Public URLs: https://arxiv.org/abs/1702.04605