Abstract
This chapter describes a construction of associative algebras that, despite starting from a commutation relation that the user may customize quite extensively, still manages to produce algebras with a number of useful properties: they have a Poincaré–Birkhoff–Witt type basis, they are equipped with a norm (actually an ultranorm) that is trivial to compute for basis elements, they are topologically complete, and they satisfy their given commutation relation. In addition, parameters can be chosen so that the algebras will in fact turn out to be skew fields and the norms become valuations. The construction is basically that of a power series algebra with given commutation relation, stated to be effective enough that the other properties can be derived. What is worked out in detail here is the case of algebras with two generators, but only the analysis of the commutation relation is specific for that case.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)
Bokut, L.A.: Embeddings into simple associative algebras (Russian). Algebra i Logika. 15(2), 117–142, 245 (1976)
Hellström, L., Silvestrov, S.D.: Commuting Elements in \(q\)-deformed Heisenberg Algebras. World Scientific Publishing Co. Inc., River Edge (2000)
Hellström, L.: The diamond lemma for power series algebras (doctorate thesis), Umeå University, xviii+228 pp. http://www.risc.jku.at/Groebner-Bases-Bibliography/details.php?details_id=1354. ISBN 91-7305-327-9 (2002)
Hellström, L.: A generic framework for diamond lemmas. arXiv:0712.1142v1 [math.RA] (2007)
Mora, T.: Seven variations on standard bases, preprint 45, Dip. Mat. Genova, 81 pp. http://www.disi.unige.it/person/MoraF/publications.html (1988)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Hellström, L. (2016). Valued Custom Skew Fields with Generalised PBW Property from Power Series Construction. In: Silvestrov, S., Rančić, M. (eds) Engineering Mathematics II. Springer Proceedings in Mathematics & Statistics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-42105-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-42105-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42104-9
Online ISBN: 978-3-319-42105-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)