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Free Products of Units in Algebras I. Quaternion Algebras

✍ Scribed by Jairo Z Gonçalves; Arnaldo Mandel; Mazi Shirvani

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
128 KB
Volume
214
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let A be a quaternion algebra over a commutative unital ring. We find sufficient conditions for pairs of units of A to generate a free group. Using the Ž . well-known isomorphism between SO 3, ‫ޒ‬ and the group of real quaternions of norm 1, we obtain free groups of rotations of the Euclidean 3-space. Specialization techniques allow us to find similar free subgroups in skew polynomial rings. A Ž consequence is the following: let kG be the group algebra of a residually torsion-. free nilpotent group G over a field k whose characteristic is not 2. If x and y are any pair of noncommuting elements of G, and c, d g k U then 1 q cx and 1 q dy generate a free subgroup of the Malcev᎐Neumann field of fractions of kG. ᮊ 1999 Academic Press

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