Biblioteca de la Universidad Complutense de Madrid

Presentations of the unit group of an order in a non-split quaternion algebra


Corrales Rodrigáñez, Carmen y Jespers, Eric y Leal, Guilherme y Rio, Ángel del (2004) Presentations of the unit group of an order in a non-split quaternion algebra. Advances in Mathematics, 186 (2). pp. 498-524. ISSN 0001-8708

[img] PDF
Restringido a Sólo personal autorizado del repositorio


URL Oficial:


We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra H(K) over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of H(Z[(1+root-7)/(2)]). As a consequence a presentation is discovered for the orthogonal group SO3(Z[(1+root-7)/(2)]). These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra.

Tipo de documento:Artículo
Palabras clave:Algorithms; Finite sets of generators; Unit groups; Orders; quaternion algebras; Presentations
Materias:Ciencias > Matemáticas > Grupos (Matemáticas)
Código ID:14942

[1] A.F. Beardon, The Geometry of Discrete Groups, springer, Berlin, 1983.

[2] L. Bianchi, Sui gruppi de sostituzioni lineari con coeficienti appartenenti a corpi quadratici imaginari, Math. Ann. 40 (1892) 332–412.

[3] J. Elstrodt, F. Grunewald, J. Mennicke, Groups Acting on Hyperbolic Space, Harmonic Analysis and Number Theory, Springer, Berlin, 1998.

[4] B. Fein, B. Gordon, J.M. Smith, On the representation of 1 as a sum of two squares in an algebraic number field, J. Number Theory 3 (1971) 310–315.

[5] B. Fine, The Algebraic Theory of the Bianchi Groups, Marcel Dekker, New York, 1989.

[6] A.J. Hahn, O.T. O’Meara, The Classical Groups and K-Theory, Grundlehren der mathematischen Wissenschaften 291, Springer, Heidelberg, 1989.

[7] E. Jespers, Units in integral group rings: a survey, Proceedings of the International Conference on Methods in Ring Theory, Trento, 1997. Lecture Notes in Pure and Applied Mathematics, Vol. 198,Marcel Dekker, New York, 1998, pp. 141–169.

[8] E. Kleinert, Units in Skew Fields, Progress in Mathematics, 186, Birkha¨ user Verlag, Basel, 2000.

[9] E. Kleinert, Units of classical orders: a survey, Enseign. Math. (2) 40 (3–4) (1994) 205–248.

[10] H. Poincare´, Me´moire sur les groupes kleine´ es, Acta. Math. 3 (1883) 49–92.

[11] R. Riley, Applications of a computer implementation of Poincare´ ’s theorem on fundamental polyhedra, Math. Comp. 40 (162) (1983) 607–632.

Depositado:20 Abr 2012 11:51
Última Modificación:15 Ene 2016 14:56

Sólo personal del repositorio: página de control del artículo