Hyperbolic unit groups and quaternion algebras

We classify the quadratic extensions K = Q[root d] and the finite groups G for which the group ring o(K)[G] of G over the ring o(K) of integers of K has the property that the group U(1)(o(K)[G]) of units of augmentation 1 is hyperbolic. We also construct units in the Z-order H(o(K)) of the quaternion algebra H(K) = (-1, -1/K), when it is a division algebra.

Quaternion orders over quadratic integer rings from arithmetic fuchsian groups

In this paper we show that the quaternion orders OZ[ √ 2] ≃ ( √ 2, −1)Z[ √ 2] and OZ[ √ 3] ≃ (3 + 2√ 3, −1)Z[ √ 3], appearing in problems related to the coding theory [4], [3], are not maximal orders in the quaternion algebras AQ( √ 2) ≃ ( √ 2, −1)Q( √ 2) and AQ( √ 3) ≃ (3 + 2√ 3, −1)Q( √ 3), respectively. Furthermore, we identify the maximal orders containing these orders.

Lattices from maximal orders into quaternion algebras

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

De oplossing van quaternion-vergelijkingen.

Thesis (doctoral)--Rijks-Universiteit te Utrecht.

Quaternion Extensions of Order 16

2000 Mathematics Subject Classification: 12F12

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE SPHERE

In this paper, we determine the lower central and derived series for the braid groups of the sphere. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is important in its own right. The braid groups of the 2-sphere S(2) were studied by Fadell, Van Buskirk and Gillette during the 1960s, and are of particular interest due to the fact that they have torsion elements (which were characterised by Murasugi). We first prove that for all n epsilon N, the lower central series of the n-string braid group B(n)(S(2)) is constant from the commutator subgroup onwards. We obtain a presentation of Gamma(2)(Bn(S(2))), from which we observe that Gamma(2)(B(4)(S(2))) is a semi-direct product of the quaternion group Q(8) of order 8 by a free group F(2) of rank 2. As for the derived series of Bn(S(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group Bn(S(2)) being finite and soluble for n <= 3, the critical case is n = 4 for which the derived subgroup is the above semi-direct product Q(8) (sic) F(2). By proving a general result concerning the structure of the derived subgroup of a semi-direct product, we are able to determine completely the derived series of B(4)(S(2)) which from (B(4)(S(2)))(4) onwards coincides with that of the free group of rank 2, as well as its successive derived series quotients.

Flatness-based control of a single qubit gate

This work considers the open-loop control problem of steering a two-level quantum system from any initial to any final condition. The model of this system evolves on the state space X = SU(2), having two inputs that correspond to the complex amplitude of a resonant laser field. A symmetry preserving flat output is constructed using a fully geometric construction and quaternion computations. Simulation results of this flatness-based open-loop control are provided.

Rotações no espaço tridimensional por meio de produtos quaterniônicos

Pós-graduação em Matemática Universitária - IGCE

Equations of hyperelliptic Shimura curves

We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefinite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld’s nonarchimedean uniformisation of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet’s bimodules and the specialization of Heegner points, as introduced in [21]. As an application, we provide a list of equations of Shimura curves and quotients of them obtained by our algorithm that had been conjectured by Kurihara.

TUḤFAT al-NAṢĀ’IḤ de Yūsuf Gadā

Dans cette copie le bāb 45 est numéroté 46 par erreur, de même que la date de composition (f. 66v) est donnée comme étant 700H.[sic] au lieu de 795H. De la main de Muḥ. Zamān, on trouve des gloses lexicographiques entre les lignes ou dans le marges. La signature d’Aussant figure au f. 1. Ce ms. (Aussant n° 38) a été acquis par la B.N. en 1798. [Anc. cote, Suppl. persan 51].

AḤVĀL-i MAWT-i ḤUSAYN-‛ALĪ HĀN VA AḤVĀL-i MAWT-i PISAR

Poème masnavī compose de deux parties, l’histoire (qiṣṣa) de la chute, puis de la mort, du vizir Sayyid ‛Alī Hān (f. 7v- 59), assassiné le 9 octobre 1720 et remplacé par Muḥ. Amīn Hān I‛timād al-Dawla. Cette partie compte 919 bayt. La seconde (f. 61- 102v) en compte 763 et raconte les circonstances de la mort de Lāla Laliyān Mal, fils de l’auteur (f. 62), survenue dix mois après, alors qu’il avait 30 ans, et les campagnes militaires de cette période. Cette qiṣṣ̣a aurait été rédigée vers 1723 (f. 102).Une préface en prose (dībāča), rédigée (f. 5v) par Zōrāvar Singh [comparer à Supplément persan 247] occupe les f. 1v à 6v. Celui-ci indique qu’il édita, après sa mort l’œuvre du poète Rāy Kirpārām (f. 3v), et dédie cette édition (f. 2v) au moghol Abū l-Fatḥ Nāṣir al-Dīn Muḥ. Šāh (m. en 1161H./1748). Ce serait Rāy Kirpārām l’auteur des deux qiṣṣa.

Modules d'endo-permutation

Dans la th´eorie des repr´esentations modulaires des groupes finis, les modules d?endo-permutation occupent une place importante. En e_et, c?est le r?ole jou´e par ces modules dans l?analyse de la structure de certains modules simples pour des groupes finis p-nilpotents, qui a amen´e E. Dade `a en introduire le concept, en 1978. Quelques ann´ees plus tard, L. Puig a d´emontr´e que la source de n?importe quel module simple pour un groupe fini p-r´esoluble quelconque est un module d?endo-permutation. Plus r´ecemment, on s?est rendu compte que ces modules interviennent aussi dans l?analyse locale des cat´egories d´eriv´ees et dans l?´etude des syst`emes de fusion. La situation que l?on consid`ere est la suivante. On se donne un nombre premier p, un p-groupe fini P, un corps alg´ebriquement clos k de caract´eristique p et on veut d´eterminer tous les kP-modules d?endo-permutation couverts ind´ecomposables de type fini, c?est-`a-dire tous les kP-modules ind´ecomposables de type fini, tels que leur alg`ebre d?endomorphismes est un kP-module de permutation ayant un facteur direct trivial. On d´efinit une relation d?´equivalence sur l?ensemble de ces kP-modules et le produit tensoriel des modules induit une structure de groupe ab´elien sur l?ensemble des classes d?´equivalence. On appelle ce groupe, le groupe de Dade de P. Ainsi, classifier les modules d?endo-permutation couverts revient `a d´eterminer le groupe de Dade de P. Le groupe de Dade d?un p-groupe fini arbitraire est encore inconnu, bien qu?E. Dade, en 1978, ´etait d´ej`a parvenu `a la classification dans le cas o`u P est ab´elien. La premi`ere partie de ce travail de th`ese est consacr´ee au probl`eme de la classification dans le cas g´en´eral et r´esoud la question dans le cas de deux familles de p-groupes finis, `a savoir celle des p-groupes m´etacycliques, pour un nombre premier p impair, et celle des 2-groupes extrasp´eciaux, de la forme D8 _ · · · _ D8. Ces deux choix ont ´et´e motiv´es par le fait que ces groupes sont "presque" ab´eliens. De plus, certains r´esultats sur la structure du groupe de Dade d?un p-groupe fini quelconque rendent le groupe de Dade des groupes de ces deux familles plus simple `a ´etudier. Dans un deuxi`eme temps, nous nous sommes int´eress´es `a deux occurrences de ces modules dans la th´eorie de la repr´esentation des groupes finis, c?est-`a-dire `a deux raisons qui motivent leur ´etude. Ainsi, nous avons r´ealis´e des modules d?endo-permutation comme sources de modules simples. En particulier, il s?av`ere que, dans le cas d?un nombre premier p impair, tout module d?endo-permutation ind´ecomposable dont la classe est un ´el´ement de torsion dans le groupe de Dade est la source d?un module simple. Finalement, nous avons d´etermin´e, parmi tous les modules d?endo-permutation connus actuellement, lesquels poss`edent une r´esolution de permutation endo-scind´ee. Nous sommes arriv´es `a la conclusion que les seuls modules d?endo-permutation qui n?ont pas de r´esolution de permutation endo-scind´ee sont les modules "exceptionnels" apparaissant pour un 2-groupe de quaternions g´en´eralis´es.<br/><br/>In modular representation theory, endo-permutation modules occupy an important position. Indeed, the role that these modules play, in the analysis of the structure of some particular simple modules for finite p-nilpotent groups, induced E. Dade, in 1978, to give them their current name. A few years later, L. Puig proved that the source of any simple module for any finite psolvable group is an endo-permutation module. More recently, the occurrence of endo-permutation modules has also been noticed in the local analysis of splendid equivalences between derived categories and in the study of fusion systems. We consider the following situation. Given a prime number p, a finite pgroup P and an algebraically closed field k of characteristic p, we are looking for all finitely generated indecomposable capped endo-permutation kP-modules. That is, all finitely generated indecomposable kP-modules such that their endomorphism algebra is a permutation kP-module having a trivial direct summand. Then, we define an equivalence relation on the set of all isomorphism classes of such modules, and it turns out that the tensor product (over k) induces a structure of abelian group on this set. We call this group the Dade group of P. Hence, classifying all indecomposable finitely generated capped endo-permutation kPmodules is equivalent to determining the Dade group of P. At present, the Dade group of an arbitrary finite p-group is still unknown. However, E. Dade computed the Dade group of all finite abelian p-groups, in 1978 already. The first part of this doctoral thesis is concerned with the problem of the classification in the general case and solve it in the case of two families of finite p-groups, namely the metacyclic p-groups, for an odd prime number p, and the extraspecial 2-groups of the shape D8 _· · ·_D8. These two choices have been motivated by the fact that these groups are not far from being abelian. Moreover, some general results concerning the Dade group of arbitrary finite p-groups suggest that the Dade group of the groups belonging to these two families is easier to study. In the second part of this thesis, we have been looking at two particular occurrences of these modules in representation theory of finite groups which motivate the interest of their classification. Thus, we realised endo-permutation modules as sources of simple modules. In particular, it turns out that, in case p is an odd prime, any indecomposable module whose class in the Dade group is a torsion element is the source of some simple module. Finally, we considered all the modules we know at present and determined which ones have an endo-split permutation resolution. We could then conclude that all but the "exceptionnal" modules occurring in the generalized quaternion case have an endo-split permutation resolution.<br/><br/>"Module d?endo-permutation" n?est pas le nom d?une maladie exotique contagieuse (du moins pas `a ma connaissance), comme vous pourriez peut-?etre l?imaginer si vous faites partie des personnes qui croient que le titre de docteur n?est destin´e qu?aux m´edecins. Dans ce cas, il se peut que le sujet dont il est question ici vous cause quelques naus´ees et r´eveille de douloureux souvenirs d?´ecole, car un module d?endo-permutation est un objet math´ematique, alg´ebrique, plus pr´ecis´ement. Ce concept a ´et´e introduit il y a un quart de si`ecle, de l?autre c?ot´e de l?Atlantique, et il s?est r´ev´el´e su_samment int´eressant pour qu?aujourd?hui il ait franchi bien des fronti`eres, celles de l?alg`ebre y compris. Mais de quoi s?agit-il ? Si vous entendez le terme "endo-permutation" probablement pour la premi`ere fois, ce n?est certainement pas le cas pour celui de "module". Cependant, sa d´efinition dans le pr´esent contexte ne co¨ýncide avec aucune de celles figurant dans les dictionnaires ordinaires. Les personnes qui ont d´ej`a entendu parler de Frobenius, Burnside, Schur, ou encore Brauer, pourront vous dire qu?un module est une repr´esentation. "De quoi ?" vous demanderezvous. "Un spectacle de marionnettes, peut-?etre ?" Bien s?ur que non ! Un module d?endo-permutation est une repr´esentation particuli`ere de certains groupes finis, o`u un groupe n?est pas un groupe de rock, comme vous pouvez vous en douter, mais d´esigne un objet math´ematique connu par tous les ´etudiants en sciences au terme de leur premi`ere ann´ee universitaire (en th´eorie, du moins). La "popularit´e" de la notion de groupe, fini ou non, est due au fait que les groupes sont fr´equemment utilis´es, aussi bien dans le domaine abstrait des math´ematiques, que dans le monde r´eel des physiciens, chimistes et autres biologistes (pour ne citer qu?eux). "Mais comment peut-on utiliser concr`etement ces objets invisibles ?" vous demanderez-vous alors. Et bien, justement, en les consid´erant par l?interm´ediaire de leurs repr´esentations, c?est-`a-dire en leur associant des matrices, de fa¸con plus ou moins naturelle. Or, comme il y a "beaucoup trop" de matrices pour un groupe donn´e, elles sont classifi´ees selon certaines de leurs propri´et´es, ce qui permet de les r´epertorier dans diverses familles (celle des modules d?endo-permutation, par exemple). Un groupe est ainsi rendu "concret", car les donn´ees matricielles sont manipulables par tous les scienti- fiques (et leurs ordinateurs), qui peuvent alors les utiliser dans leurs recherches, afin de contribuer au progr`es de la science. En toute franchise, c?est bien loin de ces soucis terre-`a-terre que ce travail de th`ese sur la classification des modules d?endo-permutation a ´et´e accompli. En fait, quitte `a choquer certaines ?ames sensibles, sa r´ealisation est surtout due au caract`ere ´epicure de son auteur, qui, avouons-le, en a ´et´e pleinement satisfait !

Computations in Relative Algebraic K-Groups

Let G be finite group and K a number field or a p-adic field with ring of integers O_K. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K_0(O_K[G],K) as an abstract abelian group. We solve the discrete logarithm problem, both in K_0(O_K[G],K) and the locally free class group cl(O_K[G]). All algorithms have been implemented in MAGMA for the case K = \IQ. In the second part of the manuscript we prove formulae for the torsion subgroup of K_0(\IZ[G],\IQ) for large classes of dihedral and quaternion groups.

The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.