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The Structure of Composition Algebras with Constants

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The Structure of Composition Algebras with Constants The Structure of Comp osition Algebras Die Struktur von Komp ositionsalgebren Dissertation zur Erlangung des akademischen Grades eines Doktors der Technischen Wissenschaften vorgelegt von DiplIng Erhard Aichinger Angefertigt am Institut f ur Algebra Sto chastik und wissensbasierte mathematische Systeme der technischnaturwissenschaftlichen Fakultat der Johannes Kepler Universitat Linz b ei OUnivProf Dr G unter Pilz Gefordert durch ein Doktorandenstip endium der osterreichischen Akademie der Wissenschaften Linz im Mai Begutachter ProfDrG unter Pilz Institut f ur Algebra Sto chastik und wissenbasierte mathematische Systeme Johannes Kepler Universitat Linz Osterreich ProfDrKalle Kaarli Institut f ur Mathematik Universitat Tartu Estland Vorwort Als ich im Mai mit der Arb eit an meiner Dissertation b egann wollte ich Komp ositionsringe untersuchen Komp ositionsringe entstehen wenn man die Menge aller Funktionen auf einem Ring mit den Op erationen der punktweisen Ad dition und Multiplikation sowie mit der Op eration der Hintereinanderausfuhrung von Funktionen versieht Ein Anliegen der Algebraiker ist es interessante und erstaunliche Eigenschaften einer Klasse von Algebren herzuleiten Verfolgen wir diese Arb eit am Beispiel der Fastkorper Fastkorper sind zunachst durch recht steril klingende Bedingungen deniert Sie sind jene algebraische Strukturen die b ei der Ko ordinatisierung b e stimmter pro jektiver Eb enen auftreten Es ist nun H Zassenhaus gelungen alle endlichen Fastkorper zu b eschreiben man erhalt diese schonen Strukturen durch Verdrehen der Korpermultiplikation eines endlichen Korpers bis auf sieb en Ausnahmefastkorper die Zassenhaus explizit angeb en konnte Komp ositionsringe hab en mit Fastringen gemeinsam da sie als Algebren von Funktionen auf einer anderen Algebra entstehen Daher hote ich den Komp osi tionsringen mit den Werkzeugen der Fastringtheorie b esonders mit den Dichte satzen erfolgreich zu Leib e r ucken Nat urlich war schon einiges uber Komp osi tionsringe b ekannt Man wute da jeder Komp ositionsring als Komp ositions ring von Funktionen auf einem Ring aufgefat werden kann was Cayleys Satz in der Grupp entheorie entspricht I Adler hatte im Jahr alle Komp ositions ringe von Funktionen auf einem Korper b estimmt die zumindest jede konstante Funktion enthalten Die Dichtesatze in der Hand glaubte ich einige weitere Fra gen klaren zu konnen Darunter war zum Beispiel die Bestimmung aller einfachen Komp ositionsringe Was mir dazu eingefallen ist erfahrt der Leser im f unften Kapitel dieser Dissertation Gleichzeitig interessierte ich mich f ur ein anderes Problem Wie lassen sich Poly nomfunktionen auf Grupp en und auf universellen Algebren b eschreiben Welche Funktionen lassen sich von Polynomfunktionen an einer gewissen Anzahl von Stellen interpolieren Eine Bedingung die eine Polynomfunktion erf ullen mu ist da sie Kongruenzen erhalt Also ist es nat urlich zu fragen wann jede kongruenzerhaltende Funktion eine Polynomfunktion ist Algebren b ei denen das der Fall ist nennen wir an vol lstandig Herrn K Kaarli verdanke ich den VORWORT Hinweis da eines der b edeutendsten Resultate zu diesem Thema in einer Ar b eit von J Hagemann und Chr Herrmann enthalten ist Da deren Arb eit ab er Konzepte verwendet die erst spater gr undlich studiert wurden so zum Beispiel die Kommutatortheorie f ur universelle Algebren lassen sich die Resultate heute eindringlicher als zur Zeit ihrer Entstehung formulieren Zum anderen sind mir die Beweise in dieser Arb eit bis heute ein Ratsel Da also kein Weg an Hagemanns und Herrmanns Resultaten vorbeif uhrte sah ich mich gezwungen sie erneut zu b eweisen Dazu hab e ich die Beweise da b estimmte Ringe und Grupp en an vollstandig sind in die Sprache der universellen Algebra ubersetzt Dort wo diese Ub ertragung den Inhalt verschleiert hat hab e ich auch die Beweise angegeb en von denen ich ausgegangen bin Ich hab e meine Einsicht in diese Resultate durch die universelle Sichtweise ab er do ch wesentlich verbessern konnen Im letzten Kapitel dieser Arb eit hab e ich die Resultate von Hagemann und Herrmann uber an vollstandige Algebren no ch einmal kurz zusammengestellt Eigentlich bin ich zum Kommutator ab er ganz anders als durch die universelle Al gebra gekommen Was man b ei der Interpolation einer Funktion auf einem Korper mithilfe der Multiplikation schat geht in der universellen Algebra mithilfe des Kommutators Multiplikationen auf Algebren die eine Korpermultiplikation im itieren wurden schon von HK Kaiser verwendet Den Dichtesatz f ur Fastringe kann man recht anschaulich b eweisen wenn man Funktionen in passender Weise multipliziert Auch SD Scott multiplizierte allerdings nicht Elemente o der Funktionen sondern Ideale in Grupp en In dieser Arb eit erklare ich wie sich diese Multiplikationen auf den Kommutator wie er in der universellen Algebra studiert wird zur uckfuhren lassen G Betsch SV Polin D Ramakotaiah und H Wielandt hab en eine sehr system atischen Strukturtheorie der Fastringe geschaen die sich auf Interpolationsre sultaten aufbauen lat Welche Auswirkungen konnen also die Interpolationsaus sagen der universellen Algebra auf die Strukturtheorie von Komp ositionsringen o der uberhaupt auf andere Algebren die den Fastringen ahnlich sind hab en Da ich Verallgemeinerungen dieser Interpolationsresultate zur Verfugung hatte konnte ich viele Resultate von Fastringen auf rechtsdistributive universellen Al gebren die dann Komp ositionsalgebren heien ubersetzen Die wesentliche Botschaft dieser Arb eit ist da sich viel aus der Fastringtheorie auf die rechts distributiven Algebren hin uberretten lat deren Addition sich zumindest in einer Hinsicht wie eine Grupp enop eration verhalt die Kongruenzen der additiven Struktur dieser Algebra m ussen b ez uglich des Relationspro dukts vertauschbar sein Das ist schon b ei Lo ops das sind nichtassoziative Grupp en immer der Fall Diese Verallgemeinerungen sind ab er nicht nur f ur jene Algebren frucht bar deren additive Struktur armer als die der Fastringe ist Was sie bringen kommt auch dann heraus wenn man die universell algebraischen Resultate auf Komp ositionsringe anwendet VORWORT Wahrend der Entwicklungen der theoretischen Resultate hab e ich immer wieder einzelne Fragestellungen am Computer untersucht Die brauchbaren Programme stehen im Paket SONATA f ur das Grupp entheoriesystem GAP und sind daher nicht in dieser Arb eit enthalten Ich danke G unter Pilz f ur die Betreuung und der osterreichischen Akademie der Wissenschaften f ur die Finanzierung dieser Arb eit eb enso danke ich Kalle Kaarli und Pawel Idziak f ur hilfreiche Anleitungen und meinen Kollegen Franz Binder Tim Boykett J urgen Ecker und Christof Nobauer f ur zahlreiche Diskussionen und Kommentare Den Dank f ur private Unterstutzung mochte ich p ersonlich abstatten lediglich meinen Eltern Gertraud und Bruno Aichinger sei die Freude nicht verwehrt hier namentlich erwahnt zu werden Linz im Mai VORWORT Preface When I started to work on this thesis four years ago my idea was to investi gate comp osition rings Those arise if one provides the set of all functions on a ring with the op erations of p ointwise addition and multiplication and with the op eration of functional comp osition An algebraists goal is to detect interesting and surprising prop erties of a class of algebras Nearelds provide a go o d example for the algebraists work reading the way they are dened they might seem as yet another structure of the quasi infra semi hemi para skew near type On a closer view one nds out that nearelds naturally arise in the co ordinatization of certain pro jective planes It was HZassenhaus who succeeded in describing all nite nearelds these b eau tiful structures can b e constructed from nite elds by doing some harm to the multiplication except for seven sp oradic nearelds that Zassenhaus managed to describ e explicitly Comp osition rings and nearrings have in common that they are algebras of func tions on another algebra This suggested to me that the to ols of nearring theory esp ecially the density theorems might prove useful also for comp osition rings What was known ab out comp osition rings It was known that every comp osi tion ring can b e embedded into a comp osition ring of functions on a ring This corresp onds to Cayleys result in group theory In IAdler determined all comp osition rings of functions on a eld that contain all constant functions Us ing the density theorems I hop ed to settle some other questions Among these is the determination of all simple comp osition rings What I have b een able to do in this direction can b e found in the fth chapter of this thesis At the same time I was interested in the following type of questions How can one describ e and distinguish p olynomial functions on groups and universal algebras Which functions can b e interpolated by a p olynomial function at a xed number of places We know that every p olynomial function preserves the congruences of an algebra So we may ask on which algebras every congruence preserving function is p olynomial Algebras where this is the case have b een called ane complete KKaarli p ointed out to me that JHagemann and ChrHerrmann have proved a central result on this topic Since they have used concepts that were to b e studied more thouroghly only later such as the theory of commutators PREFACE in universal algebra their results can b e formulated more easily to day On the other hand Hagemanns and Herrmanns pro ofs are dicult to follow Given the imp ortance of these results I have provided new pro ofs for them which
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