On the Elementary Theories of Free Nilpotent Lie Algebras and Free Nilpotent Groups by Mahmood Sohrabi A thesis submitted to The Faculty of Graduate Studies and Research In partial fulfilment of the requirements for the degree of Doctor of Philosophy School of Mathematics and Statistics Ottawa-Carleton Institute of Mathematics and Statistics Carleton University Ottawa, Ontario, Canada May, 2009 ©Copyright 2009 Mahmood Sohrabi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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Abstract In this thesis we explore certain aspects of the model theory of a free nilpotent Lie algebra of finite rank over a characteristic zero integral domain, and the model theory of a free nilpotent group of finite rank. In each case we give an algebraic description of objects elementarily equivalent to these structures. Our approach is as follows. In the case of a Lie algebra, we investigate to what extent one can recover the module structure of the algebra from the Lie ring structure. In the case of a group the question is to what extent one can recover exponentiation from knowing only the group operations. In both cases it turns out that one can not recover the extra structures completely and needs to introduce certain "quasi" objects to complete the characterization. Our work on free nilpotent Lie algebras of finite rank extends A. G. Myasnikov's work on finite dimensional algebras over a filed, in the case of free nilpotent groups of finite rank we extend 0. V. Belegradek's results on unitriangular groups and A.G. Myasnikov's work on nilpotent groups taking exponents in a characteristic zero field. This line of research takes its roots in the works of A. Tarski and A. Mal'cev from 195'0's. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments First of all I would like to thank my supervisor, Dr. Inna Bumagin, for her great advice and unconditional support. She is an extraordinary human being. I have enjoyed the insights mentorship and friendship of Prof. Alexei Myasnikov over the years. He has never stopped inspiring me. It is my pleasure to express my gratitude to him. I learnt a great deal from various members of the Ottawa-Carleton Institute of mathematics, specially from Dr. Benjamin Steinberg, Mathematics aside he has always been extremely encouraging and supportive. I owe him a thank. I would like also to thank my thesis examination committee, Prof. V. Romankov, Prof. R. Blute and Prof. A. Kranakis for their time spent on examining my thesis, their comments and insightful questions. My student life in Canada has not always been an easy ride. There are a countless of people who made it a tolerable, even an enjoyable journey and to whom I am deeply indebted. Among friends I should mention Ms. Houri Mashayekh, Dr. Khosrow Hassani, Dr. Ali Shirazi and Mohsen Nasrin. There are of course so many others. I should thank my office-mates, Behzad Omidi, Masoud Nassari and Mehrdad Kalan- tar for being such gentlemen and caring people. All those days that Mehrdad and I were hopelessly trying to break through a tough to read mathematics text, joyfully breezing through a well written one, explaining things to each other or discussing some problem are imprinted in my mind for ever. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Members of my extended family played an undeniable role in everything I have achieved. My thanks to them. My wife Roya and I have been through so much together. I could not even dream of having such an amazing partner. It is hard to imagine how someone can be as kind, caring and humble as she is. I am highly grateful for everything she has done for me. This thesis was typeset using 1^-TgX. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Abstract ii Acknowledgments iii Introduction 1 0.1 Elementary classification problem in groups 1 0.1.1 On elementary classification of nilpotent groups 3 0.1.2 On elementary classification of finite dimensional algebras . 5 0.2 Results and the structure of the thesis 6 1 Preliminaries and background 9 1.1 Preliminaries on model theory 9 1.1.1 Structures and signatures 9 1.1.2 Interpretations 11 1.1.3 Multi-sorted Vs. one-sorted structures IT 1.1.4 Types and saturation 18 1.1.5 Ultraproducts and saturated ultraproducts 18 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 Preliminaries on Lie algebras 21 1.2.1 Free Lie algebras 23 1.2.2 Free nilpotent Lie algebras 26 1.3 Preliminaries on nilpotent groups 29 1.3.1 Central series and nilpotent groups 29 1.3.2 A graded Lie ring Lie(G) associated to a nilpotent group G . 32 1.3.3 Hall-Petreseo formula and Mal'cev bases 33 1.3.4 Hall completion of r-groups and groups admitting exponents in a binomial domain . 36 1.3.5 Central and abelian extensions 40 1.3.6 A brief note on abelian groups and their model theory 43 1.3.7 Alternating bilinear maps and H2 45 2 Model theory of bilinear mappings 47 2.1 Regular Vs. absolute interpretability 48 2.2 Enrichments of bilinear mappings 50 2.3 Interpretability of the P(f) structure 53 3 Models of the Complete Theory of a Free Nilpotent Lie Algebra 59 3.1 A bilinear map associated to M and its largest ring of scalars 61 3.2 Characterization theorem 64 3.3 Conclusion, summary and some related open problems 76 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Models of the Complete Theory of a Free Nilpotent Group 79 4.1 Nr>c groups . 81 4.2 QNr>c groups 85 4.3 A bilinear map associated to Nr,c(R) and its largest ring of scalars . 88 4.4 Characterization theorem 91 4.5 Central extensions and elementary equivalence 97 r 4.6 A QA nc-group which is not Nr>c 102 4.7 Conclusion, summary and some related open problems 107 Bibliography 109 Index 114 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction 0.1 Elementary classification problem in groups Importance of the elementary (logical) classification of algebraic structures was em­ phasized in the works of A. Tarski and A. Mal'cev. In general, the problem of elementary classification requires to characterize, in somewhat algebraic terms, all algebraic structures (perhaps, from a given class) elementarily equivalent to a given one. Recall, that two algebraic structures A and B in a language L are elementarily equivalent (A = B) if they have the same first-order theories in L (indistinguishable in the first order logic in L). Tarski's results [43] on elementary theories of R and C (algebraically closed and real closed fields), as well as, the subsequent results of J. Ax and S. Kochen [1, 2, 3] and Yu. Ershov [16, 17, 18] on elementary theories of Qp (p-adically closed fields) became an algebraic classic and can be found in many text-books on model theory. One of the initial influential results on elementary classification of groups is due to W.Szmielew - she classified elementary theories of abelian groups in terms of "Szmielew's" invariants [42] (see also [14, 27, 4]). For non-abelian groups, the main inspiration, perhaps, was the famous Tarski's problem whether free non-abelian groups of finite rank are elementarily equivalent or not. This problem was open for 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.