On Amenable and Congenial Bases for Infinite Dimensional Algebras

On Amenable and Congenial Bases for Infinite Dimensional Algebras A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Rebin A. Muhammad May 2020 © 2020 Rebin A. Muhammad. All Rights Reserved. 2 This dissertation titled On Amenable and Congenial Bases for Infinite Dimensional Algebras by REBIN A. MUHAMMAD has been approved for the Department of Mathematics and the College of Arts and Sciences by Sergio R. Lopez-Permouth´ Professor of Mathematics Florenz Plassmann Dean, College of Arts and Sciences 3 Abstract MUHAMMAD, REBIN A., Ph.D., May 2020, Mathematics On Amenable and Congenial Bases for Infinite Dimensional Algebras (72 pp.) Director of Dissertation: Sergio R. Lopez-Permouth´ The study of the recently introduced notions of amenability, congeniality and simplicity of bases for infinite dimensional algebras is furthered. A basis B over an infinite dimensional F-algebra A is called amenable if FB, the direct product indexed by B of copies of the field F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules. If B is congenial to C but C is not congenial to B, then we say that B is properly congenial to C. An amenable basis B is called simple if it is not properly congenial to any other amenable basis and it is called projective if there does not exist any amenable basis which is properly congenial to B. We introduce a few families of algebras and study these notions in those contexts; in particular, we show the families to include examples of algebras without simple or projective bases. Our examples will illustrate the one-sided nature of amenability and simplicity as they include examples of bases which are amenable only on one side and, likewise, bases which have only one one-sided simple. We also consider the notions of amenability and congeniality of bases for infinite dimensional algebras in the context of the tensor product of bases for tensor products of algebras. In particular we give conditions for tensor product algebras to have simple bases in terms of the properties of the algebras being combined. Our results are then used to extend earlier results on the existence of simple bases in the algebra of single-variable polynomials to algebras of polynomials on several variables. 4 Dedication I would like to dedicate this dissertation to my grandma Aisha. 5 Acknowledgments First I would like to thank my advisor and mentor, Sergio R. Lopez-Permouth.´ His infinite support and his faith in me led me to reach the final milestone. During my research, he never tired of pushing me and guiding me in the right direction. I also have to thank my friend and fellow graduate student, Javier Ronquillo for encouraging me during my early years in graduate school. His company made grad school more fun and joyful. Another person who helped me a lot during these years was Elizabeth Story. We both were studying in different disciplines but we shared a lot of moments over these years. I would like to thank my collaborators, Pinar and Joaquin. During their visit to the Center of Ring Theory and its Applications (CRA) at Ohio University, my time with them was very productive and I enjoyed their company and friendship. I truly enjoyed my time as a graduate student and there are many graduate students who contributed to this joy, especially fellow students of Professor Lopez like Majed, Asiyeh, Ashley, Henry, Issac, Erick, and Dan. Many people from the staff at the Math Department made my time at Morton Hall fun, especially Nickie and Tammy. They were always helpful and never got tired of my demands. Finally, I have to thank Eylem for helping me immensely when I edited my dissertation. Her input made the dissertation more readable and understandable. I would like to thank everyone from my family and friends who helped me achieve this work. Without many people, I could not have continued from day one. 6 Table of Contents Page Abstract.........................................3 Dedication........................................4 Acknowledgments....................................5 1 Introduction and Preliminaries...........................7 1.1 Introduction..................................7 1.2 Preliminaries................................. 10 1.2.1 Matrices................................ 10 1.2.2 Modules............................... 10 1.2.3 Algebras Induced by Graphs..................... 15 1.2.4 Tensor Product............................ 19 2 Algebras without Simple Bases........................... 23 2.1 Basic and Simple Algebras.......................... 23 2.2 All Countable Dimensional Algebras have Non-amenable Bases...... 24 2.3 On the Symmetry of Amenability and Simplicity.............. 29 2.4 Algebras without Simple and Projective Bases................ 39 3 Tensor Product and Simplicity........................... 45 3.1 Tensor Product of Bases for Modules..................... 45 3.1.1 Sufficient Condition for Simplicity.................. 47 3.1.2 Simplicity of Tensor Products of Bases............... 49 3.2 Direct Sum and Amenability......................... 54 3.3 Laurent Polynomials............................. 57 4 On the Symmetry of Amenability of Bases and Related Properties........ 60 4.1 Kite Algebras................................. 60 4.2 Graph Algebra: Telescopic Bases....................... 67 References........................................ 72 7 1 Introduction and Preliminaries 1.1 Introduction Let A be an infinite-dimensional algebra over a field F and B be a basis for A. Let P be the F-vector space consisting of the direct product, indexed by B, of copies of the field Q B F. P may alternately be denoted BP, b2B Fb, or = F . In [2], the feasibility of a (left) A-module structure on P that is natural in the sense that it extends the module structure AA is discussed. To that avail, the notion of a (left) amenable basis B is defined by a condition that guarantees that BP has such a (left) A-module structure. When that is the case, one writes P = B M and says that B M is the basic submodule induced by the amenable basis B. The question whether amenable bases yield isomorphic module structures arises naturally and the notion of congeniality of bases is introduced in [2] in order to investigate it. Both amenability of bases and congeniality among them boil down to a requirement that certain linear transformations have representations which are row and column finite. Row and column finite have historically been the subject of much interest in the mathematical community. The survey paper [1] is a good introduction to some of the problems pertaining the topic. The remarkable result that any countable family of infinite matrices can be conjugated simultaneously to make each one of its members row and column finite (see [12]) was instrumental to prove, in [2], that every countable-dimensional algebra has at least one (left) amenable basis. That result raises the question whether it is possible to have an algebra where all bases are amenable. In Section 2.2, we include a result which was shared privately with us by Jose´ Armando Vivero (see Lemma 2.2.1) proving that there is no countable dimensional algebra such that every basis is amenable. We then use Vivero’s techniques to prove in Theorem 2.2.4 that one can find families of non amenable bases with additional requirements. In fact, we show that for any left amenable basis B, 8 there exist infinitely many non-amenable bases which are discordant to B and to one another. In section 1.2.3, we consider certain semigroup algebras where the semigroup is, in turn, induced by a graph. Our scheme to have said graphs induce the semigroup structures on their sets of vertices is an adaptation of the scheme mentioned, for example, in [9] and [13]. An adaptation of that scheme is necessary for our purposes because the structures in the literature involve exclusively finite graphs while we, on the other hand, are only interested in graphs over infinite sets of vertices. It is also important to point out that the structures in the two references mentioned (and all other literature that we are aware of) the authors use the expression graph algebras for the binary operations thus obtained (even in the general case the operations are not even expected to be associative.) We call a graph magma what [9] and [13] call a graph algebra and reserve the expression graph algebra for the semigroup algebra over an associative graph magma. Our terminology is introduced to avoid confusion and is motivated by the common trend in the literature to use the expression magma for binary operations considered without any further assupmtions. We will show that graph algebras are instrumental to answer various open questions regarding the amenability of bases. Section 2.3 is concerned with the possible left-right symmetry of the notions of amenability and simplicity bases. Can there be bases that are, say, left amenable but not right amenable? Must a left simple basis that is right amenable also be right simple? Amenable bases of polynomial and Laurent polynomial algebras over a field are investigated in [2]; the case of multivariable polynomials is addressed in [6]. Since those algebras are all commutative, there was no need to distinguish between left and right amenability there. We construct non-commutative infinite dimensional graph algebras where the notions of amenability and simplicity are not be left-right symmetric. Other 9 examples of one-sided amenable bases for algebras of infinite matrices may be found in [4]. Another question raised in [2] is whether all algebras have simple or projective bases. In the last section of this work, we construct (commutative) graph algebras which have neither simple nor projective bases. We close this section with some general notational conventions.

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