Ideals Of Identities Of Associative Algebras

This book concerns the study of the structure of identities of PI-algebras over a field of characteristic zero. In the first chapter, the author brings out the connection between varieties of algebras and finitely-generated superalgebras. The second chapter examines graded identities of finitely-generated PI-superalgebras. One of the results proved concerns the decomposition of T-ideals, which is very useful for the study of specific varieties. In the fifth section of Chapter Two, the author solves Specht's problem, which asks whether every associative algebra over a field of characteristic zero has a finite basis of identities. The book closes with an application of methods and results established earlier: the author finds asymptotic bases of identities of algebras with unity satisfying all of the identities of the full algebra of matrices of order two.

During the past forty years, a new trend in the theory of associative algebras, Lie algebras, and their representations has formed under the influence of mathematical logic and universal algebra, namely, the theory of varieties and identities of associative algebras, Lie algebras, and their representations. The last twenty years have seen the creation of the method of 2-words and *a-functions, which allowed a number of problems in the theory of groups, rings, Lie algebras, and their representations to be solved in a unified way. The possibilities of this method are far from exhausted. This book sums up the applications of the method of 2-words and *a-functions in the theory of varieties and gives a systematic exposition of contemporary achievements in the theory of identities of algebras and their representations closely related to this method. The aim is to make these topics accessible to a wider group of mathematicians.

A functional identity can be informally described as an identical relation involving arbitrary elements in an associative ring together with arbitrary (unknown) functions. The theory of functional identities is a relatively new one, and this is the first book on this subject. The book is accessible to a wide audience and touches on a variety of mathematical areas such as ring theory, algebra and operator theory.

This volume contains the talks given at the INDAM workshop entitled "Polynomial identites in algebras", held in Rome in September 2019. The purpose of the book is to present the current state of the art in the theory of PI-algebras. The review of the classical results in the last few years has pointed out new perspectives for the development of the theory. In particular, the contributions emphasize on the computational and combinatorial aspects of the theory, its connection with invariant theory, representation theory, growth problems. It is addressed to researchers in the field.

This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory. A PI-algebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finite-dimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of co-dimensions of any PI-algebra has integral exponential growth - the PI-exponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent.

With contributions derived from presentations at an international conference, Non-Associative Algebra and Its Applications explores a wide range of topics focusing on Lie algebras, nonassociative rings and algebras, quasigroups, loops, and related systems as well as applications of nonassociative algebra to geometry, physics, and natural sciences.

This book offers a panorama of recent advances in the theory of infinite groups. It contains survey papers contributed by leading specialists in group theory and other areas of mathematics. Topics include amenable groups, Kaehler groups, automorphism groups of rooted trees, rigidity, C*-algebras, random walks on groups, pro-p groups, Burnside groups, parafree groups, and Fuchsian groups. The accent is put on strong connections between group theory and other areas of mathematics.

A polynomial identity for an algebra (or a ring) A A is a polynomial in noncommutative variables that vanishes under any evaluation in A A. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

Anatolii Illarionovich Shirshov (1921–1981) was an outstanding Russian mat- maticianwhoseworksessentiallyin?uenced thetheoriesofassociative,Lie,Jordan and alternative rings. Many Shirshov’s students and students of his students had a successful research career in mathematics. AnatoliiShirshovwasbornonthe8thofAugustof1921inthevillageKolyvan near Novosibirsk. Before the II World War he started to study mathematics at Tomsk university but then went to the front to ?ght as a volunteer. In 1946 he continued his study at Voroshilovgrad (now Lugansk) Pedagogical Institute and at the same time taught mathematics at a secondary school. In 1950 Shirshov was accepted as a graduate student at the Moscow State University under the supervision of A. G. Kurosh. In 1953 he has successfully defended his Candidate of Science thesis (analog of a Ph. D. ) “Some problems in the theory of nonassociative rings and algebras” and joined the Department of Higher Algebra at the Moscow State University. In 1958 Shirshov was awarded the Doctor of Science degree for the thesis “On some classes of rings that are nearly associative”. In 1960 Shirshov moved to Novosibirsk (at the invitations of S. L. Sobolev and A. I. Malcev) to become one of the founders of the new mathematical institute of the Academy of Sciences (now Sobolev Institute of Mathematics) and to help the formation of the new Novosibirsk State University. From 1960 to 1973 he was a deputy director of the Institute and till his last days he led the research in the theory of algebras at the Institute.

This monograph is an important study of those Lie algebras which satisfy identical relations. It also deals with some of the applications of the theory. All principal results in the area are covered with the exception of those on Engel Lie algebras. The book contains basic information on Lie algebras, the varieties of Lie algebras in a general setting and the finite basis problem. An account is given of recent results on the Lie structure of associative PI algebras. The theory of identities in finite Lie algebras is also developed. In addition it contains applications to Group Theory, including some recent results on Burnside's problems.