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Author: A. Giambruno Publisher: American Mathematical Soc. ISBN: 0821838296 Category : Mathematics Languages : en Pages : 370
Book Description
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.
Author: A. Giambruno Publisher: American Mathematical Soc. ISBN: 0821838296 Category : Mathematics Languages : en Pages : 370
Book Description
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.
Author: Antonio Giambruno Publisher: CRC Press ISBN: 9780203911549 Category : Mathematics Languages : en Pages : 442
Book Description
Polynomial Identities and Combinatorial Methods presents a wide range of perspectives on topics ranging from ring theory and combinatorics to invariant theory and associative algebras. It covers recent breakthroughs and strategies impacting research on polynomial identities and identifies new concepts in algebraic combinatorics, invariant and representation theory, and Lie algebras and superalgebras for novel studies in the field. It presents intensive discussions on various methods and techniques relating the theory of polynomial identities to other branches of algebraic study and includes discussions on Hopf algebras and quantum polynomials, free algebras and Scheier varieties.
Author: Onofrio Mario Di Vincenzo Publisher: Springer Nature ISBN: 3030631117 Category : Mathematics Languages : en Pages : 421
Book Description
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.
Author: Eli Aljadeff Publisher: American Mathematical Soc. ISBN: 1470451743 Category : Education Languages : en Pages : 630
Book Description
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.
Author: Alexei Kanel-Belov Publisher: CRC Press ISBN: 1498720099 Category : Mathematics Languages : en Pages : 436
Book Description
Computational Aspects of Polynomial Identities: Volume l, Kemer's Theorems, 2nd Edition presents the underlying ideas in recent polynomial identity (PI)-theory and demonstrates the validity of the proofs of PI-theorems. This edition gives all the details involved in Kemer's proof of Specht's conjecture for affine PI-algebras in characteristic 0.The
Author: Alexei Kanel-Belov Publisher: CRC Press ISBN: 1439863725 Category : Mathematics Languages : en Pages : 400
Book Description
A comprehensive study of the main research done in polynomial identities over the last 25 years, including Kemer's solution to the Specht problem in characteristic O and examples in the characteristic p situation. The authors also cover codimension theory, starting with Regev's theorem and continuing through the Giambruno-Zaicev exponential rank. T
Author: A. Giambruno Publisher: American Mathematical Soc. ISBN: 0821847716 Category : Mathematics Languages : en Pages : 283
Book Description
Represents the proceedings of the conference on Groups, Rings and Group Rings, held July 28 - August 2, 2008, in Ubatuba, Brazil. This title contains results in active research areas in the theory of groups, group rings and algebras (including noncommutative rings), polynomial identities, Lie algebras and superalgebras.
Author: César Polcino Milies Publisher: American Mathematical Soc. ISBN: 0821852396 Category : Mathematics Languages : en Pages : 336
Book Description
Contains the proceedings of the XVIII Latin American Algebra Colloquium, held from August 3-8, 2009, in Sao Paulo, Brazil. It includes research articles as well as up-to-date surveys covering several directions of current research in algebra, such as Asymptotic Codimension Growth, Hopf Algebras, Structure Theory of both Associative and Non-Associative Algebras, Partial Actions of Groups on Rings, and contributions to Coding Theory.
Author: Helena Albuquerque Publisher: Springer Nature ISBN: 3031327071 Category : Mathematics Languages : en Pages : 305
Book Description
This proceedings volume presents a selection of peer-reviewed contributions from the Second Non-Associative Algebras and Related Topics (NAART II) conference, which was held at the University of Coimbra, Portugal, from July 18–22, 2022. The conference was held in honor of mathematician Alberto Elduque, who has made significant contributions to the study of non-associative structures such as Lie, Jordan, and Leibniz algebras. The papers in this volume are organized into four parts: Lie algebras, superalgebras, and groups; Leibniz algebras; associative and Jordan algebras; and other non-associative structures. They cover a variety of topics, including classification problems, special maps (automorphisms, derivations, etc.), constructions that relate different structures, and representation theory. One of the unique features of NAART is that it is open to all topics related to non-associative algebras, including octonion algebras, composite algebras, Banach algebras, connections with geometry, applications in coding theory, combinatorial problems, and more. This diversity allows researchers from a range of fields to find the conference subjects interesting and discover connections with their own areas, even if they are not traditionally considered non-associative algebraists. Since its inception in 2011, NAART has been committed to fostering cross-disciplinary connections in the study of non-associative structures.
Author: A. Giambruno
Author: A. Giambruno
Author: Antonio Giambruno
Author: Onofrio Mario Di Vincenzo
Author: Eli Aljadeff
Author: Alexei Kanel-Belov
Author: Alexei Kanel-Belov
Author: A. Giambruno
Author: César Polcino Milies
Author: 
Author: Helena Albuquerque