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This book is a collection of research papers and surveys on algebra
that were presented at the Conference on Groups, Rings, and Group
Rings held in Ubatuba, Brazil. This text familiarizes researchers
with the latest topics, techniques, and methodologies in several
branches of contemporary algebra. With extensive coverage, it
examines broad themes from group theory and ring theory, exploring
their relationship with other branches of algebra including actions
of Hopf algebras, groups of units of group rings, combinatorics of
Young diagrams, polynomial identities, growth of algebras, and
more. Featuring international contributions, this book is ideal for
mathematicians specializing in these areas.
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.
A polynomial identity for an algebra (or a ring) $A$ is a
polynomial in noncommutative variables that vanishes under any
evaluation in $A$. 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.
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.
Results are extended to graded algebras and algebras with
involution. The book concludes with a study of the numerical
invariants and their asymptotics in the class of Lie algebras. Even
in algebras that are close to being associative, the behavior of
the sequences of co dimensions can be wild. The material is
suitable for graduate students and research mathematicians
interested in polynomial identity algebras.
This book is a collection of research papers and surveys on algebra
that were presented at the Conference on Groups, Rings, and Group
Rings held in Ubatuba, Brazil. This text familiarizes researchers
with the latest topics, techniques, and methodologies in several
branches of contemporary algebra. With extensive coverage, it
examines broad themes from group theory and ring theory, exploring
their relationship with other branches of algebra including actions
of Hopf algebras, groups of units of group rings, combinatorics of
Young diagrams, polynomial identities, growth of algebras, and
more. Featuring international contributions, this book is ideal for
mathematicians specializing in these areas.
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