International Series of Monographs in Natural Philosophy, Volume 12: Current Algebras and their Applications provides an introduction to the underlying philosophy and to the technical methods associated with the use of the Current Algebra for the investigation of questions in elementary particle physics. This text contains 10 chapters and begins with the preliminary concepts and basic ideas of current algebras. The next chapters deal with the approximate symmetry and the dispersion theory of current algebras, as well as the current algebra sum rules with PCAC. These topics are followed by reviews of the principles of the low-energy theorems, the Schwinger terms, and the features of the dispersion theory. The last chapter examines the possible connections of current algebras and dynamics. This book will prove useful to mathematicians, physicists, teachers, and students.

This monograph is an introduction into a new and fast developing field on the crossroads of infinite-dimensional Lie algebra theory and contemporary mathematical physics. It contains a self-consistent presentation of the theory of Krichever-Novikov algebras, Lax operator algebras, their interaction, representation theory, relations to moduli spaces of Riemann surfaces and holomorphic vector bundles on them, to Lax integrable systems, and conformal field theory. For beginners, the book provides a short way to join in the investigations in these fields. For experts, it sums up the recent advances in the theory of almost graded infinite-dimensional Lie algebras and their applications. The book may serve as a base for semester lecture courses on finite-dimensional integrable systems, conformal field theory, almost graded Lie algebras. Majority of results are presented for the first time in the form of monograph.

This book is dedicated to the structure and combinatorics of classical Hopf algebras. Its main focus is on commutative and cocommutative Hopf algebras, such as algebras of representative functions on groups and enveloping algebras of Lie algebras, as explored in the works of Borel, Cartier, Hopf and others in the 1940s and 50s. The modern and systematic treatment uses the approach of natural operations, illuminating the structure of Hopf algebras by means of their endomorphisms and their combinatorics. Emphasizing notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras and Lie idempotents, the text also covers the important case of enveloping algebras of pre-Lie algebras. A wide range of applications are surveyed, highlighting the main ideas and fundamental results. Suitable as a textbook for masters or doctoral level programs, this book will be of interest to algebraists and anyone working in one of the fields of application of Hopf algebras.

The advent of quantum chromodynamics (QCD) in the early 1970s was one of the most important events in twentieth-century science. This book examines the conceptual steps that were crucial to the rise of QCD, placing them in historical context against the background of debates that were ongoing between the bootstrap approach and composite modeling, and between mathematical and realistic conceptions of quarks. It explains the origins of QCD in current algebra and its development through high-energy experiments, model-building, mathematical analysis and conceptual synthesis. Addressing a range of complex physical, philosophical and historiographical issues in detail, this book will interest graduate students and researchers in physics and in the history and philosophy of science.

Let M be a smooth manifold and G a Lie group. In this book we shall study infinite-dimensional Lie algebras associated both to the group Map(M, G) of smooth mappings from M to G and to the group of dif feomorphisms of M. In the former case the Lie algebra of the group is the algebra Mg of smooth mappings from M to the Lie algebra gof G. In the latter case the Lie algebra is the algebra Vect M of smooth vector fields on M. However, it turns out that in many applications to field theory and statistical physics one must deal with certain extensions of the above mentioned Lie algebras. In the simplest case M is the unit circle SI, G is a simple finite dimensional Lie group and the central extension of Map( SI, g) is an affine Kac-Moody algebra. The highest weight theory of finite dimensional Lie algebras can be extended to the case of an affine Lie algebra. The important point is that Map(Sl, g) can be split to positive and negative Fourier modes and the finite-dimensional piece g corre sponding to the zero mode.

Murray Gell-Mann is one of the leading physicists in the world. He was awarded the Nobel Prize in Physics in 1969 for his work on the SU(3) symmetry. His list of publications, albeit relatively short, is highly impressive — he has written mainly papers, which have become landmarks in physics. In 1953, Gell-Mann introduced the strangeness quantum number. In 1954, he proposed, together with F Low, the idea of the renormalization group. In 1958, Gell-Mann wrote, together with R Feynman, an important paper on the V-A theory of weak interactions. In 1961, Gell-Mann published his ideas on the SU(3) symmetry. In 1964, he proposed the quark model for hadrons. In 1971, Gell-Mann, together with H Fritzsch, proposed the color quantum number; and in 1972, the theory of QCD. These major publications of Gell-Mann are collected in this volume, thus providing physicists with easy access to the important publications of Gell-Mann.