Authoritative source for the interplay between mathematics and different disciplines
Addresses the most important questions on interdisciplinarity
Edited and written by leading scientists
The goal of this Handbook is to become an authoritative source with chapters that show the
origins, unification, and points of similarity between different disciplines and mathematics.
Some chapters will also show bifurcations and the development of disciplines which grow to
take on a life of their own. Science and Art are used as umbrella terms to encompass the
physical, natural and geological sciences, as well as the visual and performing arts. As arts
imagine possibilities, science attempts to generate models to test possibilities, mathematics
serves as the tool. This handbook is an indispensable collection to understand todays effort to
build bridges between disciplines. It answers questions such as: What are the origins of
interdisciplinarity in mathematics? What are cross-cultural components of interdisciplinarity
linked to mathematics? What are contemporary interdisciplinary trends? Section Editors: Michael
J. Ostwald,University of New South Wales(Australia) Kyeong-Hwa Lee, Seoul National University
(South Korea) Torsten Lindstrom, Linnaeus University (Sweden) Gizem Karaali, Pomona College
(USA)Ken Valente, Colgate University, (USA) Consulting Editors: Alexandre Borovik, Manchester
University (UK) Daina Taimina, Independent Scholar, Cornell University (USA) Nathalie Sinclair,
Simon Fraser University (Canada)
Due 2021-07-30
2021, 2500 p.
Hardcover
ISBN 978-3-319-57071-6
Product category
Handbook
July 2020
Pages: 500
This book is a sequel to the book by the same authors entitled Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras.
The presentation begins with the Dirac notation, which is illustrated by boson and fermion oscillator algebras and also Grassmann algebra. Then detailed account of finite-dimensional representations of groups SL(2, C) and SU(2) and their Lie algebras is presented. The general theory of finite-dimensional irreducible representations of simple Lie algebras based on the construction of highest weight representations is given. The classification of all finite-dimensional irreducible representations of the Lie algebras of the classical series s?(n, C), so(n, C) and sp(2r, C) is exposed.
Finite-dimensional irreducible representations of linear groups SL(N, C) and their compact forms SU(N) are constructed on the basis of the Schur?Weyl duality. A special role here is played by the theory of representations of the symmetric group algebra C[Sr] (Schur?Frobenius theory, Okounkov?Vershik approach), based on combinatorics of Young diagrams and Young tableaux. Similar construction is given for pseudo-orthogonal groups O(p, q) and SO(p, q), including Lorentz groups O(1, N-1) and SO(1, N-1), and their Lie algebras, as well as symplectic groups Sp(p, q). The representation theory of Brauer algebra (centralizer algebra of SO(p, q) and Sp(p, q) groups in tensor representations) is discussed.
Finally, the covering groups Spin(p, q) for pseudo-orthogonal groups SO↑(p, q) are studied. For this purpose, Clifford algebras in spaces Rp, q are introduced and representations of these algebras are discussed.
Dirac Notations
Finite-dimensional Representations of Lie Algebras su(2) and s?(2, ?) and Lie Groups SU(2) and SL(2, ?)
Representations of Simple Lie Algebras. Weight Theory
Finite-dimensional Representations of Algebras s?(N, ?), su(N) and Groups SL(N, ?) and SU(N)
Finite-dimensional Representations of Groups SO, Sp and Lie Algebras so, sp
Groups Spin(p, q) and their Representations
Solutions to Selected Problems
Monographs and Reviews of a General Nature
Bibliography
Index
Graduate students and researchers in theoretical physics and mathematical physics.
June 2020
Pages: 348
Back in 1982, Edward Witten noticed that classical problems of differential geometry and differential topology such as the de Rham complex and Morse theory can be described in a very simple and transparent way using the language of supersymmetric quantum mechanics. Since then, many research papers have been written on this subject. Unfortunately not all the results in this field known to mathematicians have obtained a transparent physical interpretation, even if this new physical technique has also allowed many mathematical results to be derived which are completely new, in particular, hyper-Kaehler and the so-called HKT geometry. But in almost 40 years, no comprehensive monograph has appeared on this subject. So this book written by an expert in supersymmetric quantum field theories, supersymmetric quantum mechanics and its geometrical applications, addresses this yearning gap.
It comprises three parts: The first, GEOMETRY, gives basic information on the geometry of real, complex, hyper-Kaehler and HKT manifolds, and is principally addressed to the physicist. The second part "PHYSICS" presents information on classical mechanics with ordinary and Grassmann dynamics variables. Besides, the author introduces supersymmetry and dwells in particular on the representation of supersymmetry algebra in superspace. And the last and most important part of the book "SYNTHESIS", is where the ideas borrowed from physics are used to study purely mathematical phenomena.
Geometry:
Real Manifolds
Complex Manifolds
Hyper-Kahler and HKT Manifolds
Physics:
Dynamical Systems with and without Grassmann Variables
Supersymmetry
Path Integrals and the Witten Index
Superspace and Superfields
Synthesis:
Supersymmetric Description of the de Rham Complex
Supersymmetric Description of the Dolbeault Complex
Sigma Models with Extended Supersymmetries
Taming the Zoo of Models
HK and HKT through Harmonic Glasses
Gauge Fields on the Manifolds
Atiyah-Singer Theorem
Graduate students and researchers interested in theoretical and mathematical physics.