Download or read book Extended Affine Lie Algebras and Their Root Systems written by Bruce Normansell Allison and published by American Mathematical Soc.. This book was released on 1997 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work is about extended affine Lie algebras (EALA's) and their root systems. EALA's were introduced by Høegh-Krohn and Torresani under the name irreducible quasi-simple Lie algebras. The major objective is to develop enough theory to provide a firm foundation for further study of EALA's. The first chapter of the paper is devoted to establishing some basic structure theory. It includes a proof of the fact that, as conjectured by Kac, the invariant symmetric bilinear form on an EALA can be scaled so that its restriction to the real span of the root system is positive semi-definite. The second chapter studies extended affine root systems (EARS) which are an axiomatized version of the root systems arising from EALA's. The concept of a semilattice is used to give a complete description of EARS. In the final chapter, a number of new examples of extended affine Lie algebras are given. The concluding appendix contains an axiomatic characterization of the nonisotropic roots in an EARS in a more general context than the one used in the rest of the paper.

Download or read book Geometric Representation Theory and Extended Affine Lie Algebras written by Erhard Neher and published by American Mathematical Soc.. This book was released on 2011 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lie theory has connections to many other disciplines such as geometry, number theory, mathematical physics, and algebraic combinatorics. The interaction between algebra, geometry and combinatorics has proven to be extremely powerful in shedding new light on each of these areas. This book presents the lectures given at the Fields Institute Summer School on Geometric Representation Theory and Extended Affine Lie Algebras held at the University of Ottawa in 2009. It provides a systematic account by experts of some of the exciting developments in Lie algebras and representation theory in the last two decades. It includes topics such as geometric realizations of irreducible representations in three different approaches, combinatorics and geometry of canonical and crystal bases, finite $W$-algebras arising as the quantization of the transversal slice to a nilpotent orbit, structure theory of extended affine Lie algebras, and representation theory of affine Lie algebras at level zero. This book will be of interest to mathematicians working in Lie algebras and to graduate students interested in learning the basic ideas of some very active research directions. The extensive references in the book will be helpful to guide non-experts to the original sources.

Download or read book Quantum Affine Algebras Extended Affine Lie Algebras and Their Applications written by Yun Gao, Naihuan Jing, Michael Lau, and Kailash C. Misra and published by American Mathematical Soc.. This book was released on 2010 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Quantum Affine Algebras Extended Affine Lie Algebras and Their Applications written by Yun Gao and published by American Mathematical Soc.. This book was released on 2010 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the conference on Quantum Affine Algebras, Extended Affine Lie Algebras, and Applications, which was held at the Banff International Research Station, Banff, Canada, from March 2-7, 2008. Many of the papers include new results on different aspects of quantum affine algebras, extended affine Lie algebras, and their applications in other areas of mathematics and physics. Any reader interested in learning about the recent developments in quantum affine algebras and extended affine Lie algebras will benefit from this book.

Download or read book Extended Affine Lie Algebras and Extended Affine Weyl Groups written by and published by . This book was released on 1997 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis is about extended affine Lie algebras and extended affine Weyl groups. In Chapter I, we provide the basic knowledge necessary for the study of extended affine Lie algebras and related objects. In Chapter II, we show that the well-known twisting phenomena which appears in the realization of the twisted affine Lie algebras can be extended to the class of extended affine Lie algebras, in the sense that some extended affine Lie algebras (in particular nonsimply laced extended affine Lie algebras) can be realized as fixed point subalgebras of some other extended affine Lie algebras (in particular simply laced extended affine Lie algebras) relative to some finite order automorphism. We show that extended affine Lie algebras of type A1, B, C and BC can be realized as twisted subalgebras of types A§¤(l ¡Ã 2) and D algebras. Also we show that extended affine Lie algebras of type BC can be realized as twisted subalgebras of type C algebras. In Chapter III, the last chapter, we study the Weyl groups of reduced extended affine root systems. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements which in turn leads to a presentation of the Weyl group, called a presentation by conjugation. Using a new notion, called the index, which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems. We also show that extended affine Weyl groups (of index zero) are homomorphic images of some indefinite Weyl groups where the homomorphism and its kernel are given explicitly.

Download or read book Extended Affine Root System written by Kyoji Saito and published by . This book was released on 1999 with total page 49 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Developments and Trends in Infinite Dimensional Lie Theory written by Karl-Hermann Neeb and published by Springer Science & Business Media. This book was released on 2010-10-17 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups. Contributors: B. Allison, D. Beltiţă, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

Download or read book Affine Lie Algebras Weight Multiplicities and Branching Rules written by Sam Kass and published by Univ of California Press. This book was released on 1990-01-01 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: 00 This practical treatise is an introduction to the mathematics and physics of affine Kac-Moody algebras. It is the result of an unusual interdisciplinary effort by two physicists and two mathematicians to make this field understandable to a broad readership and to illuminate the connections among seemingly disparate domains of mathematics and physics that are tantalizingly suggested by the ubiquity of Lie theory. The book will be useful to Lie algebraists, high energy physicists, statistical mechanics, and number theorists. Volume One contains a description of Kac-Moody Lie algebras, and especially the affine algebras and their representations; the results of extensive computations follow in Volume Two, which is spiral bound for easy reference. This practical treatise is an introduction to the mathematics and physics of affine Kac-Moody algebras. It is the result of an unusual interdisciplinary effort by two physicists and two mathematicians to make this field understandable to a broad readership and to illuminate the connections among seemingly disparate domains of mathematics and physics that are tantalizingly suggested by the ubiquity of Lie theory. The book will be useful to Lie algebraists, high energy physicists, statistical mechanics, and number theorists. Volume One contains a description of Kac-Moody Lie algebras, and especially the affine algebras and their representations; the results of extensive computations follow in Volume Two, which is spiral bound for easy reference.

Download or read book Lie Algebras Graded by the Root Systems BC r r geq 2 written by Bruce Normansell Allison and published by American Mathematical Soc.. This book was released on 2002 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction The $\mathfrak{g}$-module decomposition of a $\mathrm{BC}_r$-graded Lie algebra, $r\ge 3$ (excluding type $\mathrm{D}_3)$ Models for $\mathrm{BC}_r$-graded Lie algebras, $r\ge 3$ (excluding type $\mathrm{D}_3)$ The $\mathfrak{g}$-module decomposition of a $\mathrm{BC}_r$-graded Lie algebra with grading subalgebra of type $\mathrm{B}_2$, $\mathrm{C}_2$, $\mathrm{D}_2$, or $\mathrm{D}_3$ Central extensions, derivations and invariant forms Models of $\mathrm{BC}_r$-graded Lie algebras with grading subalgebra of type $\mathrm{B}_2$, $\mathrm{C}_2$, $\mathrm{D}_2$, or $\mathrm{D}_3$ Appendix: Peirce decompositions in structurable algebras References.

Download or read book Torsors Reductive Group Schemes and Extended Affine Lie Algebras written by Philippe Gille and published by American Mathematical Soc.. This book was released on 2013-10-23 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that the authors take draws heavily from the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows the authors to find a bridge between multiloop algebras and the work of F. Bruhat and J. Tits on reductive groups over complete local fields.

Download or read book Introduction to Finite and Infinite Dimensional Lie Super algebras written by Neelacanta Sthanumoorthy and published by Academic Press. This book was released on 2016-04-26 with total page 514 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theory Closely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identities Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras Focuses on Kac-Moody algebras

Download or read book Integral Bases for Affine Lie Algebras and Their Universal Enveloping Algebras written by David Mitzman and published by American Mathematical Soc.. This book was released on 1985 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: A revised version of the author's PhD thesis written under the supervision of J Lepowsky at Rutgers University in 1983.

Download or read book Lie Algebras of Finite and Affine Type written by Roger William Carter and published by Cambridge University Press. This book was released on 2005-10-27 with total page 662 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a thorough but relaxed mathematical treatment of Lie algebras.

Download or read book Infinite Dimensional Lie Algebras written by Victor G. Kac and published by Springer Science & Business Media. This book was released on 2013-11-09 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Symmetries Integrable Systems and Representations written by Kenji Iohara and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 633 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the result of two international workshops; Infinite Analysis 11 – Frontier of Integrability – held at University of Tokyo, Japan in July 25th to 29th, 2011, and Symmetries, Integrable Systems and Representations held at Université Claude Bernard Lyon 1, France in December 13th to 16th, 2011. Included are research articles based on the talks presented at the workshops, latest results obtained thereafter, and some review articles. The subjects discussed range across diverse areas such as algebraic geometry, combinatorics, differential equations, integrable systems, representation theory, solvable lattice models and special functions. Through these topics, the reader will find some recent developments in the field of mathematical physics and their interactions with several other domains.

Download or read book Infinite Dimensional Lie Algebras And Groups written by Victor G Kac and published by World Scientific. This book was released on 1989-07-01 with total page 642 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contents:Integrable Representation of Kac-Moody Algebras: Results and Open Problems (V Chari & A Pressley)Existence of Certain Components in the Tensor Product of Two Integrable Highest Weight Modules for Kac-Moody Algebras (SKumar)Frobenius Action on the B-Cohomology (O Mathieu)Certain Rank Two Subsystems of Kac-Moody Root Systems (J Morita)Lie Groups Associated to Kac-Moody Lie Algebras: An Analytic Approach (E Rodriguez-Carrington)Almost Split-K-Forms of Kac-Moody Algebras (G Rousseau)Global Representations of the Diffeomorphism Groups of the Circle (F Bien)Path Space Realization of the Basic Representation of An(1) (E Date et al)Boson-Fermion Correspondence Over (C De Concini et al)Classification of Modular Invariant Representations of Affine Algebras (V G Kac & M Wakimoto)Standard Monomial Theory for SL2 (V Lakshmibai & C S Seshadri)Some Results on Modular Invariant Representations (S Lu)Current Algebras in 3+1 Space-Time Dimensions (J Mickelson)Standard Representations of An(1) (M Primc)Representations of the Algebra Uq(sI(2)), q-Orthogonal Polynomials and Invariants of Links (A N Kirillov & N Yu Reshetikhin)Infinite Super Grassmannians and Super Plücker Equations (M J Bergvelt)Drinfeld-Sokolov Hierarchies and t-Functions (H J Imbens)Super Boson-Fermion Correspondence of Type B (V G Kac & J W van de Leur)Prym Varieties and Soliton Equations (T Shiota)Polynomial Solutions of the BKP Hierarchy and Projective Representations of Symmetric Groups (Y You)Toward Generalized Macdonald's Identities (D Bernard)Conformal Theories with Non-Linearly Extended Virasoro Symmetries and Lie Algebra Classification (A Bilal & J-LGervais)Extended Conformal Algebras from Kac-Moody Algebras (P Bouwknegt)Meromorphic Conformal Field Theory (P Goddard)Local Extensions of the U(1) Current Algebra and Their Positive Energy Representations (R R Paunov & I T Todorov)Conformal Field Theory on Moduli Family of Stable Curves with Gauge Symmetries (A Tsuchiya & Y Yamada) Readership: Mathematicians and mathematical physicists

Download or read book Moonshine The First Quarter Century and Beyond written by James Lepowsky and published by Cambridge University Press. This book was released on 2010-06-03 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume examines the impact of the 'Monstrous Moonshine' paper on mathematics and theoretical physics.