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 Extended Affine Root System written by Kyoji Saito and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 Extended Affine Root Systems written by K. Saito and published by . This book was released on 1984 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Extended Affine Root Systems III elliptic Weyl Groups written by Kyoji Saito and published by . This book was released on 1996 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We give a representation of an elliptic Weyl group W (R) (= the Weyl group for an elliptic root system*) R) in terms of the elliptic Dynkin diagram [gamma](R, G) for the elliptic root system. The representation is a generalization of a Coxeter system: the generators are in one to one correspondence with the vertices of the diagram and the relations consist of two groups: i) elliptic Coxeter relations attached to the diagram, and ii) a finiteness condition on the Coxeter transformation attached to the diagram. The group defined only by the elliptic Coxeter relations is isomorphic to the central extension W̃(R, G) of W(R) by an infinite cyclic group, called the hyperbolic extension of W(R)."

Download or read book Relations of the Weyl Groups of the Extended Affine Root Systems A i o t a 1 1 B i o t a 1 1 C i o t a 1 1 D i o t a 1 1 written by Tadayoshi Takebayashi and published by . This book was released on 1996 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Extended Affine Root Systems II written by Kyoji Saito and published by . This book was released on 1988 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Extended Affine Root System V written by Kyoji Saito and published by . This book was released on 1998 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "In the present paper, we ask for a criterion on the non-negativity of Fourier coefficients of a certain automorphic form [eta]([subscript R, G]) of one variable, called an elliptic eta-product (1.4) attached to an elliptic root system (R, G) (see Appendix 1). The goal theorem is the answer: Theorem. The Fourier coefficients at infinity of the elliptic eta-product are non-negative integers if and only if the elliptic eta-product is not a cusp form."

Download or read book Automorphisms of the Extended Affine Root System and Modular Property for Flat Theta Invariants written by Ikuo Satake and published by . This book was released on 1991 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We define the central extension [formula] of the automorphism group Aut[superscript +](R) of the extended affine root system. We give the action of [formula] on the flat theta invariants (theta functions). This describes the modular property for the flat theta invariants."

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 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 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 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 Locally Finite Root Systems written by Ottmar Loos and published by American Mathematical Soc.. This book was released on 2004 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop the basic theory of root systems $R$ in a real vector space $X$ which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: the intersection of $R$ with every finite-dimensional subspace of $X$ is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings.

Download or read book Encyclopedia of Special Functions The Askey Bateman Project Volume 2 Multivariable Special Functions written by Tom H. Koornwinder and published by Cambridge University Press. This book was released on 2020-10-15 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.

Download or read book The Monodromy Groups of Isolated Singularities of Complete Intersections written by Wolfgang Ebeling and published by Springer. This book was released on 2006-11-15 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Symmetric Functions and Orthogonal Polynomials written by Ian Grant Macdonald and published by American Mathematical Soc.. This book was released on 1998 with total page 71 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials, has long been known to be connected to combinatorics, representation theory and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures.