DOCSLIB.ORG

CONTEMPORARY MATHEMATICS 229 Trends in the Representation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CONTEMPORARY MATHEMATICS 229 Trends in the Representation CONTEMPORARY MATHEMATICS 229 Trends in the Representation Theory of Finite Dimensional Algebras 1997 Joint Summer Research Conference on Trends in the Representation Theory of Finite Dimensional Algebras July 2Q-24, 1997 Seattle, Washington Edward L. Green Birge Huisgen-Zimmermann Editors http://dx.doi.org/10.1090/conm/229 Selected Titles in This Series 229 Edward L. Green and Birge Huisgen-Zimmermann, Editors, Trends in the representation theory of finite dimensional algebras, 1998 228 Liming Ge, Huaxin Lin, Zhong-Jin Ruan, Dianzhou Zhang, and Shuang Zhang, Editors, Operator algebras and operator theory, 1999 227 John McCleary, Editor, Higher homotopy structures in topology and mathematical physics, 1999 226 Luis A. Caffarelli and Mario Milman, Editors, Monge Ampere equation: Applications to geometry and optimization, 1999 225 Ronald C. Mullin and Gary L. Mullen, Editors, Finite fields: Theory, applications, and algorithms, 1999 224 Sang Geun Hahn, Hyo Chul Myung, and Efim Zelmanov, Editors, Recent progress in algebra, 1999 223 Bernard Chazelle, Jacob E. Goodman, and Richard Pollack, Editors, Advances in discrete and computational geometry, 1999 222 Kang-Tae Kim and Steven G. Krantz, Editors, Complex geometric analysis in Pohang, 1999 221 J. Robert Dorroh, Gisele Ruiz Goldstein, Jerome A. Goldstein, and Michael Mudi Tom, Editors, Applied analysis, 1999 220 Mark Mahowald and Stewart Priddy, Editors, Homotopy theory via algebraic geometry and group representations, 1998 219 Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov, Editors, Secondary calculus and cohomological physics, 1998 218 Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, Editors, Domain decomposition methods 10, 1998 217 Eric Carlen, Evans M. Harrell, and Michael Loss, Editors, Advances in differential equations and mathematical physics, 1998 216 Akram Aldroubi and EnBing Lin, Editors, Wavelets, multiwavelets, and their applications, 1998 215 M. G. Nerurkar, D.P. Dokken, and D. B. Ellis, Editors, Topological dynamics and applications, 1998 214 Lewis A. Coburn and Marc A. Rieffel, Editors, Perspectives on quantization, 1998 213 Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter, Editors, Studies on composition operators, 1998 212 E. Ramirez de Arellano, N. Salinas, M. V. Shapiro, and N. L. Vasilevski, Editors, Operator theory for complex and hypercomplex analysis, 1998 211 Jozef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers Colloquium, 1997 210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1998 209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential equations, 1997 208 Michel L. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors, Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997 207 Yujiro Kawamata and Vyacheslav V. Shokurov, Editors, Birational algebraic geometry: A conference on algebraic geometry in memory of Wei-Liang Chow (1911-1995), 1997 206 Adam Koranyi, Editor, Harmonic functions on trees and buildings, 1997 205 Paulo D. Cordaro and Howard Jacobowitz, Editors, Multidimensional complex analysis and partial differential equations: A collection of papers in honor of Fran<;ois Treves, 1997 (Continued in the back of this publication) Trends in the Representation Theory of Finite Dimensional Algebras CoNTEMPORARY MATHEMATICS 229 Trends in the Representation Theory of Finite Dimensional Algebras 1997 Joint Summer Research Conference on Trends in the Representation Theory of Finite Dimensional Algebras July 20-24, 1997 Seattle, Washington Edward L. Green Birge Huisgen-Zimmermann Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor Andy R. Magid Michael Vogelius Clark Robinson Peter M. Winkler The 1997 AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on "Trends in the Representation Theory of Finite Dimensional Algebras" was held at the University of Washington on July 2Q-24, 1997, with support from the Na- tional Science Foundation, Grant DMS-9618514, and the National Security Agency, Grant MDA904-97-1-0071. 1991 Mathematics Subject Classification. Primary 14D15, 14130, 16D30, 16D60, 16E10, 16E30, 16E40, 16G10, 16G20, 16G30, 16G50, 16G60, 16G70, 16P10, 16P20, 16P40, 16S80, 16W30, 17B37, 81R50. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Library of Congress Cataloging-in-Publication Data Trends in the representation theory of finite dimensional algebras : proceedings of the AMS- IMS-SIAM joint summer research conference, trends in the representation theory of finite dimen- sional algebras, July 20-24, 1997, University of Washington / Edward L. Green, Birge Huisgen- Zimmermann, editors. p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 229) Includes bibliographical references. ISBN 0-8218-0928-8 (soft : alk. paper) 1. Associative algebras-Congresses. 2. Representations of algebras-Congresses. I. Green, Edward L., [DATE]. II. Huisgen-Zimmermann, Birge, [DATE]. III. Series: Contemporary mathematics (American Mathematical Society) : v. 229. QA251.5.T74 1998 512'.24--dc21 98-44526 CIP Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission~ams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) @J 1998 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http: I /www. ams. org/ 10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98 Contents Preface IX List of Talks xi Postprojective partitions for tilting torsion pairs I. ASS EM AND F. U. COELHO 1 Derived canonical algebras as one-point extensions M. BAROT AND H. LENZING 7 Special biserial algebras and their automorphisms F. M. BLEHER 17 Wild subquivers of the Auslander-Reiten quiver of a tame algebra S. BRENNER AND M. C. R. BUTLER 29 Representation theory of noetherian Hopf algebras satisfying a polynomial identity K. A. BROWN 49 Finite representation type and periodic Hochschild (co-)homology R.-0. BUCHWEITZ 81 The syzygy theorem for monomial algebras M. C. R. BUTLER 111 Algebras whose derived category is tame J. A. DELAPENA 117 Circular biextensions of tame concealed algebras P. DRAXLER 129 On the distribution of AR-components of restricted Lie algebras R. FARNSTEINER 139 Compatible deformations M. GERSTENHABER AND A. GIAQUINTO 159 Directing objects in hereditary categories D. HAPPEL AND I. REITEN 169 On subcategories associated with tilting modules D. HAPPEL AND L. UNGER 181 vii viii CONTENTS Modules of the highest homological dimension over a Gorenstein ring Y. lWANAGA AND J. MIYACHI 193 Derived equivalence of graph algebras M. KAUER 201 Basic results on wild hereditary algebras 0. KERNER 215 On minimal approximations of modules H. KRAUSE AND M. SAORIN 227 Serre duality for generalized Auslander regular algebras R. MARTINEZ-VILLA 237 Classifying finite-dimensional semisimple Hopf algebras S. MONTGOMERY 265 Geometry of modules: Degenerations CH. RIEDTMANN 281 The preprojective algebra of a tame quiver: The irreducible components of the module varieties C. M. RINGEL 293 Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders D. SIMSON 307 Degenerations in module varieties with finitely many orbits A. SKOWRONSKI AND G. ZWARA 343 Preface The American Mathematical Society, the Institute of Mathematical Statis- tics, and the Society for Industrial and Applied Mathematics sponsored a joint summer research conference at the University of Washington, from July 2D-24, 1997, on Trends in the Representation Theory of Finite Dimensional Algebras. The conference was organized by E. L. Green (Virginia Polytechnic Institute and State University) and B. Huisgen-Zimmermann (University of California at Santa Barbara). The lectures focused on the following three topics: (1) Interactions between representation theory and algebraic geometry; (2) Homological methods; (3) Applications of representation theory to the study of quantum groups. All three lines were strongly represented in the conference by lectures providing keynote reports and talks addressing current research, as well as by contributions to this volume. (1) The analysis of representations via algebraic varieties is comparatively young. It goes back to the 1970's,
Load more

Recommended publications
  • Hypercomplex Analysis Selected Topics
    Hypercomplex Analysis Selected Topics (Habilitation Thesis) Roman Lávička Field: Mathematics — Mathematical Analysis Faculty of Mathematics and Physics Charles University in Prague August 2011 Contents 1 Introduction 5 2 Preliminaries of hypercomplex analysis 11 2.1 Clifford analysis . 11 2.1.1 Euclidean Clifford analysis . 14 2.1.2 Generalized Moisil-Théodoresco systems . 14 2.1.3 Hermitian Clifford analysis . 16 2.2 Quaternionic analysis . 17 3 Complete orthogonal Appell systems 19 3.1 Spherical harmonics . 19 3.1.1 Gelfand-Tsetlin bases for spin modules . 21 3.2 Clifford algebra valued spherical monogenics . 22 3.3 Spinor valued spherical monogenics . 24 3.3.1 The generalized Appell property in dimension 3 . 27 3.4 Hodge-de Rham systems . 29 3.4.1 The Riesz system in dimension 3 . 33 3.5 Hermitian monogenics . 35 4 Finely monogenic functions 39 4.1 Finely holomorphic functions . 39 4.2 Finely monogenic functions . 41 4.3 Finely differentiable monogenic functions . 42 4.4 Open problems . 45 List of reprinted papers 47 Bibliography 49 [L1] Reversible maps in the group of quaternionic Möbius trans- formations 55 [L2] A generalization of monogenic functions to fine domains 71 [L3] A remark on fine differentiability 83 [L4] Finely continuously differentiable functions 91 [L5] The Gelfand-Tsetlin bases for spherical monogenics in di- mension 3 105 3 [L6] Canonical bases for sl(2,C)-modules of spherical monogen- ics in dimension 3 135 [L7] Orthogonal basis for spherical monogenics by step two branch- ing 149 [L8] The Fischer Decomposition for the H-action and Its Appli- cations 177 [L9] The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces 189 [L10] Gelfand-Tsetlin Bases of Orthogonal Polynomials in Her- mitean Clifford Analysis 211 4 Chapter 1 Introduction Without any doubt, complex analysis belongs to the most important areas of mathematics.
    [Show full text]
  • Basic Sets of Special Monogenic Polynomials in Fréchet Modules
    Hindawi Journal of Complex Analysis Volume 2017, Article ID 2075938, 11 pages https://doi.org/10.1155/2017/2075938 Research Article Basic Sets of Special Monogenic Polynomials in Fréchet Modules Gamal Farghaly Hassan,1,2 Lassaad Aloui,3 and Allal Bakali1 1 Department of Mathematics, Faculty of Sciences, Northern Border University, P.O. Box 1321, Arar, Saudi Arabia 2Faculty of Science, University of Assiut, Assiut 71516, Egypt 3Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis-El Manar, Tunis, Tunisia Correspondence should be addressed to Gamal Farghaly Hassan; [email protected] Received 20 June 2016; Revised 31 October 2016; Accepted 18 December 2016; Published 14 February 2017 Academic Editor: Konstantin M. Dyakonov Copyright © 2017 Gamal Farghaly Hassan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article is concerned with the study of the theory of basic sets inFrechet´ modules in Clifford analysis. The main aim of this account, which is based on functional analysis consideration, is to formulate criteria of general type for the effectiveness (convergence properties) of basic sets either in the space itself or in a subspace of finer topology. By attributing particular forms for the Frechet´ module of different classes of functions, conditions are derived from the general criteria for the convergence properties in open and closed balls. Our results improve and generalize some known results in complex and Clifford setting concerning the effectiveness of basic sets. 1. Introduction aboutthestudyofbasicsetsofpolynomialsincomplexanal- ysis, we refer to [17–20].
    [Show full text]
  • Poly Slice Monogenic Functions, Cauchy Formulas and the PS
    POLY SLICE MONOGENIC FUNCTIONS, CAUCHY FORMULAS AND THE PS-FUNCTIONAL CALCULUS DANIEL ALPAY, FABRIZIO COLOMBO, KAMAL DIKI, AND IRENE SABADINI Abstract. Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the S-spectrum have had a very fast development. This new spectral theory based on the S-spectrum has applications for example in the formulation of quaternionic quantum mechanics, in Schur analysis and in fractional diffusion problems. The notion of poly slice analytic function has been recently introduced for the quaternionic setting. In this paper we study the theory of poly slice monogenic functions and the associated functional calculus, called PS-functional calculus, which is the polyanalytic version of the S-functional calculus. Also for this poly monogenic functional calculus we use the notion of S-spectrum. AMS Classification 47A10, 47A60. Keywords: Poly slice monogenic function, Cauchy formulas, PS-resolvent operators, modified S- resolvent operators, spectral theory on the S-spectrum. Contents 1. Introduction 1 2. Preliminary material 4 3. Poly slice monogenic functions 7 4. Cauchy formulas and product of poly slice monogenic functions 15 −1 −1 4.1. Cauchy formulas with kernels PℓSL and PℓSR 15 −1 −1 4.2. Cauchy formulas with kernels ΠℓSL and ΠℓSR 18 4.3. Poly ⊛L-product and ⊛R-product 21 5. Formulations of the PS-functional calculus via the ΠS-resolvent operators 23 6. Formulations of the PS-functional calculus via the modified S-resolvent operators 29 7. Equivalence of the two definitions of the PS-functional calculus and the product rules 34 References 39 arXiv:2011.13912v1 [math.FA] 27 Nov 2020 1.
    [Show full text]
  • Hypercomplex Analysis and Applications
    Trends in Mathematics Hypercomplex Analysis and Applications Bearbeitet von Irene Sabadini, Franciscus Sommen 1. Auflage 2010. Buch. vIII, 284 S. Hardcover ISBN 978 3 0346 0245 7 Format (B x L): 15,5 x 23,5 cm Gewicht: 602 g Weitere Fachgebiete > Mathematik > Mathematische Analysis Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Bounded Perturbations of the Resolvent Operators Associated to the F-Spectrum Fabrizio Colombo and Irene Sabadini Abstract. Recently,wehaveintroducedtheF-functional calculus and the SC-functional calculus. Our theory can be developed for operators of the form T = T0 + e1T1 + ...+ enTn where (T0,T1,...,Tn)isan(n + 1)-tuple of linear commuting operators. The SC-functional calculus, which is defined for bounded but also for unbounded operators, associates to a suitable slice monogenic function f with values in the Clifford algebra Rn the operator f(T ). The F-functional calculus has been defined, for bounded operators T , by an integral transform. Such an integral transform comes from the Fueter’s mapping theorem and it associates to a suitable slice monogenic function f n−1 the operator f˘(T ), where f˘(x)=Δ 2 f(x) and Δ is the Laplace operator. Both functional calculi are based on the notion of F-spectrum that plays the role that the classical spectrum plays for the Riesz-Dunford functional calculus.
    [Show full text]
  • The Culture of Quaternions the Phoenix Bird of Mathematics
    The Culture of Quaternions The Phoenix Bird of Mathematics Herb Klitzner June 1, 2015 Presentation to: New York Academy of Sciences, Lyceum Society © 2015, Herb Klitzner http://quaternions.klitzner.org The Phoenix Bird CONTENTS 1. INTRODUCTION - new uses after a long period of neglect 2. HISTORY AND CONTROVERSIES – perceptions of quaternions 3. APPLICATIONS – advantages; how quaternions operate 4. MATH – nature of quaternions 5. MUSIC COGNITION AND 4D – potential for new uses of quaternions Introduction The Word “Quaternion” • The English word quaternion comes from a Latin word quaterni which means grouping things “four by four.” • A passage in the New Testament (Acts 12:4) refers to a Roman Army detachment of four quaternions – 16 soldiers divided into groups of four, who take turns guarding Peter after his arrest by Herod. So a quaternion was a squad of four soldiers. • In poetry, a quaternion is a poem using a poetry style in which the theme is divided into four parts. Each part explores the complementary natures of the theme or subject. [Adapted from Wikipedia] • In mathematics, quaternions are generated from four fundamental elements (1, i, j, k). • Each of these four fundamental elements is associated with a unique dimension. So math quaternions are, by nature, a 4D system. Introduction The Arc of Dazzling Success and Near-Total Obscurity Quaternions were created in 1843 by William Hamilton. Today, few contemporary scientists are familiar with, or have even heard the word, quaternion. (Mathematical physics is an exception.) And yet -- • During the 19th Century quaternions became very popular in Great Britain and in many universities in the U.S.
    [Show full text]
  • Commutative Hypercomplex Mathematics Hypercomplex Algebra 4-D Function Theory Hypercomplex Analysis Links Home
    Clyde Davenport's Commutative Hypercomplex Math Page Page 1 of 11 Commutative Hypercomplex Mathematics Hypercomplex Algebra 4-D Function Theory Hypercomplex Analysis Links Home Introduction Below, we shall give a comprehensive sketch of the 4-D commutative hypercomplex algebra ( not quaternions) and its associated function theory and analysis. The great advantage is a complete, classical 4-D function theory, something that is impossible with quaternions and other noncommutative or nonassociative systems. Fortunately, a wide audience should be able to follow the discussion, because the commutative hypercomplex math is derived directly from well-known, fundamental concepts, such as groups, rings, calculus, complex variables, matrices, complex function theory, and vector analysis. For a discussion of elementary group and ring theory, see any good introductory text on abstract algebra, such as Herstein, 1986 . There will be no occasion for deep theorems and complicated proofs. Top First, some background. How many of us know that Sir William Rowan Hamilton developed the quaternions (4-D numbers similar to those that we will discuss here, except noncommutative) in the 1830s specifically for field calculations? And that scientists and engineers of the time vehemently resisted their use? Something in the mindset at the time simply could not accept the notion that there could be a "fourth dimension," especially if it was claimed to be time. They clung tenaciously to a primitive combination of component- by-component calculation and extensive use of geometry. By the late 1800s, their calculations were accompanied by elaborate geometrical figures that looked like the 2 X 4 framing of a house. To allay their aversions, around 1880, J.
    [Show full text]
  • The H∞ Functional Calculus Based on the S- Spectrum for Quaternionic
    Chapman University Chapman University Digital Commons Mathematics, Physics, and Computer Science Science and Technology Faculty Articles and Faculty Articles and Research Research 2016 The H∞ uncF tional Calculus Based on the S- Spectrum for Quaternionic Operators and for N- Tuples of Noncommuting Operators Daniel Alpay Chapman University, [email protected] Fabrizio Colombo Politecnico di Milano Tao Qian University of Macau Irene Sabadini Politecnico di Milano Tao Qian University of Macau Follow this and additional works at: http://digitalcommons.chapman.edu/scs_articles Part of the Algebra Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation D. Alpay, F. Colombo, I. Sabadini and T. Qian. The $H^\infty$ functional calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators. Journal of functional analysis, vol. 271, (2016), 1544-1584. This Article is brought to you for free and open access by the Science and Technology Faculty Articles and Research at Chapman University Digital Commons. It has been accepted for inclusion in Mathematics, Physics, and Computer Science Faculty Articles and Research by an authorized administrator of Chapman University Digital Commons. For more information, please contact [email protected]. The H∞ uncF tional Calculus Based on the S-Spectrum for Quaternionic Operators and for N-Tuples of Noncommuting Operators Comments NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Functional Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document.
    [Show full text]
  • Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries John Anthony Emanuello
    Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2015 Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries John Anthony Emanuello Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES ANALYSIS OF FUNCTIONS OF SPLIT-COMPLEX, MULTICOMPLEX, AND SPLIT-QUATERNIONIC VARIABLES AND THEIR ASSOCIATED CONFORMAL GEOMETRIES By JOHN ANTHONY EMANUELLO A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Spring Semester, 2015 Copyright ➞ 2015 John Anthony Emanuello. All Rights Reserved. John Anthony Emanuello defended this dissertation on March 27, 2015. The members of the supervisory committee were: Craig A. Nolder Professor Directing Dissertation Samuel Tabor University Representative Bettye Anne Case Committee Member John R. Quine Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii For my lovely wife Krisztina, whose love and support made this all worthwhile. iii ACKNOWLEDGMENTS First and foremost, I would like to thank my major professor, Dr. Craig Nolder. I have appreciated his assistance, patience, and collaboration more than I could possibly convey in words. Thank you for helping me become the mathematician I am today. Dr. Bettye Anne Case has gone above and beyond the call of duty as a committee member. Her advice and guidance in all areas of my career have far exceeded any expectation one could have for a professor and administrator as busy as her.
    [Show full text]
  • 1. Introduction 1.1
    Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 10–54 (DISCRETE) ALMANSI TYPE DECOMPOSITIONS: AN UMBRAL CALCULUS FRAMEWORK BASED ON osp(1 2) SYMMETRIES | NELSON FAUSTINO AND GUANGBIN REN Abstract: We introduce the umbral calculus formalism for hypercomplex vari- ables starting from the fact that the algebra of multivariate polynomials IR[x] shall be described in terms of the generators of the Weyl-Heisenberg algebra. The ex- tension of IR[x] to the algebra of Clifford-valued polynomials give rise to an algebra of Clifford-valued operators whose canonical generatorsP are isomorphic to the orthosymplectic Lie algebra osp(1 2). This extension provides an effective| framework in continuity and discreteness that allows us to establish an alternative version of Almansi decomposition com- prising continuous and discrete versions of classical Almansi theorem in Clifford analysis (c.f. [38, 33]) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces ker(D′)k. We will discuss afterwards how the symmetries of sl2(IR) (even part of osp(1 2)) are ubiquitous on the recent approach of Render (c.f. [37]), showing that they| can be interpreted as the method of separation of variables for the Hamiltonian operator in quantum mechanics. Keywords: Almansi theorem, Clifford analysis, hypercomplex variables, orthosym- plectic Lie algebras, umbral calculus. AMS Subject Classification (2000): 30G35;35C10;39A12;70H05. 1. Introduction 1.1. The Scope of Problems. In the last two decades considerable atten- tion has been given to the study of polynomial sequences for hypercomplex variables in different contexts. For example, in the approach proposed by Faustino & Kahler¨ (c.f.
    [Show full text]
  • Arxiv:1705.07737V1 [Math.GM]
    Conformal numbers S. Ulrych Wehrenbachhalde 35, CH-8053 Z¨urich, Switzerland Abstract The conformal compactification is considered in a hierarchy of hypercomplex pro- jective spaces with relevance in physics including Minkowski and Anti-de Sitter space. The geometries are expressed in terms of bicomplex Vahlen matrices and further broken down into their structural components. The relation between two subsequent projective spaces is displayed in terms of the complex unit and three additional hypercomplex numbers. Keywords: Clifford algebras, bicomplex numbers, AdS/CFT, twistor methods, Laplace equation PACS: 02.20.Sv, 02.30.Fn, 11.25.Hf, 04.20.Gz, 41.20.Cv 2010 MSC: 15A66, 30G35, 22E70, 53C28, 81T40 1. Introduction The complex numbers are central for the representation of physical processes in terms of mathematical models. However, they cannot cover all aspects alone and generalizations are necessary. One of the possible generalizations with po- tentially underestimated relevance in physics is the bicomplex number system [1, 2, 3, 4, 5]. Recent investigations in this area have been provided for example by [6, 7, 8, 9, 10]. The number system is also known under the name of Segre num- bers [11]. The bicomplex numbers coincide with the combination of hyperbolic and complex numbers formed by two commutative imaginary units, i √ 1 ≡ − and j √+1. The hyperbolic unit carries here the second complex number of ≡ arXiv:1705.07737v1 [math.GM] 18 May 2017 the bicomplex number system. More details on hyperbolic numbers have been provided beside many other authors by Yaglom [12], Sobczyk [13], Gal [14], or in correspondence with the bicomplex numbers by Rochon and Shapiro [15].
    [Show full text]
  • Contemporary Mathematics 212
    CONTEMPORARY MATHEMATICS 212 Operator Theory for Complex and Hypercomplex Analysis Operator Theory for Complex and Hypercomplex Analysis December 12-17, 1994 Mexico City, Mexico E. Ramirez de Arellano N. Salinas M. V. Shapiro N. L. Vasilevski Editors http://dx.doi.org/10.1090/conm/212 Selected Titles in This Series 212 E. Ramirez de Arellano, N. Salinas, M. V. Shapiro, and N. L. Vasilevski, Editors, Operator theory for complex and hypercomplex analysis, 1998 211 J6zef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers Colloquium, 1997 210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1997 209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential equations, 1997 208 MichelL. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors, Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997 207 Yujiro Kawamata and Vyacheslav V. Shokurov, Editors, Birational algebraic geometry: A conference on algebraic geometry in memory of Wei-Liang Chow (1911~1995), 1997 206 Adam Koranyi, Editor, Harmonic functions on trees and buildings, 1997 205 Paulo D. Cordaro and Howard Jacobowitz, Editors, Multidimensional complex analysis and partial differential equations: A collection of papers in honor of Fran«;ois Treves, 1997 204 Yair Censor and Simeon Reich, Editors, Recent developments in optimization theory and nonlinear analysis, 1997 203 Hanna Nencka and Jean-Pierre Bourguignon, Editors, Geometry and nature: In memory of W. K. Clifford, 1997 202 Jean-Louis Loday, James D. Stasheff, and Alexander A. Voronov, Editors, Operads: Proceedings of Renaissance Conferences, 1997 201 J. R. Quine and Peter Sarnak, Editors, Extremal Riemann surfaces, 1997 200 F.
    [Show full text]
  • On Generalized Vietoris' Number Sequences-Origins, Properties And
    On generalized Vietoris’ number sequences - origins, properties and applications I. Ca¸c˜aoa M. I. Falc˜aob H. R. Maloneka [email protected] [email protected] [email protected] a CIDMA and Department of Mathematics, University of Aveiro, Aveiro, Portugal b CMAT and Department of Mathematics and Applications, University of Minho, Campus de Gualtar, Braga, Portugal Abstract Ruscheweyh and Salinas showed in 2004 the relationship of a celebrated the- orem of Vietoris (1958) about the positivity of certain sine and cosine sums with the function theoretic concept of stable holomorphic functions in the unit disc. The present paper shows that the coefficient sequence in Vietoris’ theorem is iden- tical with the number sequence that characterizes generalized Appell sequences of homogeneous Clifford holomorphic polynomials in R3. The paper studies one- parameter generalizations of Vietoris’ number sequence, their properties as well as their role in the framework of Hypercomplex Function Theory. Keywords: Vietoris’ number sequence; generating functions; combinatorial identities; monogenic Appell polynomials 1 Introduction 1.1 Vietoris’ theorem and its coefficient’s sequence In [17] Vietoris proved the positivity of two trigonometric sums with a pairwise coeffi- cient’s sequence defined by coefficients with even index of the form 1 2m a m = m 0. (1) 2 22m m ≥ We refer to Vietoris’ theorem as it is given by Askey and Steinig in [3]: Theorem 1.1 (L. Vietoris). If a m (m 0) are given by (1) and the following adjacent 2 ≥ coefficients are supposed to be the same, i.e. a2m+1 := a2m then arXiv:1608.02579v1 [math.CA] 7 Aug 2016 n σn(x) = ak sin kx > 0, 0 < x < π, (2) k=1 Xn τn(x) = ak cos kx > 0, 0 < x < π.
    [Show full text]