Invariant Lie Polynomials in Two and Three Variables
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And Hausdorff (1906)
BOOK REVIEWS 325 REFERENCES [C] A. P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sei. U.S.A. 74 (1977), 1324-1327. [CMM] R. R. Coifman, A. Mclntosh and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), 361-388. [Dl] G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sei. École Norm. Sup. (4) 17 (1984), 157-189. [D2] _, Opérateurs d'intégrale singulière sur les surfaces régulières, Ann. Sei. École Norm. Sup. (4)21 (1988), 225-258. [DS] G. David and S. Semmes, Singular integrals and rectifiable sets in M" : au-delà des graphes lipschitziens, Astérisque 193 (1991), 1-145. [D] J. R. Dorronsoro, A characterization of potential spaces, Proc. Amer. Math. Soc. 95 (1985), 21-31. [F] K. J. Falconer, Geometry of fractal sets, Cambridge Univ. Press, Cambridge, 1985. [Fe] H. Fédérer, Geometric measure theory, Springer-Verlag, Berlin and New York, 1969. [J] P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 ( 1990), 1-15. [M] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Univ. Press, Cambridge, 1995 (to appear). [S] S. Semmes, A criterion for the boundedness of singular integrals on hypersurfaces, Trans. Amer. Math. Soc. 311 (1989), 501-513. [SI] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. [S2] _, Harmonic analysis : Real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993. Pertti Mattila University of Jyvaskyla E-mail address : PMATTILAOJYLK. -
Free Lie Algebras
Last revised 3:11 p.m. April 10, 2020 Free Lie algebras Bill Casselman University of British Columbia [email protected] The purpose of this essay is to give an introduction to free Lie algebras and a few of their applications. My principal references are [Serre:1965], [Reutenauer:1993], and [de Graaf:2000]. My interest in free Lie algebras has been motivated by the well known conjecture that Kac•Moody algebras can be defined by generators and relations analogous to those introduced by Serre for finite•dimensional semi•simple Lie algebras. I have had this idea for a long time, but it was coming across the short note [de Graaf:1999] that acted as catalyst for this (alas! so far unfinished) project. Fix throughout this essay a commutative ring R. I recall that a Lie algebra over R is an R•module g together with a Poisson bracket [x, y] such that [x, x]=0 [x, [y,z]] + [y, [z, x]] + [z, [x, y]]=0 Since [x + y, x + y] = [x, x] + [x, y] + [y, x] + [y,y], the first condition implies that [x, y] = [y, x]. The − second condition is called the Jacobi identity. In a later version of this essay, I’ll discuss the Baker•Campbell•Hausdorff Theorem (in the form due to Dynkin). Contents 1. Magmas........................................... ................................ 1 2. ThefreeLiealgebra ................................ ................................ 3 3. Poincare•Birkhoff•Witt´ .................................... ......................... 5 4. FreeLiealgebrasandtensorproducts ................. .............................. 8 5. Hallsets—motivation -
On Free Conformal and Vertex Algebras
Journal of Algebra 217, 496᎐527Ž. 1999 Article ID jabr.1998.7834, available online at http:rrwww.idealibrary.com on On Free Conformal and Vertex Algebras Michael Roitman* CORE Department of Mathematics, Yale Uni¨ersity, New Ha¨en, Connecticut 06520 Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector E-mail: [email protected] Communicated by Efim Zelmano¨ Received April 1, 1998 Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representa- tions of the Monster. See, for example,wx 6, 10, 17, 15, 19 . In this paper we describe bases of free conformal and free vertex algebrasŽ as introduced inwx 6 ; see also w 20 x. All linear spaces are over a field މ- of characteristic 0. Throughout this paper ޚq will stand for the set of non-negative integers. In Sections 1 and 2 we give a review of conformal and vertex algebra theory. All statements in these sections are either inwx 9, 17, 16, 15, 18, 20 or easily follow from results therein. In Section 3 we investigate free conformal and vertex algebras. 1. CONFORMAL ALGEBRAS 1.1. Definition of Conformal Algebras We first recall some basic definitions and constructions; seew 16, 15, 18, 20x . The main object of investigation is defined as follows: DEFINITION 1.1. A conformal algebra is a linear space C endowed with a linear operator D: C ª C and a sequence of bilinear products"n : C m C ª C, n g ޚq, such that for any a, b g C one has * Partially supported by NSF Grant DMS-9704132. -
Combinatorics of Hall Trees and Hall Words
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series A 59, 285-308 (1992) Combinatorics of Hall Trees and Hall Words GUY MELANCON DPpartemenr de Mathkmatiques et d’lnformatique, UniversitP du Quebec ci Montr&al, C.P. 8888 WCC. A, Montrkal, Qukbec. Canada H3C3P8 Communicated by Gian-Carlo Rota Received May 30, 1990 We present combinatorial properties of Hall trees and Hall words. We give new proofs of the unique factorisation of w.ords into decreasing products of Hall words. An order cH on the free monoid is then constructed upon the unique factorisation of words. When the Hall set is the set of Lyndon words, the order cH coincides with the lexicographical order (with which Lyndon words are defined). Motivated by this fact, we give combinatorial properties of Hall words, related to the order c,,, which generalize known properties of Lyndon words to Hall words. 0 1992 Academic Press, Inc. INTRODUCTION Bases of the free Lie algebra appeared for the first time in an article by M. Hall [4], although they were implicit in the work of P. Hall [6] and Magnus [9] on the commutator calculus in free groups. Known as “Hall bases,” they inspired many authors and led to many generalizations: Meier-Wunderli [lo], Schiitzenberger [ 121, Shirshov [ 141, Gorchakov [3], and Ward [16]. Lyndon [8] introduced bases which were originally thought to be different. Viennot gave a generalization of all these constructions and showed that, in a sense, it was optimal [ 15, Theorem 1.21. He also showed that this generalization was equivalent to Lazard’s elimination process. -
7 Nov 2012 Deformation Theory and Rational Homotopy Type
1 Deformation Theory and Rational Homotopy Type 2 Mike Schlessinger and Jim Stasheff 3 November 8, 2012 4 Abstract 5 We regard the classification of rational homotopy types as a problem in algebraic de- 6 formation theory: any space with given cohomology is a perturbation, or deformation, 7 of the “formal” space with that cohomology. The classifying space is then a “moduli” 8 space — a certain quotient of an algebraic variety of perturbations. The description 9 we give of this moduli space links it with corresponding structures in homotopy theory, p 10 especially the classification of fibres spaces F → E → B with fixed fibre F in terms 11 of homotopy classes of maps of the base B into a classifying space constructed from 12 Aut(F ), the monoid of homotopy equivalences of F to itself. We adopt the philosophy, 13 later promoted by Deligne in response to Goldman and Millson, that any problem in 14 deformation theory is “controlled” by a differential graded Lie algebra, unique up to 15 homology equivalence (quasi-isomorphism) of dg Lie algebras. Here we extend this 16 philosophy further to control by L∞ -algebras. In memory of Dan Quillen who established the foundation on which this work rests. 17 Contents arXiv:1211.1647v1 [math.QA] 7 Nov 2012 18 1 Introduction 3 19 1.1 Background .................................... 3 20 1.2 ControlbyDGLAs ................................ 4 21 1.3 Applications.................................... 7 22 1.4 Outline....................................... 8 23 2 Models of homotopy types 10 24 2.1 The Tate–Jozefiak resolution in characteristic zero . ..... 10 25 2.2 TheHalperin–Stashefforfilteredmodel . -
Lie Algebras Theory and Algorithms [De Graaf
LIE ALGEBRAS THEORY AND ALGORITHMS North-Holland Mathematical Library Board of Honorary Editors: M. Artin, H. Bass, J. Eells, W. Feit, E J. Freyd, EW. Gehring, H. Halberstam, L.V. H6rmander, J.H.B. Kemperman, W.A.J. Luxemburg, EE Peterson, I.M. Singer and A.C. Zaanen Board of Advisory Editors." A. Bj6mer, R.H. Dijkgraaf, A. Dimca, A.S. Dow, J.J. Duistermaat, E. Looijenga, J.E May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sj6strand, J.H.M. Steenbrink, E Takens and J. van Mill VOLUME 56 ELSEVIER Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo Lie Algebras Theory and Algorithms Willem A. de Graaf University of St Andrews Scotland 2000 ELSEVIER Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 9 2000 Elsevier Science B.V. All rights reserved This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational, classroom use. Permissions may be sought directly from Elsevier Science Rights & Permissions Department, PO Box 800, Oxford OX5 IDX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected].