Algebra and Applications 2. Группа авторов

Algebra and Applications 2 - Группа авторов


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      First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

      Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

      ISTE Ltd

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       www.iste.co.uk

      John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

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      © ISTE Ltd 2021

      The rights of Abdenacer Makhlouf to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

      Library of Congress Control Number: 2021942616

      British Library Cataloguing-in-Publication Data

      A CIP record for this book is available from the British Library

      ISBN 978-1-78945–018-7

      ERC code:

      PE1 Mathematics

       PE1_2 Algebra

       PE1_5 Lie groups, Lie algebras

       PE1_12 Mathematical physics

       Abdenacer MAKHLOUF

       University of Haute Alsace, Mulhouse, France

      The aim of this series of books is to report on the new trends of research in algebra and related topics. We provide an insight into the fast development of new concepts and theories related to algebra and present self-contained chapters on various topics, with each chapter combining some of the features of both graduate-level textbooks and research-level surveys. Each chapter includes an introduction with motivations and historical remarks, as well as the basic concepts, main results and perspectives. Moreover, the authors have commented on the relevance of the results in relation to other results and applications.

      I thank Kurusch Ebrahimi-Fard for suggesting these topics, presented at Benasque Intensive School, and express my deep gratitude to all of the contributors of this volume and to ISTE for their support.

      August 2021

      Algebraic Background for Numerical Methods, Control Theory and Renormalization

       Dominique MANCHON

       University of Clermont Auvergne, Clermont-Ferrand, France

      Since the pioneering work of Cayley in the 19th century (Cayley 1857), we have known that rooted trees and vector fields on the affine space are closely related. Surprisingly enough, rooted trees were also revealed to be a fundamental tool for studying not only the integral curves of vector fields, but also their Runge–Kutta numerical approximations (Butcher 1963).

spanned by rooted trees (where k is some field of characteristic zero) can be, in a nutshell, described as follows:
is both the free pre-Lie algebra with one generator and the free non-associative permutative algebra with one generator (Chapoton and Livernet 2001; Dzhumadil’daev and Löfwall 2002), and moreover, there are two other pre-Lie structures on
, of operadic nature, which show strong compatibility with the first pre-Lie (respectively the NAP) structure (Chapoton and Livernet 2001; Calaque et al. 2011; Manchon and Saidi 2011). The Hopf algebra of coordinates on the Butcher group (Butcher 1963), that is, the graded dual of the enveloping algebra
(with respect to the Lie bracket given by the first pre-Lie structure), was first investigated in Dür (1986), and intensively studied by Kreimer for renormalization purposes in Quantum Field Theory (Connes and Kreimer 1998; Kreimer 2002), see also Brouder (2000).

      This chapter is organized as follows: the first section is devoted to general connected graded or filtered Hopf algebras, including the renormalization of their characters. The second section gives a short presentation of operads in the symmetric monoidal category of vector spaces, and the third section will treat pre-Lie algebras in some detail: in particular, we will give a “pedestrian” proof of the Chapoton-Livernet theorem on free pre-Lie algebras. In the last section, Rota-Baxter, dendriform and NAP algebras will be introduced.

      We choose a base field k of characteristic zero. Most of the material here is borrowed from Manchon (2008), to which we can refer for more details.


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