{"product_id":"algebra-and-applications-2-combinatorial-algebra-and-hopf-algebras-hardback-9781789450187","title":"Algebra and Applications 2; Combinatorial Algebra and Hopf Algebras (Hardback) 9781789450187","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eAlgebra and Applications 2\u003c\/font\u003e\u003cbr\u003e\r\n\u003cfont size=\"5\"\u003eCombinatorial Algebra and Hopf Algebras\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\u003cp\u003e\u003cfont size=\"4\"\u003eAbdenacer Makhlouf (Edited by), A Makhlouf (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9781789450187, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 4 January 2022\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e336 pages\u003cbr\u003e1 x 1 x 1 cm, 0.454 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003eThis book is part of \u003ci\u003eAlgebra and Geometry\u003c\/i\u003e, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras.\u003cbr\u003e\u003cbr\u003eThe chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory. The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and Rota–Baxter algebras are explored.\u003cbr\u003e\u003cbr\u003e\u003ci\u003eAlgebra and Applications 2\u003c\/i\u003e is of great interest to graduate students and researchers. Each chapter combines some of the features of both a graduate level textbook and of research level surveys.\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e\u003cp\u003ePrefacexi\u003cbr\u003e\u003ci\u003eAbdenacer MAKHLOUF\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 1. Algebraic Background for Numerical Methods, Control Theory and Renormalization \u003c\/b\u003e\u003cb\u003e1\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eDominique MANCHON\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e1.1. Introduction 1\u003c\/p\u003e \u003cp\u003e1.2. Hopf algebras: generalproperties 2\u003c\/p\u003e \u003cp\u003e1.2.1. Algebras 2\u003c\/p\u003e \u003cp\u003e1.2.2. Coalgebras 3\u003c\/p\u003e \u003cp\u003e1.2.3. Convolution product 6\u003c\/p\u003e \u003cp\u003e1.2.4. Bialgebras andHopf algebras 7\u003c\/p\u003e \u003cp\u003e1.2.5. Some simple examples of Hopf algebras 8\u003c\/p\u003e \u003cp\u003e1.2.6. Some basic properties of Hopf algebras 9\u003c\/p\u003e \u003cp\u003e1.3. ConnectedHopf algebras 10\u003c\/p\u003e \u003cp\u003e1.3.1. Connectedgradedbialgebras 10\u003c\/p\u003e \u003cp\u003e1.3.2. An example: the Hopf algebra of decorated rooted trees 13\u003c\/p\u003e \u003cp\u003e1.3.3. Connectedfiltered bialgebras 14\u003c\/p\u003e \u003cp\u003e1.3.4. The convolution product 15\u003c\/p\u003e \u003cp\u003e1.3.5. Characters 17\u003c\/p\u003e \u003cp\u003e1.3.6. Group schemes and the Cartier–Milnor–Moore–Quillen theorem 19\u003c\/p\u003e \u003cp\u003e1.3.7. Renormalization in connected filtered Hopf algebras 21\u003c\/p\u003e \u003cp\u003e1.4. Pre-Lie algebras 24\u003c\/p\u003e \u003cp\u003e1.4.1. Definition and general properties 24\u003c\/p\u003e \u003cp\u003e1.4.2. The groupof formalflows 25\u003c\/p\u003e \u003cp\u003e1.4.3. The pre-Lie Poincaré–Birkhoff–Witt theorem 26\u003c\/p\u003e \u003cp\u003e1.5. Algebraicoperads 28\u003c\/p\u003e \u003cp\u003e1.5.1. Manipulatingalgebraicoperations 28\u003c\/p\u003e \u003cp\u003e1.5.2. The operad of multi-linear operations 29\u003c\/p\u003e \u003cp\u003e1.5.3. A definition for linear operads 31\u003c\/p\u003e \u003cp\u003e1.5.4. Afewexamplesof operads 32\u003c\/p\u003e \u003cp\u003e1.6. Pre-Lie algebras (continued) 35\u003c\/p\u003e \u003cp\u003e1.6.1. Pre-Lie algebras and augmented operads 35\u003c\/p\u003e \u003cp\u003e1.6.2. A pedestrian approach to free pre-Lie algebra 36\u003c\/p\u003e \u003cp\u003e1.6.3. Right-sided commutative Hopf algebras and theLoday–Roncotheorem 38\u003c\/p\u003e \u003cp\u003e1.6.4. Pre-Lie algebras of vectorfields 40\u003c\/p\u003e \u003cp\u003e1.6.5. B-series, composition and substitution 42\u003c\/p\u003e \u003cp\u003e1.7. Other related algebraic structures 44\u003c\/p\u003e \u003cp\u003e1.7.1. NAPalgebras 44\u003c\/p\u003e \u003cp\u003e1.7.2. Novikovalgebras 48\u003c\/p\u003e \u003cp\u003e1.7.3. Assosymmetric algebras 48\u003c\/p\u003e \u003cp\u003e1.7.4. Dendriformalgebras 48\u003c\/p\u003e \u003cp\u003e1.7.5. Post-Lie algebras 49\u003c\/p\u003e \u003cp\u003e1.8. References 50\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 2. From Iterated Integrals and Chronological Calculus to Hopf and Rota–Baxter Algebras \u003c\/b\u003e\u003cb\u003e55\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eKurusch EBRAHIMI-FARD and Frédéric PATRAS\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e2.1. Introduction 55\u003c\/p\u003e \u003cp\u003e2.2. Generalizediterated integrals 58\u003c\/p\u003e \u003cp\u003e2.2.1. Permutations andsimplices 59\u003c\/p\u003e \u003cp\u003e2.2.2. Descents,NCSFand theBCHformula 64\u003c\/p\u003e \u003cp\u003e2.2.3. Rooted trees and nonlinear differential equations 67\u003c\/p\u003e \u003cp\u003e2.2.4. Flows and Hopf algebraic structures 71\u003c\/p\u003e \u003cp\u003e2.3. Advances in chronological calculus 74\u003c\/p\u003e \u003cp\u003e2.3.1. Chronological calculus and half-shuffles 75\u003c\/p\u003e \u003cp\u003e2.3.2. Chronological calculus and pre-Lie products 79\u003c\/p\u003e \u003cp\u003e2.3.3. Time-ordered products and enveloping algebras 81\u003c\/p\u003e \u003cp\u003e2.3.4. Formal flows and Hopf algebraic structures 83\u003c\/p\u003e \u003cp\u003e2.4. Rota–Baxter algebras 87\u003c\/p\u003e \u003cp\u003e2.4.1. Origin 87\u003c\/p\u003e \u003cp\u003e2.4.2. Definition and examples 91\u003c\/p\u003e \u003cp\u003e2.4.3. Related algebraic structures 95\u003c\/p\u003e \u003cp\u003e2.4.4. Atkinson’s factorization and Bogoliubov’s recursion 101\u003c\/p\u003e \u003cp\u003e2.4.5. Spitzer’s identity: commutative case 103\u003c\/p\u003e \u003cp\u003e2.4.6. Free commutativeRota–Baxter algebras 107\u003c\/p\u003e \u003cp\u003e2.4.7. Spitzer’s identity: noncommutative case 108\u003c\/p\u003e \u003cp\u003e2.4.8. FreeRota–Baxter algebras 111\u003c\/p\u003e \u003cp\u003e2.5. References 113\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras \u003c\/b\u003e\u003cb\u003e119\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eJean-Yves THIBON\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e3.1. Introduction 119\u003c\/p\u003e \u003cp\u003e3.2. Classical symmetric functions 120\u003c\/p\u003e \u003cp\u003e3.2.1. Symmetric polynomials 120\u003c\/p\u003e \u003cp\u003e3.2.2. The Hopf algebra of symmetric functions 122\u003c\/p\u003e \u003cp\u003e3.2.3. The \u003ci\u003eλ\u003c\/i\u003e-ringnotation 124\u003c\/p\u003e \u003cp\u003e3.2.4. Symmetric functions and formal power series 125\u003c\/p\u003e \u003cp\u003e3.2.5. Duality 126\u003c\/p\u003e \u003cp\u003e3.3. Noncommutativesymmetric functions 129\u003c\/p\u003e \u003cp\u003e3.3.1. Basic definitions 129\u003c\/p\u003e \u003cp\u003e3.3.2. Generators andlinear bases 131\u003c\/p\u003e \u003cp\u003e3.3.3. Duality 133\u003c\/p\u003e \u003cp\u003e3.3.4. Solomon’sdescent algebras 136\u003c\/p\u003e \u003cp\u003e3.4. Lie series andLie idempotents 139\u003c\/p\u003e \u003cp\u003e3.4.1. Permutational operators on tensor spaces 139\u003c\/p\u003e \u003cp\u003e3.4.2. TheHausdorff series 139\u003c\/p\u003e \u003cp\u003e3.4.3. Lie idempotents in the descent algebra 143\u003c\/p\u003e \u003cp\u003e3.5. Lie idempotents as noncommutative symmetric functions 144\u003c\/p\u003e \u003cp\u003e3.5.1. Noncommutativepower-sums 144\u003c\/p\u003e \u003cp\u003e3.5.2. The Magnus expansion 146\u003c\/p\u003e \u003cp\u003e3.5.3. The continuous BCH expansion 148\u003c\/p\u003e \u003cp\u003e3.5.4. Another proof of the Magnus expansion 150\u003c\/p\u003e \u003cp\u003e3.5.5. The (1 \u003ci\u003e− q\u003c\/i\u003e)-transform 150\u003c\/p\u003e \u003cp\u003e3.5.6. Hopf algebras enter the scene 151\u003c\/p\u003e \u003cp\u003e3.5.7. A one-parameter family of Lie idempotents 152\u003c\/p\u003e \u003cp\u003e3.5.8. The iterated \u003ci\u003eq\u003c\/i\u003e-bracketing and its diagonalization 153\u003c\/p\u003e \u003cp\u003e3.6. Decompositionsof the descent algebras 155\u003c\/p\u003e \u003cp\u003e3.6.1. Complete families of minimal orthogonal idempotents 155\u003c\/p\u003e \u003cp\u003e3.6.2. Eulerianidempotents 156\u003c\/p\u003e \u003cp\u003e3.6.3. GeneralizedEulerianidempotents 160\u003c\/p\u003e \u003cp\u003e3.7. Decompositionsof the tensor algebra 160\u003c\/p\u003e \u003cp\u003e3.8. General deformations 162\u003c\/p\u003e \u003cp\u003e3.9. Lie quasi-idempotents as Lie polynomials 163\u003c\/p\u003e \u003cp\u003e3.9.1. The left derivative 163\u003c\/p\u003e \u003cp\u003e3.9.2. Multilinear Lie polynomials 164\u003c\/p\u003e \u003cp\u003e3.9.3. Decompositions on other bases 167\u003c\/p\u003e \u003cp\u003e3.10. Permutations and free quasi-symmetric functions 169\u003c\/p\u003e \u003cp\u003e3.10.1. Free quasi-symmetricfunctions 169\u003c\/p\u003e \u003cp\u003e3.11. Packed words and word quasi-symmetric functions 171\u003c\/p\u003e \u003cp\u003e3.12. References 175\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 4. From Runge–Kutta Methods to Hopf Algebras of Rooted Trees \u003c\/b\u003e\u003cb\u003e179\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eAnder MURUA\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e4.1. Numerical integration methods for ordinary differential equations 179\u003c\/p\u003e \u003cp\u003e4.1.1. Introduction 179\u003c\/p\u003e \u003cp\u003e4.1.2. Runge–Kutta methods 180\u003c\/p\u003e \u003cp\u003e4.2. Algebraic theory of Runge–Kutta methods 182\u003c\/p\u003e \u003cp\u003e4.2.1. The order conditions of RK methods 182\u003c\/p\u003e \u003cp\u003e4.2.2. The independence of order conditions 186\u003c\/p\u003e \u003cp\u003e4.2.3. Proof of necessary and sufficient order conditions 188\u003c\/p\u003e \u003cp\u003e4.2.4. Composition of RK methods, rooted trees and forests 191\u003c\/p\u003e \u003cp\u003e4.2.5. TheButchergroup 195\u003c\/p\u003e \u003cp\u003e4.2.6. Equivalence classes of RK methods 197\u003c\/p\u003e \u003cp\u003e4.2.7. Bibliographicalcomments 198\u003c\/p\u003e \u003cp\u003e4.3. B-series and relatedformal expansions 198\u003c\/p\u003e \u003cp\u003e4.3.1. B-series 198\u003c\/p\u003e \u003cp\u003e4.3.2. Backward error analysis, the exponential and the logarithm 199\u003c\/p\u003e \u003cp\u003e4.3.3. Series of linear differentialoperators 203\u003c\/p\u003e \u003cp\u003e4.3.4. The Lie algebra of the Butcher group 205\u003c\/p\u003e \u003cp\u003e4.3.5. The pre-Lie algebra structure on g 206\u003c\/p\u003e \u003cp\u003e4.3.6. Bibliographicalcomments 209\u003c\/p\u003e \u003cp\u003e4.4. Hopf algebrasof rootedtrees 209\u003c\/p\u003e \u003cp\u003e4.4.1. The commutative Hopf algebra of rooted trees 210\u003c\/p\u003e \u003cp\u003e4.4.2. The dual algebra H\u003ci\u003e∗ \u003c\/i\u003eand the dual Hopf algebra H\u003ci\u003e◦ \u003c\/i\u003e212\u003c\/p\u003e \u003cp\u003e4.4.3. B-series and series of differential operators revisited 213\u003c\/p\u003e \u003cp\u003e4.4.4. A universal property of the commutative Hopf algebra of rootedtrees 215\u003c\/p\u003e \u003cp\u003e4.4.5. The substitution law 216\u003c\/p\u003e \u003cp\u003e4.4.6. Bibliographicalcomments 217\u003c\/p\u003e \u003cp\u003e4.5. References 217\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 5. Combinatorial Algebra in Controllability and Optimal Control \u003c\/b\u003e\u003cb\u003e221\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eMatthias KAWSKI\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e5.1. Introduction 221\u003c\/p\u003e \u003cp\u003e5.1.1. Motivation: idealized examples 223\u003c\/p\u003e \u003cp\u003e5.1.2. Controlled dynamical systems 225\u003c\/p\u003e \u003cp\u003e5.1.3. Fundamental questions in control 226\u003c\/p\u003e \u003cp\u003e5.2. Analytic foundations 228\u003c\/p\u003e \u003cp\u003e5.2.1. State-space models and vector fields on manifolds 228\u003c\/p\u003e \u003cp\u003e5.2.2. Chronological calculus 230\u003c\/p\u003e \u003cp\u003e5.2.3. Piecewise constant controls and theBaker–Campbell–Hausdorff formula 233\u003c\/p\u003e \u003cp\u003e5.2.4. Picard iterationand formal series solutions 235\u003c\/p\u003e \u003cp\u003e5.2.5. The Chen–Fliess series and abstractions 237\u003c\/p\u003e \u003cp\u003e5.3. Controllability and optimality 241\u003c\/p\u003e \u003cp\u003e5.3.1. Reachable sets and accessibility 241\u003c\/p\u003e \u003cp\u003e5.3.2. Small-time local controllability 243\u003c\/p\u003e \u003cp\u003e5.3.3. Nilpotent approximatingsystems 247\u003c\/p\u003e \u003cp\u003e5.3.4. Optimality and the maximum principle 251\u003c\/p\u003e \u003cp\u003e5.3.5. Control variations and approximating cones 255\u003c\/p\u003e \u003cp\u003e5.4. Product expansions and realizations 262\u003c\/p\u003e \u003cp\u003e5.4.1. Variation of parameters and exponential products 263\u003c\/p\u003e \u003cp\u003e5.4.2. Computations using Zinbiel products 267\u003c\/p\u003e \u003cp\u003e5.4.3. Exponential products and normal forms for nilpotent systems 269\u003c\/p\u003e \u003cp\u003e5.4.4. Logarithmof theChen series 273\u003c\/p\u003e \u003cp\u003e5.5. References 279\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 6. Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application \u003c\/b\u003e\u003cb\u003e287\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eAlexander SCHMEDING\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e6.1. The Butcher group and the Connes–Kreimer algebra 288\u003c\/p\u003e \u003cp\u003e6.1.1. The Butcher group and B-series from numerical analysis 288\u003c\/p\u003e \u003cp\u003e6.1.2. Beyond the Butcher group 291\u003c\/p\u003e \u003cp\u003e6.2. Character groups of graded and connected Hopf algebras 292\u003c\/p\u003e \u003cp\u003e6.2.1. The exponential and logarithm 294\u003c\/p\u003e \u003cp\u003e6.3. Controlled groups of characters 297\u003c\/p\u003e \u003cp\u003e6.3.1. Conventions for this section 297\u003c\/p\u003e \u003cp\u003e6.3.2. Combinatorial Hopf algebras and the inverse factorial character 304\u003c\/p\u003e \u003cp\u003e6.4. Appendix: Calculus in locally convex spaces 305\u003c\/p\u003e \u003cp\u003e6.4.1. \u003ci\u003eCr\u003c\/i\u003e-Manifolds and \u003ci\u003eCr\u003c\/i\u003e-mappingsbetween them 306\u003c\/p\u003e \u003cp\u003e6.5. References 306\u003c\/p\u003e \u003cp\u003eList of Authors 311\u003c\/p\u003e \u003cp\u003eIndex 313 \u003c\/p\u003e\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mathematics [\u003ca title=\"See our other books on Mathematics\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mathematics%20%5BPB%5D%22\"\u003ePB\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Wiley-ISTE","offers":[{"title":"Brand New","offer_id":52446801920280,"sku":"9781789450187","price":99.89,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/files\/9781789450187.jpg?v=1785114172","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/algebra-and-applications-2-combinatorial-algebra-and-hopf-algebras-hardback-9781789450187","provider":"Freshly Printed Books","version":"1.0","type":"link"}