Algebra and applications 2 : combinatorial algebra and Hopf algebras / / edited by Abdenacer Makhlouf

• Pubblicazione: Hoboken, New Jersey : , : John Wiley & Sons, Incorporated, , [2021]; ©2021
• Descrizione fisica: 1 online resource (336 pages)
• Disciplina: 512
• Soggetto topico: Algebra
• Soggetto genere / forma: Electronic books.
• Persona (resp. second.): MakhloufAbdenacer
• Nota di contenuto: Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Preface -- 1. Algebraic Background for Numerical Methods, Control Theory and Renormalization -- 1.1. Introduction -- 1.2. Hopf algebras: general properties -- 1.2.1. Algebras -- 1.2.2. Coalgebras -- 1.2.3. Convolution product -- 1.2.4. Bialgebras and Hopf algebras -- 1.2.5. Some simple examples of Hopf algebras -- 1.2.6. Some basic properties of Hopf algebras -- 1.3. Connected Hopf algebras -- 1.3.1. Connected graded bialgebras -- 1.3.2. An example: the Hopf algebra of decorated rooted trees -- 1.3.3. Connected filtered bialgebras -- 1.3.4. The convolution product -- 1.3.5. Characters -- 1.3.6. Group schemes and the Cartier-Milnor-Moore-Quillen theorem -- 1.3.7. Renormalization in connected filtered Hopf algebras -- 1.4. Pre-Lie algebras -- 1.4.1. Definition and general properties -- 1.4.2. The group of formal flows -- 1.4.3. The pre-Lie Poincaré-Birkhoff-Witt theorem -- 1.5. Algebraic operads -- 1.5.1. Manipulating algebraic operations -- 1.5.2. The operad of multi-linear operations -- 1.5.3. A definition for linear operads -- 1.5.4. A few examples of operads -- 1.6. Pre-Lie algebras (continued) -- 1.6.1. Pre-Lie algebras and augmented operads -- 1.6.2. A pedestrian approach to free pre-Lie algebra -- 1.6.3. Right-sided commutative Hopf algebras and the Loday-Ronco theorem -- 1.6.4. Pre-Lie algebras of vector fields -- 1.6.5. B-series, composition and substitution -- 1.7. Other related algebraic structures -- 1.7.1. NAP algebras -- 1.7.2. Novikov algebras -- 1.7.3. Assosymmetric algebras -- 1.7.4. Dendriform algebras -- 1.7.5. Post-Lie algebras -- 1.8. References -- 2. From Iterated Integrals and Chronological Calculus to Hopf and Rota-Baxter Algebras -- 2.1. Introduction -- 2.2. Generalized iterated integrals -- 2.2.1. Permutations and simplices. ; 2.2.2. Descents, NCSF and the BCH formula -- 2.2.3. Rooted trees and nonlinear differential equations -- 2.2.4. Flows and Hopf algebraic structures -- 2.3. Advances in chronological calculus -- 2.3.1. Chronological calculus and half-shuffles -- 2.3.2. Chronological calculus and pre-Lie products -- 2.3.3. Time-ordered products and enveloping algebras -- 2.3.4. Formal flows and Hopf algebraic structures -- 2.4. Rota-Baxter algebras -- 2.4.1. Origin -- 2.4.2. Definition and examples -- 2.4.3. Related algebraic structures -- 2.4.4. Atkinson's factorization and Bogoliubov's recursion -- 2.4.5. Spitzer's identity: commutative case -- 2.4.6. Free commutative Rota-Baxter algebras -- 2.4.7. Spitzer's identity: noncommutative case -- 2.4.8. Free Rota-Baxter algebras -- 2.5. References -- 3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras -- 3.1. Introduction -- 3.2. Classical symmetric functions -- 3.2.1. Symmetric polynomials -- 3.2.2. The Hopf algebra of symmetric functions -- 3.2.3. The λ-ring notation -- 3.2.4. Symmetric functions and formal power series -- 3.2.5. Duality -- 3.3. Noncommutative symmetric functions -- 3.3.1. Basic definitions -- 3.3.2. Generators and linear bases -- 3.3.3. Duality -- 3.3.4. Solomon's descent algebras -- 3.4. Lie series and Lie idempotents -- 3.4.1. Permutational operators on tensor spaces -- 3.4.2. The Hausdorff series -- 3.4.3. Lie idempotents in the descent algebra -- 3.5. Lie idempotents as noncommutative symmetric functions -- 3.5.1. Noncommutative power-sums -- 3.5.2. The Magnus expansion -- 3.5.3. The continuous BCH expansion -- 3.5.4. Another proof of the Magnus expansion -- 3.5.5. The (1 - q) -transform -- 3.5.6. Hopf algebras enter the scene -- 3.5.7. A one-parameter family of Lie idempotents -- 3.5.8. The iterated q-bracketing and its diagonalization. ; 3.6. Decompositions of the descent algebras -- 3.6.1. Complete families of minimal orthogonal idempotents -- 3.6.2. Eulerian idempotents -- 3.6.3. Generalized Eulerian idempotents -- 3.7. Decompositions of the tensor algebra -- 3.8. General deformations -- 3.9. Lie quasi-idempotents as Lie polynomials -- 3.9.1. The left derivative -- 3.9.2. Multilinear Lie polynomials -- 3.9.3. Decompositions on other bases -- 3.10. Permutations and free quasi-symmetric functions -- 3.10.1. Free quasi-symmetric functions -- 3.11. Packed words and word quasi-symmetric functions -- 3.12. References -- 4. From Runge-Kutta Methods to Hopf Algebras of Rooted Trees -- 4.1. Numerical integration methods for ordinary differential equations -- 4.1.1. Introduction -- 4.1.2. Runge-Kutta methods -- 4.2. Algebraic theory of Runge-Kutta methods -- 4.2.1. The order conditions of RK methods -- 4.2.2. The independence of order conditions -- 4.2.3. Proof of necessary and sufficient order conditions -- 4.2.4. Composition of RK methods, rooted trees and forests -- 4.2.5. The Butcher group -- 4.2.6. Equivalence classes of RK methods -- 4.2.7. Bibliographical comments -- 4.3. B-series and related formal expansions -- 4.3.1. B-series -- 4.3.2. Backward error analysis, the exponential and the logarithm -- 4.3.3. Series of linear differential operators -- 4.3.4. The Lie algebra of the Butcher group -- 4.3.5. The pre-Lie algebra structure on ɡ -- 4.3.6. Bibliographical comments -- 4.4. Hopf algebras of rooted trees -- 4.4.1. The commutative Hopf algebra of rooted trees -- 4.4.2. The dual algebra Н* and the dual Hopf algebra H° -- 4.4.3. B-series and series of differential operators revisited -- 4.4.4. A universal property of the commutative Hopf algebra of rooted trees -- 4.4.5. The substitution law -- 4.4.6. Bibliographical comments -- 4.5. References. ; 5. Combinatorial Algebra in Controllability and Optimal Control -- 5.1. Introduction -- 5.1.1. Motivation: idealized examples -- 5.1.2. Controlled dynamical systems -- 5.1.3. Fundamental questions in control -- 5.2. Analytic foundations -- 5.2.1. State-space models and vector fields on manifolds -- 5.2.2. Chronological calculus -- 5.2.3. Piecewise constant controls and the Baker-Campbell-Hausdorff formula -- 5.2.4. Picard iteration and formal series solutions -- 5.2.5. The Chen-Fliess series and abstractions -- 5.3. Controllability and optimality -- 5.3.1. Reachable sets and accessibility -- 5.3.2. Small-time local controllability -- 5.3.3. Nilpotent approximating systems -- 5.3.4. Optimality and the maximum principle -- 5.3.5. Control variations and approximating cones -- 5.4. Product expansions and realizations -- 5.4.1. Variation of parameters and exponential products -- 5.4.2. Computations using Zinbiel products -- 5.4.3. Exponential products and normal forms for nilpotent systems -- 5.4.4. Logarithm of the Chen series -- 5.5. References -- 6. Algebra is Geometry is Algebra - Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application -- 6.1. The Butcher group and the Connes-Kreimer algebra -- 6.1.1. The Butcher group and B-series from numerical analysis -- 6.1.2. Beyond the Butcher group -- 6.2. Character groups of graded and connected Hopf algebras -- 6.2.1. The exponential and logarithm -- 6.3. Controlled groups of characters -- 6.3.1. Conventions for this section -- 6.3.2. Combinatorial Hopf algebras and the inverse factorial character -- 6.4. Appendix: Calculus in locally convex spaces -- 6.4.1. Cr -Manifolds and Cr -mappings between them -- 6.5. References -- List of Authors -- Index -- EULA.
• Titolo autorizzato: Algebra and Applications 2
• ISBN: 1-119-88090-4; 1-119-88091-2; 1-119-88089-0
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910555013003321
• Lo trovi qui: Univ. Federico II