Hopf algebras in dynamical systems theory
| dc.creator | Carinena, J. F. | |
| dc.creator | Ebrahimi-Fard, K. | |
| dc.creator | Figueroa, H. | |
| dc.creator | Gracia-Bondia, J. M. | |
| dc.date | 2006-12-30 | |
| dc.date | 2007-04-17 | |
| dc.date.accessioned | 2026-07-07T10:38:19Z | |
| dc.date.available | 2026-07-07T10:38:19Z | |
| dc.description | The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota-Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern approach involving Lie idempotents. Approximate solutions to differential equations involve, on the one hand, series of iterated integrals solving the corresponding integral equations; on the other hand, exponential solutions. Equating those solutions yields identities among products of iterated Riemann integrals. Now, the Riemann integral satisfies the integration-by-parts rule with the Leibniz rule for derivations as its partner; and skewderivations generalize derivations. Thus we seek an algebraic theory of integration, with the Rota-Baxter relation replacing the classical rule. The methods to deal with noncommutativity are especially highlighted. We find new identities, allowing for an extensive embedding of Dyson-Chen series of time- or path-ordered products (of generalized integration operators); of the corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota-Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew)differential equations. | |
| dc.description | International Journal of Geometric Methods in Modern Physics, in press | |
| dc.identifier | https://arxiv.org/abs/math/0701010 | |
| dc.identifier | http://arxiv.org/abs/math/0701010 | |
| dc.identifier | Int.J.Geom.Meth.Mod.Phys.4:577-646,2007 | |
| dc.identifier | doi:10.1142/S0219887807002211 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/180681 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Mathematical Physics | |
| dc.subject | 16W25, 16W30, 37B55, 37C10 | |
| dc.title | Hopf algebras in dynamical systems theory | |
| dc.type | text |