دورية أكاديمية
Hopf algebras in dynamical systems theory
| العنوان: | Hopf algebras in dynamical systems theory |
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| المؤلفون: | Carinena, J. F., Ebrahimi-Fard, K., Figueroa, H., Gracia-Bondia, J. M. |
| سنة النشر: | 2006 |
| المجموعة: | ArXiv.org (Cornell University Library) |
| مصطلحات موضوعية: | Mathematics - Classical Analysis and ODEs, High Energy Physics - Theory, Mathematical Physics, 16W25, 16W30, 37B55, 37C10 |
| الوصف: | 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. ; Comment: International Journal of Geometric Methods in Modern Physics, in press |
| نوع الوثيقة: | text |
| اللغة: | unknown |
| العلاقة: | http://arxiv.org/abs/math/0701010Test; Int.J.Geom.Meth.Mod.Phys.4:577-646,2007 |
| DOI: | 10.1142/S0219887807002211 |
| الإتاحة: | https://doi.org/10.1142/S0219887807002211Test http://arxiv.org/abs/math/0701010Test |
| رقم الانضمام: | edsbas.F1A85341 |
| قاعدة البيانات: | BASE |
| DOI: | 10.1142/S0219887807002211 |
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