March  2014, 34(3): 1021-1040. doi: 10.3934/dcds.2014.34.1021

The tridendriform structure of a discrete Magnus expansion

1. 

ICMAT, Nicolás Cabrera 13-15, 28049 Madrid, Spain

2. 

Laboratoire de Mathématiques, UMR 6620, CNRS-Université Blaise Pascal, BP 80026, F63171 Aubière Cedex, France

Received  December 2012 Revised  May 2013 Published  August 2013

The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow these ideas by further developing the observation that the logarithm of the solution of a linear first-order finite-difference equation can be written in terms of the Magnus expansion taking place in a pre-Lie algebra. By using basic combinatorics on planar reduced trees we derive a closed formula for the Magnus expansion in the context of free tridendriform algebra. The tridendriform algebra structure on word quasi-symmetric functions permits us to derive a discrete analogue of the Mielnik--Plebański--Strichartz formula for this logarithm.
Citation: Kurusch Ebrahimi-Fard, Dominique Manchon. The tridendriform structure of a discrete Magnus expansion. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1021-1040. doi: 10.3934/dcds.2014.34.1021
References:
[1]

A. Agrachev and R. Gamkrelidze, Chronological algebras and nonstationary vector fields, J. Sov. Math., 11 (1980), 135-176.  Google Scholar

[2]

A. Agrachev and R. Gamkrelidze, The shuffle product and symmetric groups, in "Differential Equations, Dynamical Systems and Control Science" (editors, K. D. Elworthy, W. N. Everitt and E. B. Lee), Lecture Notes in Pure and Appl. Math., 152, 365-82. Marcel Dekker, Inc., New York (1994).  Google Scholar

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C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Commun. Mathematical Physics, 297 (2010), 553-596. doi: 10.1007/s00220-010-0998-7.  Google Scholar

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S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

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D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Central European Journal of Mathematics, 4 (2006), 323-357. doi: 10.2478/s11533-006-0014-9.  Google Scholar

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E. Burgunder and M. Ronco, Tridendriform structure on combinatorial Hopf algebras, J. Algebra, 324 (2010), 2860-2883. doi: 10.1016/j.jalgebra.2010.07.010.  Google Scholar

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J. C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972), 79-106. doi: 10.1090/S0025-5718-1972-0305608-0.  Google Scholar

[9]

J. C. Butcher, "Numerical Methods for Ordinary Differential Equations," Second edition, John Wiley & Sons, Chichester, 2008. Google Scholar

[10]

E. Celledoni, R. I. McLachlan, B. Owren and R. Quispel, Energy-preserving integrators and the structure of B-series, Found. Comput. Math., 10 (2010), 673-693. doi: 10.1007/s10208-010-9073-1.  Google Scholar

[11]

F. Chapoton, Rooted trees and an exponential-like series,, , ().   Google Scholar

[12]

F. Chapoton, Hyperarbres, arbres enracinés et partitions pointées, Homol., Homot. and Appl., 9 (2007), 193-212.  Google Scholar

[13]

F. Chapoton, A rooted-trees q-series lifting a one-parameter family of Lie idempotents, Algebra & Number Theory, 3 (2009), 611-636. doi: 10.2140/ant.2009.3.611.  Google Scholar

[14]

F. Chapoton and F. Patras, Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula,, , ().  doi: 10.1142/S0218196713400134.  Google Scholar

[15]

F. Chapoton, F. Hivert, J. C. Novelli and J. Y. Thibon, An operational calculus for the mould operad, Int. Math. Res. Not., 2008 (2008), 22 pp. doi: 10.1093/imrn/rnn018.  Google Scholar

[16]

F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not., 2001 (2001), 395-408. doi: 10.1155/S1073792801000198.  Google Scholar

[17]

Ph. Chartier and A. Murua, An algebraic theory of order, M2AN, 43 (2009), 607-630. doi: 10.1051/m2an/2009029.  Google Scholar

[18]

K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras, Found. Comput. Math., 9 (2009), 295-316. doi: 10.1007/s10208-008-9023-3.  Google Scholar

[19]

K. Ebrahimi-Fard and D. Manchon, Dendriform equations, J. Algebra, 322 (2009), 4053-4079. doi: 10.1016/j.jalgebra.2009.06.002.  Google Scholar

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K. Ebrahimi-Fard and D. Manchon, Twisted dendriform algebras and the pre-Lie Magnus expansion, Journal of Pure and Applied Algebra, 215 (2011), 2615-2627. doi: 10.1016/j.jpaa.2011.03.004.  Google Scholar

[21]

K. Ebrahimi-Fard and D. Manchon, On an extension of Knuth's rotation correspondence to reduced planar trees, Accepted for Publication in Journal of Noncommutative Geometry, arXiv:1203.0425 (2012). Google Scholar

[22]

K. Ebrahimi-Fard and D. Manchon, The Magnus expansion, trees and Knuth's rotation correspondence, Accepted for Publication in J. Found. Comput. Math., arXiv:1203.2878 (2012). Google Scholar

[23]

I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh and J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math., 112 (1995), 218-348. doi: 10.1006/aima.1995.1032.  Google Scholar

[24]

E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing, 13 (1974), 1-15.  Google Scholar

[25]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, 31, Springer Series in Computational Mathematics. Springer-Verlag, Second edition, Berlin, 2006.  Google Scholar

[26]

F. Hivert, "Combinatoire des Fonctions Quasi-Symétriques," Ph.D. Thesis, Université de Marne la Vallée, 1999. Google Scholar

[27]

A. Iserles and S. P. Nørsett, On the solution of linear differential equations in lie groups, Philosophical Transactions of the Royal Society A, 357 (1999), 983-1019. doi: 10.1098/rsta.1999.0362.  Google Scholar

[28]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365. doi: 10.1017/S0962492900002154.  Google Scholar

[29]

A. Iserles, Expansions that grow on trees, Notices of the AMS, 49 (2002), 430-440.  Google Scholar

[30]

N. Jacobson, "Lie Algebras," Dover, Second Edition, 1979.  Google Scholar

[31]

D. E. Knuth, "The Art of Computer Programming I. Fundamental Algorithms," Addison-Wesley, 1975, xxii+634 pp.  Google Scholar

[32]

J.-L. Loday, Dialgebras, Lect. Notes Math., 1763 (2001), 7-66. doi: 10.1007/3-540-45328-8_2.  Google Scholar

[33]

J.-L. Loday and M. Ronco, Hopf algebra of the planar binary trees, Adv. Math., 139 (1998), 293-309. doi: 10.1006/aima.1998.1759.  Google Scholar

[34]

J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, in "Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology and Algebraic K-Theory," Contemporary Mathematics, 346 (2004), 369-398. doi: 10.1090/conm/346/06296.  Google Scholar

[35]

A. Lundervold and H. Munthe-Kaas, Hopf algebras of formal diffeomorphisms and numerical integration on manifolds, AMS, Contemporary Mathematics, 539 (2011), 295-324. doi: 10.1090/conm/539/10641.  Google Scholar

[36]

A. Lundervold and H. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifolds,, , ().   Google Scholar

[37]

A. Lundervold and H. Munthe-Kaas, On post-Lie algebras, Lie-Butcher series and moving frames,, , ().   Google Scholar

[38]

W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673. doi: 10.1002/cpa.3160070404.  Google Scholar

[39]

C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177 (1995), 967-982. doi: 10.1006/jabr.1995.1336.  Google Scholar

[40]

D. Manchon, A short survey on pre-Lie algebras, Institut Lectures in Math. Phys., "Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory" (Ed. A. Carey and E. Schrödinger), Eur. Math. Soc., (2011). doi: 10.4171/008-1/3.  Google Scholar

[41]

B. Mielnik and J. Plebański, Combinatorial approach to Baker-Campbell-Hausdorff exponents, Ann. Inst. Henri Poincaré A, 12 (1970), 215-254.  Google Scholar

[42]

A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math., 6 (2006), 387-426. doi: 10.1007/s10208-003-0111-0.  Google Scholar

[43]

J.-C. Novelli and J.-Y. Thibon, Polynomial realizations of some trialgebras, Proceedings FPSAC, San Diego, (2006). Google Scholar

[44]

J.-C. Novelli, F. Patras and J.-Y. Thibon, Natural endomorphisms of quasi-shuffle Hopf algebras, Bull. Soc. Math. France, 141 (2013), 107–-130. arXiv:1101.0725v1 [math.CO].  Google Scholar

[45]

P. Palacios and M. Ronco, Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra, 299 (2006), 648-678. doi: 10.1016/j.jalgebra.2005.09.042.  Google Scholar

[46]

D. Segal, Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff-Witt-theorem, J. Algebra, 164 (1994), 750-772. doi: 10.1006/jabr.1994.1088.  Google Scholar

[47]

F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc., 82 (1956), 323-339. doi: 10.1090/S0002-9947-1956-0079851-X.  Google Scholar

[48]

R. S. Strichartz, The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, J. Func. Anal., 72 (1987), 320-345. doi: 10.1016/0022-1236(87)90091-7.  Google Scholar

[49]

B. Vallette, Homology of generalized partition posets, J. of Pure and Appl. Algebra, 208 (2007), 699-725. doi: 10.1016/j.jpaa.2006.03.012.  Google Scholar

show all references

References:
[1]

A. Agrachev and R. Gamkrelidze, Chronological algebras and nonstationary vector fields, J. Sov. Math., 11 (1980), 135-176.  Google Scholar

[2]

A. Agrachev and R. Gamkrelidze, The shuffle product and symmetric groups, in "Differential Equations, Dynamical Systems and Control Science" (editors, K. D. Elworthy, W. N. Everitt and E. B. Lee), Lecture Notes in Pure and Appl. Math., 152, 365-82. Marcel Dekker, Inc., New York (1994).  Google Scholar

[3]

C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Commun. Mathematical Physics, 297 (2010), 553-596. doi: 10.1007/s00220-010-0998-7.  Google Scholar

[4]

G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10 (1960), 731-742. doi: 10.2140/pjm.1960.10.731.  Google Scholar

[5]

S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[6]

D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Central European Journal of Mathematics, 4 (2006), 323-357. doi: 10.2478/s11533-006-0014-9.  Google Scholar

[7]

E. Burgunder and M. Ronco, Tridendriform structure on combinatorial Hopf algebras, J. Algebra, 324 (2010), 2860-2883. doi: 10.1016/j.jalgebra.2010.07.010.  Google Scholar

[8]

J. C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972), 79-106. doi: 10.1090/S0025-5718-1972-0305608-0.  Google Scholar

[9]

J. C. Butcher, "Numerical Methods for Ordinary Differential Equations," Second edition, John Wiley & Sons, Chichester, 2008. Google Scholar

[10]

E. Celledoni, R. I. McLachlan, B. Owren and R. Quispel, Energy-preserving integrators and the structure of B-series, Found. Comput. Math., 10 (2010), 673-693. doi: 10.1007/s10208-010-9073-1.  Google Scholar

[11]

F. Chapoton, Rooted trees and an exponential-like series,, , ().   Google Scholar

[12]

F. Chapoton, Hyperarbres, arbres enracinés et partitions pointées, Homol., Homot. and Appl., 9 (2007), 193-212.  Google Scholar

[13]

F. Chapoton, A rooted-trees q-series lifting a one-parameter family of Lie idempotents, Algebra & Number Theory, 3 (2009), 611-636. doi: 10.2140/ant.2009.3.611.  Google Scholar

[14]

F. Chapoton and F. Patras, Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula,, , ().  doi: 10.1142/S0218196713400134.  Google Scholar

[15]

F. Chapoton, F. Hivert, J. C. Novelli and J. Y. Thibon, An operational calculus for the mould operad, Int. Math. Res. Not., 2008 (2008), 22 pp. doi: 10.1093/imrn/rnn018.  Google Scholar

[16]

F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not., 2001 (2001), 395-408. doi: 10.1155/S1073792801000198.  Google Scholar

[17]

Ph. Chartier and A. Murua, An algebraic theory of order, M2AN, 43 (2009), 607-630. doi: 10.1051/m2an/2009029.  Google Scholar

[18]

K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras, Found. Comput. Math., 9 (2009), 295-316. doi: 10.1007/s10208-008-9023-3.  Google Scholar

[19]

K. Ebrahimi-Fard and D. Manchon, Dendriform equations, J. Algebra, 322 (2009), 4053-4079. doi: 10.1016/j.jalgebra.2009.06.002.  Google Scholar

[20]

K. Ebrahimi-Fard and D. Manchon, Twisted dendriform algebras and the pre-Lie Magnus expansion, Journal of Pure and Applied Algebra, 215 (2011), 2615-2627. doi: 10.1016/j.jpaa.2011.03.004.  Google Scholar

[21]

K. Ebrahimi-Fard and D. Manchon, On an extension of Knuth's rotation correspondence to reduced planar trees, Accepted for Publication in Journal of Noncommutative Geometry, arXiv:1203.0425 (2012). Google Scholar

[22]

K. Ebrahimi-Fard and D. Manchon, The Magnus expansion, trees and Knuth's rotation correspondence, Accepted for Publication in J. Found. Comput. Math., arXiv:1203.2878 (2012). Google Scholar

[23]

I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh and J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math., 112 (1995), 218-348. doi: 10.1006/aima.1995.1032.  Google Scholar

[24]

E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing, 13 (1974), 1-15.  Google Scholar

[25]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, 31, Springer Series in Computational Mathematics. Springer-Verlag, Second edition, Berlin, 2006.  Google Scholar

[26]

F. Hivert, "Combinatoire des Fonctions Quasi-Symétriques," Ph.D. Thesis, Université de Marne la Vallée, 1999. Google Scholar

[27]

A. Iserles and S. P. Nørsett, On the solution of linear differential equations in lie groups, Philosophical Transactions of the Royal Society A, 357 (1999), 983-1019. doi: 10.1098/rsta.1999.0362.  Google Scholar

[28]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365. doi: 10.1017/S0962492900002154.  Google Scholar

[29]

A. Iserles, Expansions that grow on trees, Notices of the AMS, 49 (2002), 430-440.  Google Scholar

[30]

N. Jacobson, "Lie Algebras," Dover, Second Edition, 1979.  Google Scholar

[31]

D. E. Knuth, "The Art of Computer Programming I. Fundamental Algorithms," Addison-Wesley, 1975, xxii+634 pp.  Google Scholar

[32]

J.-L. Loday, Dialgebras, Lect. Notes Math., 1763 (2001), 7-66. doi: 10.1007/3-540-45328-8_2.  Google Scholar

[33]

J.-L. Loday and M. Ronco, Hopf algebra of the planar binary trees, Adv. Math., 139 (1998), 293-309. doi: 10.1006/aima.1998.1759.  Google Scholar

[34]

J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, in "Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology and Algebraic K-Theory," Contemporary Mathematics, 346 (2004), 369-398. doi: 10.1090/conm/346/06296.  Google Scholar

[35]

A. Lundervold and H. Munthe-Kaas, Hopf algebras of formal diffeomorphisms and numerical integration on manifolds, AMS, Contemporary Mathematics, 539 (2011), 295-324. doi: 10.1090/conm/539/10641.  Google Scholar

[36]

A. Lundervold and H. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifolds,, , ().   Google Scholar

[37]

A. Lundervold and H. Munthe-Kaas, On post-Lie algebras, Lie-Butcher series and moving frames,, , ().   Google Scholar

[38]

W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673. doi: 10.1002/cpa.3160070404.  Google Scholar

[39]

C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177 (1995), 967-982. doi: 10.1006/jabr.1995.1336.  Google Scholar

[40]

D. Manchon, A short survey on pre-Lie algebras, Institut Lectures in Math. Phys., "Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory" (Ed. A. Carey and E. Schrödinger), Eur. Math. Soc., (2011). doi: 10.4171/008-1/3.  Google Scholar

[41]

B. Mielnik and J. Plebański, Combinatorial approach to Baker-Campbell-Hausdorff exponents, Ann. Inst. Henri Poincaré A, 12 (1970), 215-254.  Google Scholar

[42]

A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math., 6 (2006), 387-426. doi: 10.1007/s10208-003-0111-0.  Google Scholar

[43]

J.-C. Novelli and J.-Y. Thibon, Polynomial realizations of some trialgebras, Proceedings FPSAC, San Diego, (2006). Google Scholar

[44]

J.-C. Novelli, F. Patras and J.-Y. Thibon, Natural endomorphisms of quasi-shuffle Hopf algebras, Bull. Soc. Math. France, 141 (2013), 107–-130. arXiv:1101.0725v1 [math.CO].  Google Scholar

[45]

P. Palacios and M. Ronco, Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra, 299 (2006), 648-678. doi: 10.1016/j.jalgebra.2005.09.042.  Google Scholar

[46]

D. Segal, Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff-Witt-theorem, J. Algebra, 164 (1994), 750-772. doi: 10.1006/jabr.1994.1088.  Google Scholar

[47]

F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc., 82 (1956), 323-339. doi: 10.1090/S0002-9947-1956-0079851-X.  Google Scholar

[48]

R. S. Strichartz, The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, J. Func. Anal., 72 (1987), 320-345. doi: 10.1016/0022-1236(87)90091-7.  Google Scholar

[49]

B. Vallette, Homology of generalized partition posets, J. of Pure and Appl. Algebra, 208 (2007), 699-725. doi: 10.1016/j.jpaa.2006.03.012.  Google Scholar

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