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 and 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.

[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).

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[11]

F. Chapoton, Rooted trees and an exponential-like seriesarXiv:math/0209104.

[12]

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

[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.

[14]

F. Chapoton and F. Patras, Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formulaarXiv:1201.2159v1 [math.QA]. doi: 10.1142/S0218196713400134.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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).

[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).

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[29]

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

[30]

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

[31]

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

[32]

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

[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.

[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.

[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.

[36]

A. Lundervold and H. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifoldsarXiv:1112.4465v1 [math.NA].

[37]

A. Lundervold and H. Munthe-Kaas, On post-Lie algebras, Lie-Butcher series and moving framesarXiv:1203.4738v1 [math.NA].

[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.

[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.

[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.

[41]

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

[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.

[43]

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

[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].

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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).

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[11]

F. Chapoton, Rooted trees and an exponential-like seriesarXiv:math/0209104.

[12]

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

[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.

[14]

F. Chapoton and F. Patras, Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formulaarXiv:1201.2159v1 [math.QA]. doi: 10.1142/S0218196713400134.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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).

[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).

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[29]

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

[30]

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

[31]

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

[32]

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

[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.

[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.

[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.

[36]

A. Lundervold and H. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifoldsarXiv:1112.4465v1 [math.NA].

[37]

A. Lundervold and H. Munthe-Kaas, On post-Lie algebras, Lie-Butcher series and moving framesarXiv:1203.4738v1 [math.NA].

[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.

[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.

[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.

[41]

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

[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.

[43]

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

[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].

[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.

[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.

[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.

[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.

[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.

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