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What Is a Post-Lie Algebra and Why Is It Useful in Geometric Integration

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Numerical Mathematics and Advanced Applications ENUMATH 2017 (ENUMATH 2017)

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Abstract

We explain the notion of a post-Lie algebra and outline its role in the theory of Lie group integrators.

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Notes

  1. 1.

    Neglecting convergence of infinite series at this point.

References

  1. C. Bai, L. Guo, X. Ni, Nonabelian generalized Lax pairs, the classical Yang–Baxter equation and PostLie algebras. Commun. Math. Phys. 297(2), 553–596 (2010)

    Article  MathSciNet  Google Scholar 

  2. Ch. Brouder, Runge-Kutta methods and renormalization. Eur. Phys. J. C12, 512–534 (2000)

    Google Scholar 

  3. F. Chapoton, M. Livernet,Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 2001, 395–408 (2001)

    Article  MathSciNet  Google Scholar 

  4. F. Chapoton, F. Patras, Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula. Int. J. Algebra Comput. 23(4), 853–861 (2013)

    Article  Google Scholar 

  5. Ph. Chartier, E. Hairer, G. Vilmart, Algebraic structures of B-series. Found. Comput. Math. 10, 407–427 (2010)

    Article  MathSciNet  Google Scholar 

  6. M.T. Chu, L.K. Norris, Isospectral flows and abstract matrix factorizations. SIAM J. Numer. Anal. 25, 1383–1391 (1988)

    Article  MathSciNet  Google Scholar 

  7. P. Deift, L.C. Li, C. Tomei, Matrix factorizations and integrable systems. Commun. Pure Appl. Math. 42(4), 443–521 (1989)

    Article  MathSciNet  Google Scholar 

  8. K. Ebrahimi-Fard, I. Mencattini, Post-Lie algebras, factorization theorems and Isospectral flows, arXiv:1711.02694

    Google Scholar 

  9. K. Ebrahimi-Fard, A. Lundervold, H.Z. Munthe-Kaas, On the Lie enveloping algebra of a post-Lie algebra. J. Lie Theory 25(4), 1139–1165 (2015)

    MathSciNet  MATH  Google Scholar 

  10. K. Ebrahimi-Fard, A. Lundervold, I. Mencattini, H.Z. Munthe-Kaas, Post-Lie algebras and Isospectral flows. SIGMA 25(11), 093 (2015)

    Google Scholar 

  11. K. Ebrahimi-Fard, I. Mencattini, H.Z. Munthe-Kaas,Post-Lie algebras and factorization theorems. J. Geom. Phys. 119, 19–33 (2017)

    Article  MathSciNet  Google Scholar 

  12. G. Fløystad, H. Munthe-Kaas, Pre- and Post-Lie Algebras: The Algebro-Geometric View. Abel Symposium Series (Springer, Berlin, 2018)

    Google Scholar 

  13. R. Grossman, R. Larson, Hopf algebraic structures of families of trees. J. Algebra 26, 184–210 (1989)

    Article  MathSciNet  Google Scholar 

  14. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31 (Springer, Berlin, 2002)

    Google Scholar 

  15. A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, A. Zanna, Lie-group methods. Acta Numer. 9, 215–365 (2000)

    Article  MathSciNet  Google Scholar 

  16. A. Lundervold, H.Z. Munthe-Kaas, On post-Lie algebras, Lie–Butcher series and moving frames. Found. Comput. Math. 13(4), 583–613 (2013)

    MATH  Google Scholar 

  17. D. Manchon, A short survey on pre-lie algebras, in Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory, ed. by A. Carey. E. Schrödinger Institut Lectures in Mathematics and Physics (European Mathematical Society, Helsinki, 2011)

    Google Scholar 

  18. R. McLachlan, K. Modin, H. Munthe-Kaas, O. Verdier, Butcher series: a story of rooted trees and numerical methods for evolution equations. Asia Pac. Math. Newsl. 7, 1–11 (2017)

    MathSciNet  Google Scholar 

  19. H. Munthe-Kaas, Lie–Butcher theory for Runge–Kutta methods. BIT Numer. Math. 35, 572–587 (1995)

    Article  MathSciNet  Google Scholar 

  20. H. Munthe-Kaas, Runge–Kutta methods on Lie groups. BIT Numer. Math. 38(1), 92–111 (1998)

    Article  MathSciNet  Google Scholar 

  21. H. Munthe-Kaas, W. Wright, On the Hopf Algebraic Structure of Lie Group Integrators. Found. Comput. Math. 8(2), 227–257 (2008)

    Article  MathSciNet  Google Scholar 

  22. B. Vallette, Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The research on this paper was partially supported by the Norwegian Research Council (project 231632).

Author information

Authors and Affiliations

  1. Norwegian University of Science and Technology (NTNU), Institutt for matematiske fag, Trondheim, Norway

    Charles Curry & Kurusch Ebrahimi-Fard

  2. University of Bergen, Department of Mathematics, Bergen, Norway

    Hans Munthe-Kaas

Authors
  1. Charles Curry
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  2. Kurusch Ebrahimi-Fard
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  3. Hans Munthe-Kaas
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Corresponding author

Correspondence to Charles Curry .

Editor information

Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Florin Adrian Radu

  2. University of Bergen, Bergen, Norway

    Kundan Kumar

  3. University of Bergen, Bergen, Norway

    Inga Berre

  4. University of Bergen, Bergen, Norway

    Jan Martin Nordbotten

  5. Hasselt University, Diepenbeek, Belgium

    Iuliu Sorin Pop

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Curry, C., Ebrahimi-Fard, K., Munthe-Kaas, H. (2019). What Is a Post-Lie Algebra and Why Is It Useful in Geometric Integration. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_38

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