Advertisement

Extension of the Product of a Post-Lie Algebra and Application to the SISO Feedback Transformation Group

  • Loïc FoissyEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)

Abstract

We describe both post- and pre-Lie algebra \(\mathfrak {g}_{SISO}\) associated to the affine SISO feedback transformation group. We show that it is a member of a family of post-Lie algebras associated to representations of a particular solvable Lie algebra. We first construct the extension of the magmatic product of a post-Lie algebra to its enveloping algebra, which allows to describe free post-Lie algebras and is widely used to obtain the enveloping of \(\mathfrak {g}_{SISO}\) and its dual.

Notes

Acknowledgements

The research leading these results was partially supported by the French National Research Agency under the reference ANR-12-BS01-0017.

References

  1. 1.
    Cartier, P.: Vinberg algebras, Lie groups and combinatorics. In: Quanta of Maths, Clay Mathematics Proceedings, vol. 11, pp. 107–126. American Mathematical Society, Providence (2010)Google Scholar
  2. 2.
    Ebrahimi-Fard, K., Lundervold, A., Munthe-Kaas, H.Z.: On the Lie enveloping algebra of a post-Lie algebra. J. Lie Theory 25(4), 1139–1165 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Foissy, L.: A pre-Lie algebra associated to a linear endomorphism and related algebraic structures. Eur. J. Math. 1(1), 78–121 (2015). arXiv:1309.5318MathSciNetCrossRefGoogle Scholar
  4. 4.
    Manchon, D.: A short survey on pre-Lie algebras. In: Carey, A.L. (ed.) Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory. Lectures in Mathematics and Physics, pp. 89–102. European Mathematical Society, Zürich (2011)CrossRefGoogle Scholar
  5. 5.
    Munthe-Kaas, H.Z., Lundervold, A.: On post-Lie algebras, Lie-Butcher series and moving frames. Found. Comput. Math. 13(4), 583–613 (2013). arXiv:1203.4738MathSciNetCrossRefGoogle Scholar
  6. 6.
    Oudom, J.-M., Guin, D.: Sur l’algèbre enveloppante d’une algèbre pré-Lie. C. R. Math. Acad. Sci. Paris 340(5), 331–336 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Oudom, J.-M., Guin, D.: On the Lie enveloping algebra of a pre-Lie algebra. J. K-Theory 2(1), 147–167 (2008). arXiv:math/0404457Google Scholar
  8. 8.
    Sloane, N.J.A.: On-line encyclopedia of integer sequences. http://oeis.org/
  9. 9.
    Steven Gray, W., Ebrahimi-Fard, K.: SISO output affine feedback transformation group and its Faà di Bruno Hopf algebra. SIAM J. Control Optim. 55(2), 885–912 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vallette, B.: Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007). arXiv:math/0405312MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Fédération de Recherche Mathématique du Nord Pas de Calais FR 2956, Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniversité du Littoral Côte d’Opale-Centre Universitaire de la Mi-VoixCalais CedexFrance

Personalised recommendations