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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
Foissy, L.: A pre-Lie algebra associated to a linear endomorphism and related algebraic structures. Eur. J. Math. 1(1), 78–121 (2015). arXiv:1309.5318
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)
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:1203.4738
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)
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/0404457
Sloane, N.J.A.: On-line encyclopedia of integer sequences. http://oeis.org/
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)
Vallette, B.: Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007). arXiv:math/0405312
Acknowledgements
The research leading these results was partially supported by the French National Research Agency under the reference ANR-12-BS01-0017.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Foissy, L. (2018). Extension of the Product of a Post-Lie Algebra and Application to the SISO Feedback Transformation Group. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-01593-0_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01592-3
Online ISBN: 978-3-030-01593-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)