Skip to main content
SpringerLink
Log in

Algebraic Structures

  • Chapter
  • First Online:
Nonsymmetric Operads in Combinatorics

  • 264 Accesses

Abstract

This chapter deals with vector spaces obtained from graded collections. A general framework for algebraic structures having products and coproducts is presented. Most of the algebraic structures encountered in algebraic combinatorics like associative, dendriform, pre-Lie algebras, and Hopf bialgebras fit into this framework. This chapter contains classical examples of such structures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Aguiar, Pre-Poisson algebras. Lett. Math. Phys. 54(4), 263–277 (2000)

    Article  MathSciNet  Google Scholar 

  2. M. Aguiar, J.-L. Loday, Quadri-algebras. J. Pure Appl. Algebra 191(3), 205–221 (2004)

    Article  MathSciNet  Google Scholar 

  3. E. Burgunder, B. Delcroix-Oger, Confluence laws and Hopf-Borel type theorem for operads (2018). arXiv:1701.01323v2

  4. C. Brouder, A. Frabetti, QED Hopf algebras on planar binary trees. J. Algebra 1, 298–322 (2003)

    Article  MathSciNet  Google Scholar 

  5. F. Chapoton, Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces. J. Pure Appl. Algebra 168(1), 1–18 (2002)

    Article  MathSciNet  Google Scholar 

  6. F. Chapoton, Operads and algebraic combinatorics of trees. Sém. Lothar. Combin. B58c, 27 (2008)

    Google Scholar 

  7. F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 8, 395–408 (2001)

    Article  MathSciNet  Google Scholar 

  8. G. Châtel, V. Pilaud, Cambrian Hopf algebras. Adv. Math. 311, 598–633 (2017)

    Article  MathSciNet  Google Scholar 

  9. G. Duchamp, F. Hivert, J.-Y. Thibon, Noncommutative symmetric functions VI: free quasi- symmetric functions and related algebras. Int. J. Algebra Comput. 12, 671–717 (2002)

    Article  MathSciNet  Google Scholar 

  10. K. Ebrahimi-Fard, D. Manchon, Dendriform equations. J. Algebra 322(11), 4053–4079 (2009)

    Article  MathSciNet  Google Scholar 

  11. K. Ebrahimi-Fard, D. Manchon, F. Patras, New identities in dendriform algebras. J. Algebra 320(2), 708–727 (2008)

    Article  MathSciNet  Google Scholar 

  12. L. Foissy, Bidendriform bialgebras, trees, and free quasi-symmetric functions. J. Pure Appl. Algebra 209(2), 439–459 (2007)

    Article  MathSciNet  Google Scholar 

  13. L. Foissy, Ordered forests and parking functions. Int. Math. Res. Not. 2012(7), 1603–1633 (2012)

    Article  MathSciNet  Google Scholar 

  14. L. Foissy, Examples of Com-PreLie Hopf algebras. arXiv:1501.06375v1 (2015)

  15. M. Gerstenhaber, The cohomology structure of an associative ring. Ann. Math. 78, 267–288 (1963)

    Article  MathSciNet  Google Scholar 

  16. S. Giraudo, Algebraic and combinatorial structures on pairs of twin binary trees. J. Algebra 360, 115–157 (2012)

    Article  MathSciNet  Google Scholar 

  17. S. Giraudo, Pluriassociative algebras I: the pluriassociative operad. Adv. Appl. Math. 77, 1–42 (2016)

    Article  MathSciNet  Google Scholar 

  18. S. Giraudo, Pluriassociative algebras II: the polydendriform operad and related operads. Adv. Appl. Math. 77, 43–85 (2016)

    Article  MathSciNet  Google Scholar 

  19. I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions I. Adv. Math. 112, 218–348 (1995)

    Article  MathSciNet  Google Scholar 

  20. F. Hivert, Combinatoire des fonctions quasi-symétriques, Ph.D. thesis, Université de Marne-la-Vallée (1999)

    Google Scholar 

  21. F. Hivert, An Introduction to Combinatorial Hopf Algebras (IOS Press, Amsterdam, 2003)

    Google Scholar 

  22. F. Hivert, J.-C. Novelli, J.-Y. Thibon, The algebra of binary search trees. Theor. Comput. Sci. 339(1), 129–165 (2005)

    Article  MathSciNet  Google Scholar 

  23. P. Leroux, Ennea-algebras. J. Algebra 281(1), 287–302 (2004)

    Article  MathSciNet  Google Scholar 

  24. P. Leroux, A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers. J. Integer Seq. 10(4), 1–23 (2007), Article 07.4.7

    Google Scholar 

  25. M. Livernet, A rigidity theorem for pre-Lie algebras. J. Pure Appl. Algebra 207(1), 1–18 (2006)

    Article  MathSciNet  Google Scholar 

  26. J.-L. Loday, Dialgebras, in Dialgebras and Related Operads. Lecture Notes in Mathematics, vol. 1763 (Springer, Berlin, 2001), pp. 7–66

    Google Scholar 

  27. J.-L. Loday, Arithmetree. J. Algebra 258(1), 275–309 (2002); Special issue in celebration of Claudio Procesi’s 60th birthday

    Google Scholar 

  28. J.-L. Loday, Generalized bialgebras and triples of operads. Astérisque 320, 1–114 (2008)

    MathSciNet  MATH  Google Scholar 

  29. J.-L. Loday, M. Ronco, Hopf algebra of the planar binary trees. Adv. Math. 139, 293–309 (1998)

    Article  MathSciNet  Google Scholar 

  30. J.-L. Loday, M. Ronco, Trialgebras and families of polytopes, in Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory. Contemporary Mathematics, vol. 346 (American Mathematical Society, Providence, 2004), pp. 369–398

    Google Scholar 

  31. S. Law, N. Reading, The Hopf algebra of diagonal rectangulations. J. Comb. Theory A 119(3), 788–824 (2012)

    Article  MathSciNet  Google Scholar 

  32. J.-L. Loday, B. Vallette, Algebraic Operads. Grundlehren der mathematischen Wissenschaften, vol. 346 (Springer, Heidelberg, 2012), pp. xxiv+634

    Book  Google Scholar 

  33. I.G. Macdonald, Symmetric Functions and Hall Polynomials. Oxford Classic Texts in the Physical Sciences, 2nd edn. (Oxford University Press, Oxford, 2015)

    Google Scholar 

  34. D. Manchon, A short survey on pre-Lie algebras, in Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory. ESI Lectures in Mathematics and Physics. (European Mathematical Society, Zürich, 2011), pp. 89–102

    Google Scholar 

  35. C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995)

    Article  MathSciNet  Google Scholar 

  36. J.-C. Novelli, m-dendriform algebras (2014). arXiv:1406.1616v1

  37. J.-C. Novelli, J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions. Fundam. Math. 193, 189–241 (2007)

    Article  MathSciNet  Google Scholar 

  38. S. Poirier, C. Reutenauer, Algèbres de Hopf de tableaux. Ann. Sci. Math. Québec 19, 79–90 (1995)

    MathSciNet  MATH  Google Scholar 

  39. R. Ree, Lie elements and an algebra associated with shuffles. Ann. Math. 68(2), 210–220 (1958)

    Article  MathSciNet  Google Scholar 

  40. M. Rey, Algebraic constructions on set partitions, in Formal Power Series and Algebraic Combinatorics (2007)

    Google Scholar 

  41. G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeit 2, 340–368 (1964)

    MATH  Google Scholar 

  42. R.P. Stanley, Enumerative Combinatorics, 2nd edn., vol. 1 (Cambridge University Press, Cambridge, 2011)

    Book  Google Scholar 

  43. È.B. Vinberg, The theory of homogeneous convex cones. Trudy Moskov. Mat. Obšč. 12, 303–358 (1963)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Paris-Est, Marne-la-Vallee, France

    Samuele Giraudo

Authors
  1. Samuele Giraudo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Giraudo, S. (2018). Algebraic Structures. In: Nonsymmetric Operads in Combinatorics. Springer, Cham. https://doi.org/10.1007/978-3-030-02074-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-02074-3_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-02073-6

  • Online ISBN: 978-3-030-02074-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation