Advertisement

Higher Homotopy Hopf Algebras Found: A Ten Year Retrospective

  • Ronold N. UmbleEmail author
Chapter
Part of the Progress in Mathematics book series (PM, volume 287)

Abstract

At the 1996 conference honoring Jim Stasheff in the year of his 60th birthday, I initiated the search for A -bialgebras in a talk entitled “In Search of Higher Homotopy Hopf Algebras.” The idea in that talk was to think of a DG bialgebra as some (unknown) higher homotopy structure with trivial higher order structure and apply a graded version of Gerstenhaber and Schack’s bialgebra deformation theory. Indeed, deformation cohomology, which detects some (but not all) A -bialgebra structure, motivated the definition given by S. Saneblidze and myself in 2004.

Associahedron A(n)-algebra A-bialgebra Bialgebradeformation Biderivative Deformation cohomology Matrad Operad Permutahedron 

References

  1. Bau98.
    Baues, H.J.: The cobar construction as a Hopf algebra and the Lie differential. Inv. Math. 132, 467–489 (1998)CrossRefzbMATHGoogle Scholar
  2. BU10.
    Berciano, A., Umble, R.: Some naturally occurring examples of A -bialgebras. J. Pure Appl. Algebra (2010 in press)Google Scholar
  3. Car55.
    Cartier, P.: Cohomologie des cogèbres. In: Séminaire Sophus Lie, Exposé 5 (1955–1956)Google Scholar
  4. CS04.
    Chas, M., Sullivan, D.: Closed string operators in topology leading to Lie bialgebras and higher string algebra. The legacy of Niels Henrik Abel, pp. 771–784. Springer, Berlin (2004)Google Scholar
  5. Ger63.
    Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78(2), 267–288 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  6. GS92.
    Gerstenhaber, M., Schack, S.D.: Algebras, bialgebras, quantum groups, and algebraic deformations. In: Contemp. Math. 134. AMS, Providence, RI (1992)Google Scholar
  7. HKR62.
    Hochschild, G., Kostant, B., Rosenberg, A.: Differential forms on regular affine varieties. Trans. Am. Math. Soc. 102, 383–408 (1962)CrossRefzbMATHGoogle Scholar
  8. LM91.
    Lazarev, A., Movshev, M.: Deformations of Hopf algebras. Russ. Math. Surv. (translated from Russian) 42, 253–254 (1991)Google Scholar
  9. LM96.
    Lazarev, A., Movshev, M.: On the cohomology and deformations of diffrential graded algebras. J. Pure Appl. Algebra 106, 141–151 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Lod08.
    Loday, J.-L.: Generalized bialgebras and triples of operads. Astérisque No. 320, x–116 (2008)Google Scholar
  11. Lod10.
    Loday, J.-L.: The diagonal of the Stasheff polytope. International Conference in honor of Murray Gerstenhaber and Jim Stasheff (Paris 2007); this volume (2010)Google Scholar
  12. LU92.
    Lupton, G., Umble, R.: Rational homotopy types with the rational cohomology of stunted complex projective space. Canadian J. Math. 44(6), 1241–1261 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Mar06.
    Markl, M.: A resolution (minimal model) of the PROP for bialgebras. J. Pure Appl. Algebra 205, 341–374 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Mar08.
    Markl, M.: Operads and PROPs. Handbook of algebra, vol. 5, pp. 87–140. Elsevier/North-Holland, Amsterdam (2008)Google Scholar
  15. MS06.
    Markl, M., Shnider, S.: Associahedra, cellular W-construction and products of A -algebras. Trans. Am. Math. Soc. 358(6), 2353–2372 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. MSS02.
    Markl, M., Shnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, 96. AMS, Providence, RI (2002)Google Scholar
  17. McC98.
    McCleary, J. (ed.): Higher Homotopy Structures in Topology and Mathematical Physics. Contemp. Math., 227. AMS, Providence, RI (1998)Google Scholar
  18. Pir02.
    Pirashvili, T.: On the PROP coresponding to bialgebras. Cah. Topol. Géom. Différ. Catég. 43(3), 221–239 (2002)MathSciNetzbMATHGoogle Scholar
  19. San96.
    Saneblidze, S.: The formula determining an A -coalgebra structure on a free algebra. Bull. Georgian Acad. Sci. 154, 351–352 (1996)MathSciNetzbMATHGoogle Scholar
  20. San99.
    Saneblidze, S.: On the homotopy classification of spaces by the fixed loop space homology. Proc. A. Razmadze Math. Inst. 119, 155–164 (1999)MathSciNetzbMATHGoogle Scholar
  21. SU00.
    Saneblidze, S., Umble, R.: A diagonal on the associahedra (unpublished manuscript 2000). Preprint math.AT/0011065Google Scholar
  22. SU04.
    Saneblidze, S., Umble, R.: Diagonals on the permutahedra, multiplihedra and associahedra. J. Homol. Homotopy Appl. 6(1), 363–411 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. SU05.
    Saneblidze, S., Umble, R.: The biderivative and A -bialgebras. J. Homol. Homotopy Appl. 7(2), 161–177 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. SU10.
    Saneblidze, S., Umble, R.: Matrads, biassociahedra and A -bialgebras. J. Homol. Homotopy Appl (in press 2010). Preprint math.AT/0508017Google Scholar
  25. SU11.
    Saneblidze, S., Umble, R.: Morphisms of A -bialgebras and Applications (unpublished manuscript)Google Scholar
  26. Sta63.
    Stasheff, J.: Homotopy associativity of H-spaces I, II. Trans. Am. Math. Soc. 108, 275–312 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Sul07.
    Sullivan, D.: String Topology: Background and Present State. Current developments in mathematics, 2005, 41–88, Int. Press, Somerville, MA (2007)Google Scholar
  28. Ton97.
    Tonks, A.: Relating the associahedron and the permutohedron. In: Operads: Proceedings of the Renaissance Conferences (Hartford CT / Luminy Fr 1995). Contemp. Math. 202, 33–36 (1997)Google Scholar
  29. Umb89.
    Umble, R.: Homotopy conditions that determine rational homotopy type. J. Pure Appl. Algebra 60, 205–217 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Umb96.
    Umble, R.: The deformation complex for differential graded Hopf algebras. J. Pure Appl. Algebra 106, 199–222 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Umb08.
    Umble, R.: Structure relations in special A -bialgebras. Sovremennya Matematika i Ee Prilozheniya (Contemporary Mathematics and its Applications), 43, Topology and its Applications, 136–143 (2006). Translated from Russian in J. Math. Sci. 152(3), 443–450 (2008)Google Scholar
  32. Val04.
    Vallette, B.: Koszul duality for PROPs. Trans. Am. Math. Soc. 359, 4865–4943 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsMillersville University of PennsylvaniaMillersvilleUSA

Personalised recommendations