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Drinfel$'$d algebra deformations, homotopy comodules and the associahedra


Authors: Martin Markl and Steve Shnider
Journal: Trans. Amer. Math. Soc. 348 (1996), 3505-3547
MSC (1991): Primary 17B37; Secondary 18G60
DOI: https://doi.org/10.1090/S0002-9947-96-01489-4
MathSciNet review: 1321583
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Abstract: The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra $A$ and thus finish the program which began in [13], [14]. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfel'd algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in [12].


References [Enhancements On Off] (What's this?)

  • 1. J. Beloff, The relentless question. Reflections on the paranormal, McFarland Publishers, NC, USA, 1990.
  • 2. P. Cartier, Cohomologie des coalgèbres, Séminaire Sophus Lie 1955-1956, exp. 4&5.
  • 3. V.G. Drinfel$'$d, Quasi-Hopf algebras, Algebra i Analiz, 1 (1989), no. 6, 114--148; English transl. in Leningrad Math. J. 1 (1990). MR 91b:17016
  • 4. T.F. Fox, An introduction to algebraic deformation theory Journal of Pure and Appl. Algebra, 84 (1993), 17--41. MR 93k:16055
  • 5. M. Gerstenhaber, On the deformation of rings and algebras, I Annals of Mathematics, 79 (1964), 59--104; II, 84 (1966), 1-19; III, 88 (1968), 1-34; IV, 99 (1974), 257--276. MR 30:2034; MR 34:7608; MR 39:1521; MR 52:10807
  • 6. M. Gerstenhaber, The cohomology structure of an associative ring, Annals of Mathematics, (2) 78 (1963), 268--288. MR 28:5102
  • 7. M. Gerstenhaber and S.D. Schack, Algebraic cohomology and deformation theory, In Deformation Theory of Algebras and Structures and Applications, pages 11--264. Kluwer, Dordrecht, 1988. MR 90c:16016
  • 8. M. Gerstenhaber and S. D. Schack, Bialgebra cohomology, deformations, and quantum groups, Proc. Nat. Acad. Sci., 87 (1990), 478--481. MR 90j:16062
  • 9. S. Mac Lane, Homology, Springer-Verlag, 1963. MR 30:1160
  • 10. M. Markl, Cotangent cohomology of a theory and deformations, Journal of Pure Appl. Algebra, to appear.
  • 11. M. Markl, Models for operads, Submitted. Available as preprint hep-th/9411208.
  • 12. M. Markl and S. Shnider, Drinfel$'$d algebra deformations and the associahedra, IMRN, 1994, no. 4. Math. Journal, submitted. MR 95f:16050;
  • 13. M. Markl and J.D. Stasheff, Deformation theory via deviations, Journal of Algebra, 70 (1994), 122--155. CMP 95:04
  • 14. S. Shnider and S. Sternberg, The cobar construction and a restricted deformation theory for Drinfeld algebras, Journal of Algebra, 169 (1994), 343--366. CMP 95:02
  • 15. S. Shnider and S. Sternberg, Quantum groups: from coalgebras to Drinfeld algebras, Graduate texts in mathematical physics, ed. Elliot Lieb, International Press, 1993. MR 95e:17022
  • 16. J.D. Stasheff, Homotopy associativity of H-spaces I, II, Trans. Amer. Math. Soc., 108 (1963), 275--312. MR 28:1623

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Additional Information

Martin Markl
Affiliation: Mathematical Institute of the Academy, Žitná 25, 115 67 Praha 1, Czech Republic
Email: mark@earn.cvut.cz

Steve Shnider
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel
Email: shnider@bimacs.cs.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9947-96-01489-4
Keywords: Quasi-bialgebra, formal deformations, multicomplex, associativity constraints
Received by editor(s): October 3, 1994
Additional Notes: The first author partially supported by the National Research Counsel, USA
The second author partially supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities
Article copyright: © Copyright 1996 American Mathematical Society

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