Algebras, Coalgebras, Homology

  • Jean-Louis Loday
  • Bruno Vallette
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 346)


In this chapter we recall elementary facts about algebras and homological algebra, essentially to establish the terminology and the notation. We first review the notions of associative, commutative and Lie algebra. Then we deal with the notion of coalgebra, which is going to play a key role in this book. This leads to the notion of convolution. The last sections cover bialgebras, pre-Lie algebras, differential graded objects and convolution algebra.


Hopf Algebra Chain Complex Associative Algebra Grade Vector Space Convolution Algebra 
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  1. BT82.
    Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York, 1982. MR 658304 (83i:57016) zbMATHGoogle Scholar
  2. Bur06.
    Dietrich Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4 (2006), no. 3, 323–357 (electronic). MathSciNetzbMATHCrossRefGoogle Scholar
  3. Car56.
    Pierre Cartier, Hyperalgèbres et groupes formels, Séminaire “Sophus Lie” 2 (1955–56), Exposé No. 2, 1–6. Google Scholar
  4. DGMS75.
    P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MathSciNetzbMATHCrossRefGoogle Scholar
  5. Ger63.
    M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MathSciNetzbMATHCrossRefGoogle Scholar
  6. Gri04.
    P.-P. Grivel, Une histoire du théorème de Poincaré-Birkhoff-Witt, Expo. Math. 22 (2004), no. 2, 145–184. MathSciNetzbMATHCrossRefGoogle Scholar
  7. JR82.
    S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Umbral calculus and Hopf algebras (Norman, Okla., 1978), Contemp. Math., vol. 6, Amer. Math. Soc., Providence, RI, 1982, pp. 1–47. CrossRefGoogle Scholar
  8. Kos47.
    Jean-Louis Koszul, Sur les opérateurs de dérivation dans un anneau, C. R. Acad. Sci. Paris 225 (1947), 217–219. MR 0022345 (9,196h) MathSciNetzbMATHGoogle Scholar
  9. Ler46.
    Jean Leray, Structure de l’anneau d’homologie d’une représentation, C. R. Acad. Sci. Paris 222 (1946), 1419–1422. MathSciNetzbMATHGoogle Scholar
  10. Lod93.
    —, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3-4, 269–293. MathSciNetGoogle Scholar
  11. Lod94.
    —, Série de Hausdorff, idempotents eulériens et algèbres de Hopf, Exposition. Math. 12 (1994), no. 2, 165–178. MathSciNetGoogle Scholar
  12. Lod98.
    —, Cyclic homology, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by María O. Ronco, Chap. 13 by the author in collaboration with Teimuraz Pirashvili. Google Scholar
  13. Lod08.
    —, Generalized bialgebras and triples of operads, Astérisque (2008), no. 320, x+116. Google Scholar
  14. ML95.
    —, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1975 edition. zbMATHGoogle Scholar
  15. MM65.
    J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MathSciNetzbMATHCrossRefGoogle Scholar
  16. Qui69.
    —, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MathSciNetzbMATHCrossRefGoogle Scholar
  17. Ree58.
    R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. (2) 68 (1958), 210–220. MathSciNetzbMATHCrossRefGoogle Scholar
  18. Reu93.
    Ch. Reutenauer, Free Lie algebras, xviii+269, Oxford Science Publications. Google Scholar
  19. Vin63.
    È. B. Vinberg, The theory of homogeneous convex cones, Trudy Moskov. Mat. Obšč. 12 (1963), 303–358. MathSciNetGoogle Scholar
  20. Wei94.
    C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. zbMATHGoogle Scholar
  21. Wig89.
    D. Wigner, An identity in the free Lie algebra, Proc. Amer. Math. Soc. 106 (1989), no. 3, 639–640. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jean-Louis Loday
    • 1
  • Bruno Vallette
    • 2
  1. 1.IRMA, CNRS et Université de StrasbourgStrasbourgFrance
  2. 2.Lab. de Mathématiques J.A. DieudonnéUniversité de Nice-Sophia AntipolisNiceFrance

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