Koszul Duality for Associative Algebras

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


A minimal model for the associative algebra A is a quasi-free resolution (T(W),d) such that the differential map d maps W into ⊕ n≥2 W n . We would like to find a method to construct this minimal model when A is quadratic, that is A=T(V)/(R) where the ideal (R) is generated by RV ⊗2. We will see that the quadratic data (V,R) permits us to construct explicitly a coalgebra Open image in new window and a twisting morphism Open image in new window . Then, applying the theory of Koszul morphisms given in the previous chapter, we obtain a simple condition which ensures that the cobar construction on the Koszul dual coalgebra, that is Open image in new window , is the minimal model of A.

The quadratic hypothesis RV ⊗2 can be weakened by only requiring RV ⊗2V. In this case, we say that the algebra is inhomogeneous quadratic. We show how to modify the preceding method to handle the inhomogeneous quadratic case. Two examples are: the universal enveloping algebra \(U(\mathfrak{g})\) of a Lie algebra \(\mathfrak{g}\) (original example due to J.-L. Koszul) and the Steenrod algebra.


Associative Algebra Symmetric Algebra Koszul Complex Quadratic Algebra Weight Grade 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. BCK+66.
    A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, and J. W. Schlesinger, The mod−p lower central series and the Adams spectral sequence, Topology 5 (1966), 331–342. MathSciNetMATHCrossRefGoogle Scholar
  2. BG96.
    A. Braverman and D. Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996), no. 2, 315–328. MathSciNetMATHCrossRefGoogle Scholar
  3. CE48.
    C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MathSciNetMATHCrossRefGoogle Scholar
  4. CE56.
    H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MATHGoogle Scholar
  5. Frö99.
    R. Fröberg, Koszul algebras, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., vol. 205, Dekker, New York, 1999, pp. 337–350. Google Scholar
  6. Kos50.
    —, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65–127. MathSciNetMATHGoogle Scholar
  7. Löf86.
    Clas Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 291–338. CrossRefGoogle Scholar
  8. Man87.
    Yu. I. Manin, Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 4, 191–205. MathSciNetMATHCrossRefGoogle Scholar
  9. Man88.
    —, Quantum groups and noncommutative geometry, Université de Montréal Centre de Recherches Mathématiques, Montreal, QC, 1988. Google Scholar
  10. May66.
    J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123–146. MathSciNetMATHCrossRefGoogle Scholar
  11. ML95.
    —, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1975 edition. MATHGoogle Scholar
  12. Pio01.
    D. I. Piontkovskiĭ, On Hilbert series of Koszul algebras, Funktsional. Anal. i Prilozhen. 35 (2001), no. 2, 64–69, 96. MathSciNetGoogle Scholar
  13. Pir02b.
    —, Polynomial functors over finite fields (after Franjou, Friedlander, Henn, Lannes, Schwartz, Suslin), Astérisque (2002), no. 276, 369–388, Séminaire Bourbaki, Vol. 1999/2000. Google Scholar
  14. Pos95.
    L. E. Positsel’skiĭ, The correspondence between Hilbert series of quadratically dual algebras does not imply their having the Koszul property, Funktsional. Anal. i Prilozhen. 29 (1995), no. 3, 83–87. MathSciNetGoogle Scholar
  15. PP05.
    A. Polishchuk and L. Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. MATHGoogle Scholar
  16. Pri70.
    S. B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. MathSciNetMATHCrossRefGoogle Scholar
  17. Roo95.
    Jan-Erik Roos, On the characterisation of Koszul algebras. Four counterexamples, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 1, 15–20. MathSciNetMATHGoogle Scholar
  18. Ufn95.
    V. A. Ufnarovski, Combinatorial and asymptotic methods in algebra, Algebra VI, Encyclopedia Math. Sci., vol. 57, Springer, Berlin, 1995, pp. 1–196. Google Scholar
  19. Wan67.
    J. S. P. Wang, On the cohomology of the mod−2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Illinois J. Math. 11 (1967), 480–490. MathSciNetMATHGoogle Scholar
  20. Wei94.
    C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MATHGoogle 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

Personalised recommendations