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 
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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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