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Koszul Duality for Associative Algebras

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

Abstract

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.

Keywords

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.

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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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