Koszul Duality for Associative Algebras

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 R⊂V ^⊗2. We will see that the quadratic data (V,R) permits us to construct explicitly a coalgebra and a twisting morphism . 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 , is the minimal model of A.

The quadratic hypothesis R⊂V ^⊗2 can be weakened by only requiring R⊂V ^⊗2⊕V. 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.

In the process of its internal development and prompted by its inner logic, mathematics, too, creates virtual worlds of great complexity and internal beauty which defy any attempt to describe them in natural language but challenge the imagination of a handful of professionals in many successive generations.

Yuri I. Manin in “Mathematics as metaphor”