In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) as well as topology (e.g., equivariant cohomology). The prototype example, due to Bernstein, Gelfand and Gelfand, is the rough duality between the derived category of a symmetric algebra and that of an exterior algebra. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature.
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Koszul dual of a Koszul algebra
Koszul duality, as treated by Beilinson, Ginzburg, and Soergel can be formulated using the notion of a Koszul algebra. An example of such a Koszul algebra A is the symmetric algebra S(V) on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic ring, i.e., of the form
where
where
Koszul duality
If
Variants
As an alternative to passing to certain subcategories of the derived categories of
An extension of Koszul duality to D-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra
Koszul duality for operads
An extension of the above concept of Koszul duality was formulated by Ginzburg and Kapranov who introduced the notion of a quadratic operad and defined the quadratic dual of such an operad. Very roughly, an operad is an algebraic structure consisting of an object of n-ary operations for all n. An algebra over an operad is an object on which these n-ary operations act. For example, there is an operad called the associative operad whose algebras are associative algebras, i.e., depending on the precise context, non-commutative rings (or, depending on the context, non-commutative graded rings, differential graded rings). Algebras over the so-called commutative operad are commutative algebras, i.e., commutative (possibly graded, differential graded) rings. Yet another example is the Lie operad whose algebras are Lie algebras. The quadratic duality mentioned above is such that the associative operad is self-dual, while the commutative and the Lie operad correspond to each other under this duality.
Koszul duality for operads states an equivalence between algebras over dual operads. The special case of associative algebras gives back the functor