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)[1] and topology (e.g., equivariant cohomology[2]).
The prototypical example of Koszul duality was introduced by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand,.
[3] It establishes a duality between the derived category of a symmetric algebra and that of an exterior algebra, as well as the BGG correspondence, which links the stable category of finite-dimensional graded modules over an exterior algebra to the bounded derived category of coherent sheaves on projective space.
The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature.
[citation needed] The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space V over a field k, with dual vector space
, serve to build a two-step chain complex whose differential is induced by natural evaluation map Choosing a basis of V,
, and the previous chain complex becomes isomorphic to the complex whose differential is multiplication by t. This computation shows that the cohomology of the above complex is 0 at the left hand term, and is k at the right hand term.
In other words, k (regarded as a chain complex concentrated in a single degree) is quasi-isomorphic to the above complex, which provides a close link between the exterior algebra of V and the symmetric algebra of its dual.
Koszul duality, as treated by Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel[4] can be formulated using the notion of Koszul algebra.
is the tensor algebra on a finite-dimensional vector space, and
Its opposite ring is given by the graded ring of self-extensions of the underlying field k, thought of as an A-module: If an algebra
is Koszul, there is an equivalence between certain subcategories of the derived categories of graded
These subcategories are defined by certain boundedness conditions on the grading vs. the cohomological degree of a complex.
As an alternative to passing to certain subcategories of the derived categories of
[5] Usually these quotients are larger than the derived category, as they are obtained by factoring out some subcategory of the category of acyclic complexes, but they have the advantage that every complex of modules determines some element of the category, without needing to impose boundedness conditions.
An extension of Koszul duality to D-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra
of Kähler differentials on a smooth algebraic variety X and the
[6][7][8] 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.
[9] 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).
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.
The special case of associative algebras gives back the functor