Koszul duality

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 prototype example, due to Joseph Bernstein, Israel Gelfand, and Sergei Gelfand,[3] 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.[citation needed]

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.

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). 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.