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) 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.

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 ${\displaystyle T(V)}$ is the tensor algebra on a finite-dimensional vector space, and R is a submodule of ${\displaystyle V\otimes V(=T^{2}(V))}$. The Koszul dual then coincides with the quadratic dual

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