A Survey on Koszul Algebras and Koszul Duality

Abstract

In this survey article, we discuss the theory of Koszul algebras and Koszul duality. Koszul property plays an important role in various topics such as commutative and noncommutative algebra, geometry, topology, Lie algebras, representation theory, combinatorics and semigroup rings. Most of the results in this survey article are already known or are easy corollaries of known results. We collect these results and we discuss some proofs. Several survey articles, notably [22, 13, 14, 15, 31], and a book on quadratic algebras [36] contain valuable material to the theory of Koszul algebras or Koszul duality. Keeping these survey articles in mind, the focus has been to collect new results on Koszulness of diagonal subalgebras, its interaction with Polya frequency sequence, and monomial projective curves.