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The dual braid monoid was introduced by Bessis in his work on complexreflection arrangements. The goal of this work is to show that Koszul dualityprovides a nice interplay between the dual braid monoid and the cluster complexintroduced by Fomin and Zelevinsky. Firstly, we prove koszulity of the dualbraid monoid algebra, by building explicitly the minimal free resolution of theground field. This is done explicitly using some chains complexes defined interms of the positive part of the cluster complex. Secondly, we derive variousproperties of the quadratic dual algebra. We show that it is naturally gradedby the noncrossing partition lattice. We get an explicit basis, naturallyindexed by positive faces of the cluster complex. Moreover, we find thestructure constants via a geometric rule in terms of the cluster fan.Eventually, we realize this dual algebra as a quotient of a Nichols algebra.This latter fact makes a connection with results of Zhang, who used the samealgebra to compute the homology of Milnor fibers of reflection arrangements.

Koszulity of dual braid monoid algebras via cluster complexes