Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties

For a root system of type $A$, we introduce and study a certain extension of the quadratic algebra invented by S. Fomin and the first author, to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application, a generalization of the equivariant Pieri rule for double Schubert polynomials is described. For a general finite Coxeter system, an extension of the corresponding Nichols–Woronowicz algebra is constructed. In the case of finite crystallographic Coxeter systems, a construction is presented of an extended Nichols–Woronowicz algebra model for the equivariant cohomology of the corresponding flag variety.