Nichols–Woronowicz algebra model for Schubert calculus on Coxeter groups

Abstract

We realise the cohomology ring of a flag manifold, more generally the coinvariant algebra of an arbitrary finite Coxeter group W, as a commutative subalgebra of a certain Nichols–Woronowicz algebra in the Yetter–Drinfeld category over W. This gives a braided Hopf algebra version of the corresponding Schubert calculus. The nilCoxeter algebra and its action on the coinvariant algebra by divided difference operators are also realised in the Nichols–Woronowicz algebra. We discuss the relationship between Fomin–Kirillov quadratic algebras, Kirillov–Maeno bracket algebras and our construction.