  Let $\mathcal{E}_m$ be the Fomin-Kirillov algebra, and let
$\mathcal{B}_{\mathbb{S}_m}$ be the Nichols-Woronowicz algebra model for
Schubert calculus on the symmetric group $\mathbb{S}_m$ which is a quotient of
$\mathcal{E}_m$, i.e. the Nichols algebra associated to a Yetter-Drinfeld
$\mathbb{S}_m$-module defined by the set of reflections of $\mathbb{S}_m$ and a
specific one-dimensional representation of a subgroup of $\mathbb{S}_m$. It is
a famous open problem to prove that $\mathcal{E}_m$ is infinite dimensional for
all $m\geq 6$. In this work, as a step towards a solution of this problem, we
introduce a subalgebra of $\mathcal{B}_{\mathbb{S}_m}$, and prove, under the
assumption of finite dimensionality of $\mathcal{B}_{\mathbb{S}_m}$, that this
subalgebra admits unique integrals in a strong sense, and we relate these
integrals to integrals in $\mathcal{B}_{\mathbb{S}_m}$. The techniques we use
rely on braided differential calculus as developed by Bazlov and Liu, and on
the notion of integrals for Hopf algebras as introduced by Sweedler.

Quantum Algebra;Rings and Algebras