Quantized nilradicals of parabolic subalgebras of $\mathfrak{sl}(n)$ and algebras of coinvariants

TLDR: In this paper, it was shown that the smash product algebra is isomorphic to a quantum Schubert cell algebra, which is a quantum analogue of the coordinate ring coordinate ring.

Abstract: Let $P_J$ be the standard parabolic subgroup of $SL_n$ obtained by deleting a subset $J$ of negative simple roots, and let $P_J = L_JU_J$ be the standard Levi decomposition. Following work of the first author, we study the quantum analogue $\theta: {\mathcal O}_q(P_J) \to{\mathcal O}_q(L_J) \otimes {\mathcal O}_q(P_J)$ of an induced coaction and the corresponding subalgebra ${\mathcal O}_q(P_J)^{\operatorname{co} \theta} \subseteq {\mathcal O}_q(P_J)$ of coinvariants. It was shown that the smash product algebra ${\mathcal O}_q(L_J)\# {\mathcal O}_q(P_J)^{\operatorname{co} \theta}$ is isomorphic to ${\mathcal O}_q(P_J)$. In view of this, ${\mathcal O}_q(P_J)^{\operatorname{co} \theta}$ -- while it is not a Hopf algebra -- can be viewed as a quantum analogue of the coordinate ring ${\mathcal O}(U_J)$. In this paper we prove that when $q\in \mathbb{K}$ is nonzero and not a root of unity, ${\mathcal O}_q(P_J)^{\operatorname{co} \theta}$ is isomorphic to a quantum Schubert cell algebra ${\mathcal U}_q^+[w]$ associated to a parabolic element $w$ in the Weyl group of $\mathfrak{sl}(n)$. An explicit presentation in terms of generators and relations is found for these quantum Schubert cells.