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

TL;DR

This paper establishes that certain quantum coinvariant algebras associated with parabolic subalgebras of sl(n) are isomorphic to quantum Schubert cell algebras, providing explicit presentations for these structures.

Contribution

It proves the isomorphism between quantum coinvariant algebras and quantum Schubert cells for non-root of unity q, with explicit generators and relations.

Findings

Quantum coinvariant algebras are isomorphic to quantum Schubert cell algebras.

Explicit presentations for these quantum Schubert cells are provided.

The results hold for q not a root of unity.

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.