Quantized nilradicals of parabolic subalgebras of and algebras of coinvariants

Abstract

Let P_J be the standard parabolic subgroup of SL_n associated to a subset J of simple roots, and let $P_J=L_J{U}_J$ be the standard Levi decomposition. We let ${\mathcal{O}}_q(P_J)$ and ${\mathcal{O}}_q(L_J)$ denote the quantized algebras of regular functions on P_J and L_J, respectively. 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 ${\mathcal{O}}_q{(P_J)}^{\text{co\hspace{0.17em}}\theta }\subseteq {\mathcal{O}}_q(P_J)$ of coinvariants. It was previously shown that the smash product algebra ${\mathcal{O}}_q(L_J)\#{\mathcal{O}}_q{(P_J)}^{\text{co\hspace{0.17em}}\theta }$ is isomorphic to ${\mathcal{O}}_q(P_J).$ In view of this, ${\mathcal{O}}_q{(P_J)}^{\text{co\hspace{0.17em}}\theta }$ – while it is not a Hopf algebra – can be viewed as a quantum analogue of the coordinate ring $\mathcal{O}(U_J).$ In this article we prove that when $q\in \mathbb{K}$ is nonzero and not a root of unity, ${\mathcal{O}}_q{(P_J)}^{\text{co\hspace{0.17em}}\theta }$ is isomorphic to a quantum Schubert cell algebra ${\mathcal{U}}_q^{+}[w]$ associated to a certain parabolic element w in the Weyl group of $\mathfrak{s}\mathfrak{l}(n).$ In this setting, the quantum Schubert cell algebra ${\mathcal{U}}_q^{+}[w]$ is a q-deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of $\mathfrak{s}{\mathfrak{l}}_n.$ We also compute explicit commutation relations among the Lusztig root vectors for these particular quantized nilradicals and we give explicit algebra isomorphisms from ${\mathcal{U}}_q^{+}[w]$ to ${\mathcal{O}}_q{(P_J)}^{\text{co\hspace{0.17em}}\theta }.$

Keywords:

• Algebras of coinvariants
• levi decomposition
• quantum Schubert cell algebras
• quantum algebras
• quantized coordinate rings
• smash products

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17B37
• Secondary: 16T15
• 16T20

Additional information

Funding

The second author was supported in part by NSF grant DMS-1900823 and Division of Mathematical Sciences.