On injective modules and support varieties for the small quantum group

TL;DR

This paper establishes new criteria for detecting injectivity of modules over small quantum groups and their Frobenius kernels using support and restriction to root subalgebras, generalizing previous results.

Contribution

It extends injectivity detection methods to modules with torus actions and introduces a rank variety approach for support varieties in the quantum group setting.

Findings

Injectivity can be detected via restriction to root subalgebras.

The criterion applies to modules with compatible torus actions.

Supports varieties can be characterized by restriction to a single root subalgebra.

Abstract

Let $u_\zeta(g)$ denote the small quantum group associated to the simple complex Lie algebra $g$, with parameter $q$ specialized to a primitive $\ell$-th root of unity $\zeta$ in the field $k$. Generalizing a result of Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional $u_\zeta(g)$-module admitting a compatible torus action, then the injectivity of $M$ as a module for $u_\zeta(g)$ can be detected by the restriction of $M$ to certain root subalgebras of $u_\zeta(g)$. If the characteristic of $k$ is positive, then this injectivity criterion also holds for the higher Frobenius--Lusztig kernels $U_\zeta(G_r)$ of the quantized enveloping algebra $U_\zeta(g)$. Now suppose that $M$ lifts to a $U_\zeta(g)$-module. Using a new rank variety type result for the support varieties of $u_\zeta(g)$, we prove that the injectivity of $M$ for $u_\zeta(g)$ can be detected by the restriction of $M$ to a single root subalgebra.