Extensions of simple modules for quantum groups at complex roots of $1$

TL;DR

This paper investigates the extension groups between simple modules for quantum groups at roots of unity, revealing their structure via Kazhdan-Lusztig polynomials and relating to classical algebraic group results.

Contribution

It establishes a connection between extension groups in quantum groups and Kazhdan-Lusztig polynomials, providing explicit dimension formulas and relating quantum and algebraic group theories.

Findings

Extension groups are determined by finitely many simple modules with small weights.

Dimensions of extension groups match top degree coefficients of Kazhdan-Lusztig polynomials.

Results generalize classical algebraic group extension findings to quantum groups at roots of unity.

Abstract

Let $U_q$ be the quantum group corresponding to a complex simple Lie algebra $\mathfrak g$ with root system $R$. Assume the quantum parameter $q\in \C$ is a root of unity. In this paper we study the extensions between simple modules in the category consisting of the finite dimensional modules for $U_q$. We first prove that this problem is equivalent to finding the extensions between the finitely many simple modules for the small quantum group $u_q$ in $U_q$. Then we show that the extension groups in question are determined by a finite subset with small highest weights. When the order of $q^2$ is at least the Coxeter number for $R$ we prove that the dimensions of such extension groups equal the top degree coefficients of some associated Kazhdan-Lusztig polynomial for the affine Weyl group for $R$. We relate all this to similar (old) results for almost simple algebraic groups and their Frobenius subgroup schemes over fields of large prime characteristics.