Verma Modules for Restricted Quantum Groups at a Fourth Root of Unity

TL;DR

This paper extends the understanding of representations of restricted quantum groups at a fourth root of unity, generalizing tensor decompositions and characterizing conditions for isomorphisms, with implications for braid group representations.

Contribution

It generalizes tensor decomposition formulas for representations of restricted quantum groups to all semisimple Lie algebras and describes projective covers in the $ ext{sl}_3$ case.

Findings

Generalized tensor decomposition for all semisimple Lie algebras.

Described projective covers of representations in $ ext{sl}_3$.

Characterized conditions for isomorphisms of tensor products.

Abstract

For a semisimple Lie algebra $\mathfrak{g}$ of rank $n$, let $\overline{U}_\zeta(\mathfrak{g})$ be the restricted quantum group of $\mathfrak{g}$ at a primitive fourth root of unity. This quantum group admits a natural Borel-induced representation $V({\boldsymbol{t}})$, with ${\boldsymbol{t}}\in(\mathbb{C}^\times)^n$ determined by a character on the Cartan subalgebra. Ohtsuki showed that for $\mathfrak{g}=\mathfrak{sl}_2$, the braid group representation determined by tensor powers of $V({\boldsymbol{t}})$ is the exterior algebra of the Burau representation. In this paper, we generalize the tensor decomposition of $V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})$ used in Ohtsuki's proof to any semisimple $\mathfrak{g}$. Upon specializing to the $\mathfrak{sl}_3$ case, we describe all projective covers of $V({\boldsymbol{t}})$ in terms of induced representations. The above decomposition formula for $V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})$ is then extended to more general $\boldsymbol{t}$ and $\boldsymbol{s}$ where these projective covers occur as indecomposable summands. We also define a stratification of $(\mathbb{C}^\times)^{4}$ whose points $({\boldsymbol{t}},{\boldsymbol{s}})$ in the lower strata are associated with representations $V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})$ that do not have a homogeneous cyclic generator. With this information, we characterize under what conditions the isomorphism ${V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})\cong V({\boldsymbol{\lambda t}})\otimes V({\boldsymbol{\lambda^{-1} s}})}$ holds.