Indecomposable U_q(sl_n) modules for q^h = -1 and BRS intertwiners

TL;DR

This paper constructs and analyzes indecomposable modules of quantum groups U_q(sl_n) at roots of unity, revealing structures similar to conformal field theory modules and introducing BRS intertwiners for n=2,3.

Contribution

It introduces new indecomposable representations of U_q(sl_n) at q^h = -1 and constructs BRS intertwiners, expanding understanding of quantum group modules at roots of unity.

Findings

Constructed indecomposable modules for n=2,3 at q^h=-1.

Established BRS intertwiners analogous to conformal field theory.

Connected module structures with tensor product expansions.

Abstract

A class of indecomposable representations of U_q(sl_n) is considered for q an even root of unity (q^h = -1) exhibiting a similar structure as (height h) indecomposable lowest weight Kac-Moody modules associated with a chiral conformal field theory. In particular, U_q(sl_n) counterparts of the Bernard-Felder BRS operators are constructed for n=2,3. For n=2 a pair of dual d_2(h) = h dimensional U_q(sl_2) modules gives rise to a 2h-dimensional indecomposable representation including those studied earlier in the context of tensor product expansions of irreducible representations. For n=3 the interplay between the Poincare'-Birkhoff-Witt and (Lusztig) canonical bases is exploited in the study of d_3(h) = h(h+1)(2h+1)/6 dimensional indecomposable modules and of the corresponding intertwiners.