PBW basis from Jimbo’s quantum root vectors

Determine whether Jimbo’s quantum root vectors in the positive and negative parts of the quantized enveloping algebra can be used to construct a basis of the quantized enveloping algebra U_q(gl_n) analogous to the Poincaré–Birkhoff–Witt basis of the enveloping algebra of gl_n.

Background

The paper constructs families of negative root vectors through the recursively defined Ψ operators and compares them with Jimbo’s recursively defined positive and negative root vectors \hat{E}{ij} and \hat{F}{ij}. The authors note that explicit commutation relations for Jimbo’s root vectors are known, but it is unresolved whether these vectors provide a PBW-type basis for the quantized enveloping algebra. Establishing such a basis would give a quantum analogue of the classical PBW basis and clarify the structural role of Jimbo’s root-vector construction.