2025/08/28 by Chepurnoi, Maxim, Sharov, Mikhail
#FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · doi:10.48550/arxiv.2508.20966
Non-perturbative partition functions of quantum theories constitute a class of τ-functions, which are distinguished satisfying Hirota's bilinear identities(BI). To make this statement general, there must be a proper definition of τ-function that gives rise to a set of bilinear identities. In the classical definition of τ-function for integrable Toda or KP hierarchies, there is a restriction on matrix elements to be based on group-like elements with the comultiplication Δ(g)=g ⊗ g. This restriction can not be straightforwardly transferred to the q-deformed case, because there are no group-like elements in q-deformed universal enveloping algebra (UEA), except for its Cartan subalgebra. The new approach to the τ-function is to remove the restriction on g to be obligatory the group-like element. The main result of this work is a derivation of the set of bilinear identities and τ-functions for Uq(\mathfraksl3) in the fundamental representations for non-group-like elements. We consider difference operators which lead to the basic bilinear identities. Also, we provide an analysis of the ways of obtaining BI for higher rank algebras Uq(sln).