Towards the τ-function of the quantum groups

The theory of τ -functions is receiving attention, due to their appearance as non-perturbative partition functions of certain quantum field theories. These τ − functions satisfy 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 constructed from group-like elements with the comultiplication Δ ( g ) = g ⊗ g . This restriction cannot be transferred straightforwardly 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 requirement that g be a group-like element. The main result of this work is a derivation of the set of bilinear identities and τ − functions for U q ( sl 3 ) in the fundamental representations for non-group-like elements. We consider different operators that lead to the basic bilinear identities. Also, we provide an analysis of possible approaches to obtaining BI for higher rank algebras U q ( sl N ) .