# Towards the {\tau}-function of the quantum groups

## TL;DR

This paper develops a new framework for defining and deriving bilinear identities for $ au$-functions in quantum groups, specifically for $U_q(rak{sl}_3)$, extending classical integrable system concepts to the q-deformed setting.

## Contribution

It introduces a novel approach to $ au$-functions in quantum groups by removing the restriction to group-like elements, deriving bilinear identities for $U_q(rak{sl}_3)$ in fundamental representations.

## Key findings

- Derived bilinear identities for $U_q(\mathfrak{sl}_3)$
- Extended the framework to non-group-like elements in quantum groups
- Analyzed methods for higher rank algebras $U_q(sl_n)$

## Abstract

Non-perturbative partition functions of quantum theories constitute a class of $\tau-$functions, which are distinguished satisfying Hirota's bilinear identities(BI). To make this statement general, there must be a proper definition of $\tau-$function that gives rise to a set of bilinear identities. In the classical definition of $\tau-$function for integrable Toda or KP hierarchies, there is a restriction on matrix elements to be based on group-like elements with the comultiplication $\Delta(g)=g \otimes g$. 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 $\tau-$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 $\tau-$functions for $U_q(\mathfrak{sl}_3)$ 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 $U_q(sl_n)$.