Towards the τ-function of the quantum groups

Abstract

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