This work derives bilinear identities for tau-functions in quantum groups, indicating a novel approach with implications for higher rank algebras.
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(sl₃) 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).
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Chepurnoi et al. (2025) studied this question.
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