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arXiv · 2508.20966

Towards the τ-function of the quantum groups

Maxim Chepurnoi·Mikhail Sharov
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Abstract

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 $Δ(g)=g \otimes g$. 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 $τ-$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 $U_q(\mathfrak{sl}_3)$ 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 $U_q(sl_n)$.

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Maxim Chepurnoi, Mikhail Sharov. 2025-08-28. Towards the τ-function of the quantum groups. https://arxiv.org/abs/2508.20966

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