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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.20966 |
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| _version_ | 1866918132102201344 |
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| author | Chepurnoi, Maxim Sharov, Mikhail |
| author_facet | Chepurnoi, Maxim Sharov, Mikhail |
| contents | 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 $U_q(sl_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards the τ-function of the quantum groups Chepurnoi, Maxim Sharov, Mikhail High Energy Physics - Theory 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 $U_q(sl_n)$. |
| title | Towards the τ-function of the quantum groups |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2508.20966 |