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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.00695 |
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| _version_ | 1866914707840958464 |
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| author | Mironov, A. Mishnyakov, V. Morozov, A. |
| author_facet | Mironov, A. Mishnyakov, V. Morozov, A. |
| contents | Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $Δ(X) = X\otimes X$, then their generating functions obey bilinear Hirota equations and hence are named $τ$-functions. However, dealing with group elements is not always easy, especially for non-commutative algebras of functions, and this slows down the development of $τ$-function theory and the study of integrability properties of non-perturbative functional integrals. A simple way out is to use arbitrary elements of the universal enveloping algebra, and not just the group elements. Then the Hirota equations appear to interrelate a whole system of generating functions, which one may call $\textit{generalized}$ $τ$-functions. It was recently demonstrated that this idea can be applicable even to a somewhat sophisticated case of the quantum toroidal algebra. We consider a number of simpler examples, including ordinary and quantum groups, to explain how the method works and what kind of solutions one can obtain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00695 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tau-functions beyond the group elements Mironov, A. Mishnyakov, V. Morozov, A. High Energy Physics - Theory Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $Δ(X) = X\otimes X$, then their generating functions obey bilinear Hirota equations and hence are named $τ$-functions. However, dealing with group elements is not always easy, especially for non-commutative algebras of functions, and this slows down the development of $τ$-function theory and the study of integrability properties of non-perturbative functional integrals. A simple way out is to use arbitrary elements of the universal enveloping algebra, and not just the group elements. Then the Hirota equations appear to interrelate a whole system of generating functions, which one may call $\textit{generalized}$ $τ$-functions. It was recently demonstrated that this idea can be applicable even to a somewhat sophisticated case of the quantum toroidal algebra. We consider a number of simpler examples, including ordinary and quantum groups, to explain how the method works and what kind of solutions one can obtain. |
| title | Tau-functions beyond the group elements |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2312.00695 |