Difference hierarchies for \(nT\) \(\tau\)-functions. (English) Zbl 1447.17019
Summary: We introduce hierarchies of difference equations (referred to as \(nT\)-systems) associated to the action of a (centrally extended, completed) infinite matrix group \(\mathrm{GL}_\infty^{(n)}\) on \(n\)-component fermionic Fock space. The solutions are given by matrix elements (\(\tau\)-functions) for this action. We show that the \(\tau\)-functions of type \(nT\) satisfy bilinear equations of length \(3, 4, \dots, n + 1\). The \(2T\)-system is, after a change of variables, the usual \(3\) term \(T\)-system of type \(A\). Restriction from \(\mathrm{GL}_\infty^{(n)}\) to a subgroup isomorphic to the loop group \(\mathrm{LGL}_n\), defines \(nQ\)-systems, studied earlier in [SIGMA, Symmetry Integrability Geom. Methods Appl. 15, Paper 023, 42 p. (2019; Zbl 1447.17020), see also arXiv:1605.00192] by the present authors for \(n = 2, 3\).
MSC:
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 39A30 | Stability theory for difference equations |
Citations:
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