<p>We present two involutivity theorems in the context of Poisson quasi-Nijenhuis manifolds. The second one stems from recursion relations that generalize the so-called Lenard–Magri relations on a bi-Hamiltonian manifold. We apply these results to the closed (or periodic) Toda lattices of type <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1970_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$A_n^{(1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1970_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$C_n^{(1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>C</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1970_Article_IEq3.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$A_{2n}^{(2)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, and, for the ones of type <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1970_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$A^{(1)}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, we show how this geometrical setting relates to their bi-Hamiltonian representation and to their recursion relations.</p>

Poisson quasi-Nijenhuis manifolds, closed Toda lattices, and generalized recursion relations

期刊论文
发表日期：

2025年7月26日

查看全文

E. Chuño Vizarreta,
G. Falqui,
I. Mencattini,
M. Pedroni

摘要

We present two involutivity theorems in the context of Poisson quasi-Nijenhuis manifolds. The second one stems from recursion relations that generalize the so-called Lenard–Magri relations on a bi-Hamiltonian manifold. We apply these results to the closed (or periodic) Toda lattices of type $A_n^{(1)}$ $A_{n}^{(1)}$ , $C_n^{(1)}$ $C_{n}^{(1)}$ , $A_{2n}^{(2)}$ $A_{2 n}^{(2)}$ , and, for the ones of type $A^{(1)}_n$ $A_{n}^{(1)}$ , we show how this geometrical setting relates to their bi-Hamiltonian representation and to their recursion relations.