BiHom-Lie brackets and the Toda equation
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914205641211904 |
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| author | Gai, Botong Li, Chuanzhong Sun, Jiacheng Wang, Shuanhong Zhu, Haoran |
| author_facet | Gai, Botong Li, Chuanzhong Sun, Jiacheng Wang, Shuanhong Zhu, Haoran |
| contents | We introduce a BiHom-type skew-symmetric bracket on $\mathfrak{gl}(V)$ built from two commuting inner automorphisms $α=Ad_ψ$ and $β=Ad_ϕ$ with $ψ,ϕ\in \mathfrak{gl}(V)$ and integers $i,j$. We prove that $(\mathfrak{gl}(V),[\cdot,\cdot]^{(i,j)}_{(ψ,ϕ)},α,β)$ is a BiHom--Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $ϕ=ψ$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\widetilde L=ψ^{-1}Lψ$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times2$ blocks, we derive explicit trigonometric and hyperbolic formulas that make symmetry constraints (e.g. tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | BiHom-Lie brackets and the Toda equation Gai, Botong Li, Chuanzhong Sun, Jiacheng Wang, Shuanhong Zhu, Haoran Exactly Solvable and Integrable Systems Rings and Algebras We introduce a BiHom-type skew-symmetric bracket on $\mathfrak{gl}(V)$ built from two commuting inner automorphisms $α=Ad_ψ$ and $β=Ad_ϕ$ with $ψ,ϕ\in \mathfrak{gl}(V)$ and integers $i,j$. We prove that $(\mathfrak{gl}(V),[\cdot,\cdot]^{(i,j)}_{(ψ,ϕ)},α,β)$ is a BiHom--Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $ϕ=ψ$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\widetilde L=ψ^{-1}Lψ$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times2$ blocks, we derive explicit trigonometric and hyperbolic formulas that make symmetry constraints (e.g. tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values. |
| title | BiHom-Lie brackets and the Toda equation |
| topic | Exactly Solvable and Integrable Systems Rings and Algebras |
| url | https://arxiv.org/abs/2512.15523 |