BiHom-Lie brackets and the Toda equation

TL;DR

This work constructs a BiHom-Lie algebra on $\mathfrak{gl}(V)$ by deforming the Lie bracket with two commuting inner automorphisms and studies the resulting BiHom–Toda Lax dynamics. It analyzes three canonical one-parameter deformations (scalar dilations, planar rotations, planar hyperbolic rotations), deriving explicit $2\times 2$ block dynamics, trace and isospectrality properties, and gauge reductions that connect to Toda-type flows in symmetric cases. The paper extends to weakly coupled Toda lattices with indefinite metrics, providing explicit 2×2 solutions via inverse scattering, spectral data classification, and Miura-type transformations that relate deformed and undeformed systems. Special parameter choices recover classical Toda dynamics, situating the deformations as a unified framework for non-standard Lax flows and symmetry breaking phenomena in integrable lattices.

Abstract

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.

Theorems & Definitions (34)

• Theorem 1
• Remark 2.1: Specializations
• Remark 3.1
• Proposition 4.1
• proof
• Proposition 4.2: Traces and equilibria in the $2\times 2$ block
• proof
• Remark 4.3
• Proposition 4.4
• proof