2025/12/17 by Gai, Botong, Li, Chuanzhong, Sun, Jiacheng +2
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2512.15523
We introduce a BiHom-type skew-symmetric bracket on \mathfrakgl(V) built from two commuting inner automorphisms α=Adψ and β=Adϕ with ψ,ϕ∈ \mathfrakgl(V) and integers i,j. We prove that (\mathfrakgl(V),[⋅,⋅](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-1Lψ, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded 2×2 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 2×2 solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values.