Unimodularity and Invariant Volume Forms for Hamiltonian Dynamics on Poisson-Lie Groups

The paper’s main statements are: (i) if the dual Lie algebra g* is unimodular, then every Hamiltonian vector field on a Poisson–Lie group (G, Π) preserves the volume form √f0 ν^l (hence a multiple of any left-invariant volume), and (ii) conversely, if H is Morse at the identity and X^Π_H preserves some volume form, then g* is unimodular. These are developed via the modular vector field formula M^Π_{ν^l} = 1/2(M^l_{g*} + M^r_{g*} + Π♯(d log f0)) and the change-of-volume relation M^Π_{e^σΦ} = M^Π_Φ − X^Π_σ, leading to the criterion X^Π_H(σ − log√f0) + 1/2(M^l_{g*}(H)+M^r_{g*}(H)) = 0 for invariance of e^σν^l (equation (9) ⇒ Theorem 3.1, then Corollary 3.3) . The converse (Theorem 3.6) differentiates this invariance condition at e to obtain (Hess H)(e)(ξ, M^Π_{ν^l}(e)) = 0 for all ξ, hence M^Π_{ν^l}(e)=M_{g*}=0 by nondegeneracy of the Hessian . The candidate solution reproduces exactly this modular-field argument: it derives the same criterion, sets σ = (1/2) log f0 in the unimodular case to kill the modular vector field, and in the converse differentiates at e to conclude Hess H(e)(ξ, M_{g*})=0, hence M_{g*}=0. This matches the paper’s logic and uses the same ingredients (modular vector field formula, the function f0=det(Ad^n_g), and the Hessian at a Morse critical point) . Any differences are presentational (the candidate directly inserts log√f0 into the criterion before differentiating), not substantive.