Bialgebra Structures on Flat Lie Algebras and their Poisson-Lie Groups

We study Lie bialgebra structures on \emph{flat metric Lie algebras}, that is, Lie algebras $(\mathfrak{g},⟨\cdot ,\cdot ⟩)$ whose associated left-invariant Riemannian metric on the simply connected Lie group $G$ has zero curvature. By Milnor’s structure theorem, such $\mathfrak{g}$ splits orthogonally as $$\mathfrak{g}=\mathfrak{a}\oplus \mathfrak{u},\mathfrak{u}=[\mathfrak{g},\mathfrak{g}]\text{ abelian and even dimensional},\mathfrak{a}:=\mathfrak{s}\oplus \mathfrak{z},$$ where $\mathfrak{z}$ is the center and $\mathfrak{s}$ is an abelian subalgebra that acts on $\mathfrak{u}$ by commuting infinitesimal rotations; this yields a decomposition of $\mathfrak{u}$ into $2$-dimensional weight planes $P_{\mathrm{\ell }}$. Under a generic \emph{nondegeneracy} (nonresonance) condition on the weights, we establish a normal form for Lie-bialgebra $1$-cocycles $\xi \text{\hspace{0.17em}⁣}:\mathfrak{g}\to {\wedge }^2\mathfrak{g}$: each $\xi$ admits a decomposition $\xi =\text{ad}r+R$, where $\text{ad}r$ is a coboundary and $R$ is a normalized cocycle with tightly controlled components. Using the Big Bracket (Maurer–Cartan) formalism together with the rotation geometry of the weight planes, we split the co-Jacobi condition into two independent equations: a reduced co-Jacobi equation $\{R,R\}=0$ for the normalized cocycle, and an invariant-trivector condition $[r,r]+2\{r,R\}\in ({\wedge }^3\mathfrak{g}{)}^{\mathfrak{g}}$ for the coupling term. We then describe the quasi-triangular (classical Yang–Baxter) locus via invariant Schouten squares. Finally, we integrate $\xi$ to explicit multiplicative Poisson tensors on $G$, producing concrete families of flat Poisson–Lie groups with polynomial formulas along the abelian normal subgroup $\exp (\mathfrak{z}\oplus \mathfrak{u})$.