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

Abstract

We study Lie bialgebra structures on \emph{flat metric Lie algebras}, that is, Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ 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 \[\mathfrak{g}=\mathfrak{a}\oplus\mathfrak{u},\qquad \mathfrak{u}=[\mathfrak{g},\mathfrak{g}]\ \text{abelian and even dimensional},\quad\mathfrak{a}:=\mathfrak{s}\oplus\mathfrak{z},\] 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 $P_\ell$. Under a generic \emph{nondegeneracy} (nonresonance) condition on the weights, we establish a normal form for Lie-bialgebra $1$-cocycles $\xi\colon\mathfrak{g}\to \wedge^2\mathfrak{g}$: each $\xi$ admits a decomposition $\xi=\ad r+R$, where $\ad r$ 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 $\exp(\mathfrak{z}\oplus\mathfrak{u})$.