2023/10/19 by Amine Bahayou, Bahayou, Amine · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2310.12966
We study Lie bialgebra structures on flat metric Lie algebras, that is, Lie algebras (\mathfrakg,⟨⋅,⋅⟩) whose associated left-invariant Riemannian metric on the simply connected Lie group G has zero curvature. By Milnor's structure theorem, such \mathfrakg splits orthogonally as \mathfrakg=\mathfraka⊕\mathfraku, \mathfraku=[\mathfrakg,\mathfrakg] abelian and even dimensional, \mathfraka:=\mathfraks⊕\mathfrakz, where \mathfrakz is the center and \mathfraks is an abelian subalgebra that acts on \mathfraku by commuting infinitesimal rotations; this yields a decomposition of \mathfraku into 2-dimensional weight planes P_ℓ. Under a generic nondegeneracy (nonresonance) condition on the weights, we establish a normal form for Lie-bialgebra 1-cocycles ξ\colon\mathfrakg→ \wedge2\mathfrakg: each ξ admits a decomposition ξ=\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\∈(\wedge3\mathfrakg)^\mathfrakg for the coupling term. We then describe the quasi-triangular (classical Yang--Baxter) locus via invariant Schouten squares. Finally, we integrate ξ to explicit multiplicative Poisson tensors on G, producing concrete families of flat Poisson--Lie groups with polynomial formulas along the abelian normal subgroup exp(\mathfrakz⊕\mathfraku).