arXiv subjects

Amine Bahayou

Publications and source records attributed to Amine Bahayou.

4 recordsLinked to original sources

arXiv2026

Toric geometry of Schouten squares on rotational double extensions

Let $\g=\fb\oplus V\oplus\fb^{*}$ be a Medina-Revoy rotational double extension in which an abelian Lie algebra $\fb$ acts by scalar rotations on an orthogonal sum of oriented Euclidean two-planes. We analyse the restriction of the Schouten square $r\mapsto[r,r]$ to the block-rank-one locus $\Rone\subset\fb\wedge V$ and to its nondegenerate open part $\Rplus$. On every marked slice $S_{H}\subset\Rplus$ the Schouten square splits orthogonally into a radial and an angular component. The radial component factors through the standard toric moment map: its image is a simplicial cone, it satisfies a sharp quadratic coercive estimate with an intrinsic optimal constant, and the restricted Schouten map is proper with compact semialgebraic affine fibres. The angular component becomes Laurent monomial in joint eigencoordinates; a marker-cancellation criterion determines its signed exponent configuration, whose integral lattice controls the differential rank, the compact isotropy, its component group, and the Laurent binomial ideal defining the Zariski closure of the complexified image. Smith normal form makes each invariant algorithmic. All statements hold for arbitrary $\dim\fb$ and impose no nonresonance assumption. A corollary of the underlying disjoint-support decomposition classifies the classical Yang-Baxter equation on $\Rone$: for any bivector whose $\fb\wedge V$ component lies in $\Rone$, triangularity forces every block detected by its rotation weight to vanish, regardless of the $\Lambda^{2}\fb$, $\Lambda^{2}V$, or central-wedge terms.

math.DG↗

arXiv2026

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 $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ 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 $ξ\colon\mathfrak{g}\to \wedge^2\mathfrak{g}$: 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 $[r,r]+2\{r,R\}\in(\wedge^3\mathfrak{g})^{\mathfrak{g}}$ for the coupling term. We then describe the quasi-triangular (classical Yang--Baxter) locus via invariant Schouten squares. Finally, we integrate $ξ$ to explicit multiplicative Poisson tensors on $G$, producing concrete families of flat Poisson--Lie groups with polynomial formulas along the abelian normal subgroup $\exp(\mathfrak{z}\oplus\mathfrak{u})$.

math.DG↗

arXiv2022

Triangular structures on flat Lie algebras

In this work we study a large class of exact Lie bialgebras arising from noncommutative deformations of Poisson-Lie groups endowed with a left invariant Riemannian metric. We call these structures \emph{exact metaflat Lie bialgebras}. We give a complete classification of these structures. We show that given the metaflatness geometrical condition, these exact bialgebra structures arise necessarily from a nontrivial solution of the classical Yang-Baxter equation. Moreover, the dual Lie bialgebra is also flat and metaflat constituting an important kind of symmetry.

math.DG↗

arXiv2009

Multiplicative deformations of spectrale triples associated to left invariant metrics on Lie groups

We study the triple $(G,π,\prs)$ where $G$ is a connected and simply connected Lie group, $π$ and $\prs$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on $G$ such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the direction of $π$) of the spectrale triple associated to $\prs$ are satisfied. We show that the geometric problem of the classification of such triple $(G,π,\prs)$ is equivalent to an algebraic one. We solve this algebraic problem in low dimensions and we give the list of all $(G,π,\prs)$ satisfying Hawkins's conditions, up to dimension four.

math.DG↗