Born Lie algebras
Alejandro Gil-García, Paula Naomi Pilatus
TL;DR
The paper develops a systematic framework for Born Lie algebras by showing every Born Lie algebra arises from a bicross product of two real pseudo-Riemannian Lie algebras connected by an isometry $Q$ with compatible representations $\varphi,\rho$. This approach enables a complete classification of Lie algebras admitting integrable Born structures in dimensions $2$, $4$, and in six-dimensional nilpotent cases, revealing explicit lists: in 2D all algebras are Born; in 4D the non-abelian algebras are precisely $\mathfrak{rh}_3,\ \mathfrak{rr}_{3,0},\ \mathfrak{r}_2\mathfrak{r}_2,\ \mathfrak{r}'_2,\ \mathfrak{r}_{4,-1,-1},\ \mathfrak{d}_{4,1},\ \mathfrak{d}_{4,2},\ \mathfrak{d}_{4,1/2}$; in 6D, the nilpotent algebras $\mathfrak{h}_4,\ \mathfrak{h}_7,\ \mathfrak{h}_8,\ \mathfrak{h}_9,\ \mathfrak{h}_{10},\ \mathfrak{h}_{11},\ \mathfrak{h}_{13}$ arise. The work also analyzes curvature properties of the resulting metrics and provides constructions yielding pseudo-hyperkähler and hypersymplectic examples, highlighting the geometric richness of integrable Born structures on low-dimensional Lie groups.
Abstract
We show that every Born Lie algebra can be obtained by the bicross product construction starting from two pseudo-Riemannian Lie algebras. We then obtain a classification of all Lie algebras up to dimension four and all six-dimensional nilpotent Lie algebras admitting an integrable Born structure. Finally, we study the curvature properties of the pseudo-Riemannian metrics of the integrable Born structures obtained in our classification results.