Infinitely many left-symmetric structures on nilpotent Lie algebras
Naoki Kato
TL;DR
The paper addresses the problem of constructing and classifying left-symmetric structures on cotangent Lie algebras of almost abelian nilpotent Lie algebras. It develops a parametric family $\Delta_{(\alpha,\beta)}$ of complete left-symmetric products on $T^*\mathfrak{g}$ for $\mathfrak{g}=\mathbb{R} \ltimes_{J_n(0)} \mathbb{R}^n$, providing an explicit isomorphism criterion (Theorem 3.1) that yields uncountably many pairwise non-isomorphic structures. The paper further connects these LS structures to symplectic geometry by constructing symplectic forms $\omega_{\lambda}$ on $T^*\mathfrak{g}$ whose induced LS structures coincide with $\Delta_{(\alpha,\beta)}$ (with $\alpha=(\lambda-1)/\lambda$, $\beta=-(\lambda-n+2)/(\lambda-n+1)$), and shows that, up to homothety, there are uncountably many non-equivalent symplectic structures. Overall, the work demonstrates a rich landscape of complete left-symmetric and symplectic structures on cotangent nilpotent algebras, with precise isomorphism and symplectomorphism criteria enabling classification and continuous families.
Abstract
Dekimpe and Ongenae constructed infinitely many pairwise non-isomorphic complete left-symmetric structures on $\mathbb{R}^n$ for $n\geq 6$. In this paper, we construct a family of complete left-symmetric structures on the cotangent Lie algebra $T^*\mathfrak{g}$ of a certain $n$-dimensional almost abelian nilpotent Lie algebra $\mathfrak{g}$ and give a condition under which two left-symmetric structures in this family are isomorphic. As a consequence of this result, we obtain infinitely many pairwise non-isomorphic left-symmetric structures on $T^{*}\mathfrak{g}$. As an application of this construction, we also obtain infinitely many symplectic structures on $T^{*}\mathfrak{g}$ which are pairwise non-symplectomorphic up to homothety.