Compatible root graded anti-pre-Lie algebraic structures on finite-dimensional complex simple Lie algebras
Chengming Bai, Dongfang Gao
TL;DR
The paper investigates compatible root graded anti-pre-Lie algebraic structures on finite-dimensional complex simple Lie algebras through the lens of $\mathfrak{sl}_2(\mathbb{C})$ representation theory and root space decompositions. It establishes that a unique compatible root graded anti-pre-Lie structure exists on $\mathfrak{sl}_2(\mathbb{C})$, explicitly described by the products $h_1\circ e_{12}=-2 e_{12}$, $e_{12}\circ h_1=-4 e_{12}$, $h_1\circ e_{21}=2 e_{21}$, $e_{21}\circ h_1=4 e_{21}$, $e_{12}\circ e_{21}=\tfrac{1}{2} h_1$, $e_{21}\circ e_{12}=-\tfrac{1}{2} h_1$, with $h_1\circ h_1=e_{12}\circ e_{12}=e_{21}\circ e_{21}=0$. Conversely, it proves there are no such structures on any other finite-dimensional complex simple Lie algebra. The arguments combine root graded constraints, weight representations of $\mathfrak{sl}_2(\mathbb{C})$, and a constructive contradiction using a family of algebras $\mathfrak{b}_n$, together with a case-by-case analysis across the classical and exceptional types. This result substantially narrows the landscape of compatible anti-pre-Lie structures on simple Lie algebras and highlights the special role of $\mathfrak{sl}_2(\mathbb{C})$ in this context.
Abstract
We investigate the compatible root graded anti-pre-Lie algebraic structures on any finite-dimensional complex simple Lie algebra by the representation theory of ${\rm sl_2(\C)}$. We show that there does not exist a compatible root graded anti-pre-Lie algebraic structure on a finite-dimensional complex simple Lie algebra except ${\rm sl_2(\C)}$, whereas there is exactly one compatible root graded anti-pre-Lie algebraic structure on ${\rm sl_2(\C)}$.