关于q-预李代数
On $q$-pre-Lie algebras
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中文总结 AI 辅助
本文从李代数表示角度引入q-预李代数概念，统一预李代数和反预李代数，介绍相关概念及关系并给出构造，对Witt代数等进行分类，证明特定条件下某些代数上q-预李代数结构的存在性与不存在性。
中文摘要 AI 辅助
本文从李代数表示的角度引入q-预李代数的概念，给出一个参数化推广，统一了预李代数和反预李代数。对于q-预李代数(A,∘)，∘的换位子是一个李括号，q倍缩放的左乘算子给出相关换位子李代数的一个表示。还引入q-$\mathcal{O}$-算子和q-诺维科夫代数的概念并研究它们与q-预李代数的关系，给出q-预李代数的几种显式构造。给出Witt代数上分次q-预李代数结构的完全分类，证明当q≠1时Virasoro代数上不存在此类结构。最后表明，对于有限维复单李代数，$\mathfrak{sl}_2(\mathbb{C})$上恰在q = 2或q = -1时存在相容根分次q-预李代数，其他单李代数上不存在。
英文摘要
In this paper, we introduce the notion of $q$-pre-Lie algebras from the perspective of representations of Lie algebras, providing a parametrized generalization that unifies pre-Lie algebras and anti-pre-Lie algebras. For a $q$-pre-Lie algebra $(A,\circ)$, the commutator of $\circ$ is a Lie bracket and the left multiplication operator scaled by $q$ gives a representation of the associated commutator Lie algebra. We also introduce the notions of $q$-$\mathcal{O}$-operators and $q$-Novikov algebras, and investigate their relationships with $q$-pre-Lie algebras. Several explicit constructions of $q$-pre-Lie algebras are provided. Moreover, we give a complete classification of graded $q$-pre-Lie algebra structures on the Witt algebra and prove the existence of such structures on the Virasoro algebra only when $q=1$ and $q=2$. Finally, we classify compatible root-graded $q$-pre-Lie algebra structures on finite-dimensional complex simple Lie algebras.
