AI 中文总结
研究格尔范德 - 多尔夫曼代数的幂零性与可解性，给出构造方法及单李代数上该代数结构示例，通过与其他代数对比，对低维复GD代数进行完整代数分类，确定其相关性质。
AI 中文摘要
本文刻画了格尔范德 - 多尔夫曼（GD）代数的幂零性和可解性。与泊松代数和转置泊松代数不同，给出例子表明GD代数的幂零性和可解性不由其基础代数的幂零性和可解性决定。给出多种构造方法并判断所得代数是否特殊。研究了单李代数上的GD代数结构，给出例子说明其不一定平凡。最后对低维复GD代数进行了完整代数分类并确定其幂零性、可解性和特殊性。
英文摘要
In this paper, we characterize the nilpotency and solvability of Gelfand--Dorfman (GD) algebras. In contrast with Poisson algebras and transposed Poisson algebras, we give examples show the nilpotency and solvability of a GD algebra are not determined by the nilpotency and solvability of its underlying algebras. To obtain more examples of special and non-special GD algebras, we give several construction methods and determine whether the resulting algebras are special. Futhermore, we study GD algebra structures on simple Lie algebras. We provide examples demonstrating that GD algebra structures on simple Lie algebras are not necessarily trivial, distinguishing them from Poisson and transposed Poisson algebras. Finally, we provide a complete algebraic classification of low-dimensional complex GD algebras, and determine their nilpotency, solvability and speciality.
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