有限可解李共形代数的结构理论
Structure theory of finite solvable Lie conformal algebras
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中文总结 AI 辅助
本文建立有限可解李共形代数的结构理论，证明其Cartan子代数的共轭性，将结果应用于有限顶点代数回答相关问题，还完成零幂形李共形代数分类并证明其极大共形环面的性质。
中文摘要 AI 辅助
我们建立了有限可解李共形代数的结构理论。证明了每一个这类代数都存在Cartan子代数，且任意两个Cartan子代数可通过导出代数中元素的幂零零模的有限次指数乘积实现共轭。将该结果应用于有限顶点代数，可得到Cartan子代数的内共轭性，从而回答了D'Andrea和Marchei提出的问题。还证明了任意有限李共形代数的幂零根基和可解根基都是饱和的，且在普通导子和共形导子下不变。对于有限自由李共形代数，我们证明了共形环面的同时对角化定理，并通过常数权的显式恒等式刻画了非零环面的存在性。一个例子表明，有限幂零李共形代数的极大共形环面可以有不同的秩，因此共轭性一般不成立。我们对所有容许非零共形环面的完全饱和零幂形李共形代数进行了分类。对于该分类中的每一个代数，我们证明其极大共形环面的秩为1，且具有指定幂零根基的极大可解扩张是唯一的（同构意义下），即它是与一个极大共形环面的半直积。
英文摘要
We develop a structure theory of finite solvable Lie conformal algebras. It is proved that every such algebra admits a Cartan subalgebra, and any two Cartan subalgebras are conjugate by a finite product of exponentials of nilpotent zero modes of elements of the derived algebra. Applied to finite vertex algebras, this yields inner conjugacy of Cartan subalgebras, answering a question of D'Andrea and Marchei. The nilradical and solvable radical of an arbitrary finite Lie conformal algebra are shown to be saturated and invariant under ordinary and conformal derivations. For finite free Lie conformal algebras, we prove a simultaneous diagonalization theorem for conformal tori and characterize the existence of a nonzero torus by explicit identities for constant weights. An example demonstrates that maximal conformal tori of a finite nilpotent Lie conformal algebra can have different ranks, so that conjugacy fails in general. The totally saturated null-filiform Lie conformal algebras admitting a nonzero conformal torus are classified. For each algebra in this classification, we prove that maximal conformal tori have rank one and that the maximal solvable extension with the prescribed nilradical is unique (up to isomorphism) and is the semidirect product with a maximal conformal torus.
