自由 Novikov 代数中 Lie 元素的消没判据
A vanishing criterion for Lie elements in a free Novikov algebra
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中文总结 AI 辅助
本文给出自由 Novikov 代数中多重线性元素属于 Lie 子代数的有限消没判据，利用微分实现与微分算子刻画，并应用于维数恢复与实例验证。
中文摘要 AI 辅助
每个 Novikov 代数都是 Lie 可容许的：换位运算使其成为 Lie 代数。我们给出了自由 Novikov 代数中一个多重线性元素属于由自由生成元生成的 Lie 子代数的有限判据。借助自由 Novikov 代数的微分实现，$n$ 次多重线性分量等同于 $n$ 个变量中 $n-1$ 次齐次多项式的空间。我们证明：一个多重线性元素是 Lie 元素当且仅当其符号在满足 $a_i\le 1$ 且 $a_1+\cdots+a_n\ge 2$ 的每个整数点 $(a_1,\ldots,a_n)$ 处消没。等价地，该符号被两个显式的二阶和三阶线性微分算子零化。证明结合了 Witt 代数、对称变量块的专门化论证、齐次插值以及 Molev 关于由自由生成元生成的 Lie 代数的多重线性分量作为对称群模的描述。作为应用，我们证明非零多重线性 Lie 元素绝不可能是全导数，恢复了多重线性 Lie 分量的维数，并通过 $4$、$5$ 和 $6$ 次的例子说明了该判据。
英文摘要
Every Novikov algebra is Lie-admissible: the commutator turns it into a Lie algebra. We give a finite criterion for a multilinear element of the free Novikov algebra to belong to the Lie subalgebra generated by the free generators. By the differential realization of free Novikov algebras, the multilinear component of degree $n$ is identified with the space of homogeneous polynomials of degree $n-1$ in $n$ variables. We prove that a multilinear element is a Lie element if and only if its symbol vanishes at every integer point $(a_1,\ldots,a_n)$ with $a_i\le 1$ and $a_1+\cdots+a_n\ge 2$. Equivalently, the symbol is annihilated by two explicit linear differential operators of orders two and three. The proof combines the Witt algebra, a specialization argument for a symmetric block of variables, homogeneous interpolation and Molev's description of the multilinear component of the Lie algebra generated by the free generators as a module over the symmetric group. As applications we show that a nonzero multilinear Lie element is never a total derivative, recover the dimension of the multilinear Lie component, and illustrate the criterion by examples in degrees $4$, $5$ and $6$.
