标题： Wronskians 作为有限维 $sl(2)$类似物中的 $N$元括号 显示英文标题

标题： Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$

摘要： 雅可比类型二次恒等式表对于强同伦李代数的高阶微分算子系数满足——即对于$L_\infty$变形的一个特殊情况——该一维仿射流形上的向量场李代数。我们证明了$\mathfrak{sl}(2)$通过二次系数向量场的标准实现是有限维多项式代数$\Bbbk_N[x]$的底层结构，其中 Wronskian 作为$N$元括号；结构常数被显式计算。关键词：Wronskian 行列式，$N$元括号，$L_\infty$-代数，强同伦李代数，$sl(2)$，Witt 代数，Vandermonde 行列式。 显示英文摘要

摘要： The Wronskian determinants (for coefficients of higher-order differential operators on the affine real line or circle) satisfy the table of Jacobi-type quadratic identities for strong homotopy Lie algebras -- i.e. for a particular case of $L_\infty$-deformations -- for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of $\mathfrak{sl}(2)$ by quadratic-coefficient vector fields is the bottom structure in a sequence of finite-dimensional polynomial algebras $\Bbbk_N[x]$ with the Wronskians as $N$-ary brackets; the structure constants are calculated explicitly. Key words: Wronskian determinant, $N$-ary bracket, $L_\infty$-\/algebra, strong homotopy Lie algebra, $sl(2)$, Witt algebra, Vandermonde determinant.