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

Abstract

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.