Explicit classification of all finite-dimensional polynomial SH-Lie algebras over R^d or C^d using complete generalized Wronskians of order k as N-ary brackets, together with a factorization formula for the associated generalized Vandermonde determinants.
Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
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.
fields
math.RA 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
citing papers explorer
-
Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$
Explicit classification of all finite-dimensional polynomial SH-Lie algebras over R^d or C^d using complete generalized Wronskians of order k as N-ary brackets, together with a factorization formula for the associated generalized Vandermonde determinants.