All finite-dimensional polynomial SH-Lie algebras over k[x^1,...,x^d] with complete generalized Wronskians of order k as N-ary brackets (N=binom(d+k,d)) are explicitly described, with a factorization of the associated generalized Vandermonde determinants.
Wronskians as n-Lie multiplications
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abstract
Filipov proved that Jacobian algebra is n-Lie. In our paper we consider algebras defined on associative commutative algebra U with derivation $\der$ by (k+1)-multiplication $V^{0,1,...,k}=\der^0\wedge\der^1\wedge...\wedge \der^k$ (Wronskian). We study whether they have (k+1)-Lie, k-left commutative and homotopical (k+1)-Lie structures.
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2026 1verdicts
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Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$
All finite-dimensional polynomial SH-Lie algebras over k[x^1,...,x^d] with complete generalized Wronskians of order k as N-ary brackets (N=binom(d+k,d)) are explicitly described, with a factorization of the associated generalized Vandermonde determinants.