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

Arthemy V. Kiselev

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.

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 -deformations -- for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of by quadratic-coefficient vector fields is the bottom structure in a sequence of finite-dimensional polynomial algebras with the Wronskians as -ary brackets; the structure constants are calculated explicitly. Key words: Wronskian determinant, -ary bracket, -\/algebra, strong homotopy Lie algebra, , Witt algebra, Vandermonde determinant.

Paper Structure

This paper contains 5 sections, 8 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Wronskians as $N$-ary brackets
  3. Finite-dimensional algebras $\Bbbk_N[x]$ with Wronskian brackets
  4. Strong homotopy deformation of the Witt algebra by Wronskians
  5. The conformal weight of the Wronskian determinant

Key Result

Theorem 1

The subspace ${\mathrm D}_p(M^1)$ of differential operators $w_j=w_j(x)\cdot \partial_x^p$ of strict order $p\in\mathbb{N}$ is closed under the alternated composition $[{\cdot},\ldots,{\cdot}]_{N=2p}$ of twice as many arguments $w_1,\ldots,w_N\in{\mathrm D}_p(M^1)$. Moreover, the structure constants where $W^{0,1,\ldots,N-1} = \mathbf{1}\wedge \partial_x\wedge\ldots\wedge \partial_x^{N-1}$ is the

Theorems & Definitions (19)

  • Example 1
  • Definition 1
  • Theorem 1
  • Remark 1
  • Proposition 2: DzhFAPHanlonWachs
  • Corollary 3
  • Proposition 4: DzhFAP
  • proof
  • Proposition 5: see Dzhuma2002
  • proof
  • ...and 9 more