Wronskians over multidimension: From sl(2) to (in)finite-dimensional polynomial homotopy Lie algebras

par Arthemy V. Kiselev (Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, The Netherlands)

mercredi 13 mai 2026, 11:00 → 12:30 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

By commuting three vector fields on the line ℝ ∋ x with monomial coefficients 1, −2x, and −x², we realise the Lie algebra sl(2) in its Chevalley basis; the bracket acts on the coefficients as the Wronskian determinant. Let us extend this model to a class of polynomial homotopy Lie algebras in which the N-ary brackets are given by the Wronskian determinants over multidimension; the generalised Vandermonde determinants then express the structure constants.

The alternated composition of N = 2p differential operators w_j(x) ∂^p_x of strict order p on the line ℝ ∋ x is again a differential operator of strict order p; its coefficient is the constant c(p) times the Wronskian determinant of the coefficients w₁, …, w_N. At p = 1, the sl(2) case fixes c(1) = 1; easy is c(2) = 2, then c(3) = 90. In a recent joint work with K. C. Shah, we reach the exact values c(p = 4) = 586 656, c(p = 5) ≈ 1.9 · 10¹², and c(p = 6) ≈ 7.9 · 10²¹. The positive integer sequence c(p) seems to be new; to know c(p ⩾ 7) is an open problem.

Deform the binary Lie bracket to a formal sum of Wronskians with purely even (N = 2p) or arbitrary (N ∈ ℕ_⩾2) arities, see arXiv:2510.02145 [math.RA]. Not only does the full bracket Δ satisfy the Jacobi identity Δ[Δ] = 0 for homotopy Lie algebra, but for every pair of arities ℓ, m ⩾ 2 the respective (ℓ, m)-term in the identity vanishes separately. Over base dimension d = 1, we spot an infinite sequence of finite-dimensional polynomial homotopy Lie algebras starting at sl(2) and with the Wronskians as the brackets; all the structure constants, unless zero due to repetitions, equal ±1 in a suitable basis.

Let the base dimension d ⩾ 1 be arbitrary: ℝ^d ∋ (x₁,…,x_d). We proved in arXiv:math.RA/04110185 that the complete generalised Wronskians – involving all the derivatives up to a given differential order k ⩾ 1 – still satisfy the table of Jacobi identities for strong homotopy Lie algebras. The arity N = (d+k  d) of such brackets grows with dimension d and order k but the steps, as k ↦ k + 1, grow as well: over d > 1 the gaps get larger and larger. In a recent work arXiv:2511.03848 [math.RA] we prove that by allowing the multivariate Wronskians be incomplete in their top differential order k > 1, we do preserve all the SH-Lie Jacobi identities.

For complete Wronskians of orders k ⩾ 1 over (multi)dimension d ⩾ 1 as the brackets, in a work in progress (joint with M. G. Ķēniņš) we exhaustively describe all the finite-dimensional polynomial N-ary SH-Lie algebra generalisations of sl(2); we express their structure constants in terms of the multivariate Vandermonde determinants. Relaxing the finite-dimensionality assumption and taking the (Laurent-)monomials in d variables for the generators, we obtain multivariate analogues of the Witt algebra from CFT.

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