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SUMMARY:Wronskians over multidimension: From sl(2) to (in)finite-dimension
 al polynomial homotopy Lie algebras
DTSTART:20260513T090000Z
DTEND:20260513T103000Z
DTSTAMP:20260523T230800Z
UID:indico-event-16555@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr\;jasserand@ihes.fr
DESCRIPTION:Speakers: Arthemy V. Kiselev (Bernoulli Institute for Mathemat
 ics\, Computer Science and Artificial Intelligence\, University of Groning
 en\, The Netherlands)\n\nBy commuting three vector fields on the line ℝ 
 ∋ x with monomial coefficients 1\, −2x\, and −x2\, we realise the Li
 e algebra sl(2) in its Chevalley basis\; the bracket acts on the coefficie
 nts as the Wronskian determinant. Let us extend this model to a class of p
 olynomial homotopy Lie algebras in which the N-ary brackets are given by t
 he Wronskian determinants over multidimension\; the generalised Vandermond
 e determinants then express the structure constants.\nThe alternated compo
 sition of N = 2p differential operators wj(x) ∂px of strict order p on t
 he 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 c
 oefficients w1\, …\, wN. 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\, an
 d 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.\nDeform the binary Lie brack
 et 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 al
 gebra\, but for every pair of arities ℓ\, m ⩾ 2 the respective (ℓ\, 
 m)-term in the identity vanishes separately. Over base dimension d = 1\, w
 e spot an infinite sequence of finite-dimensional polynomial homotopy Lie 
 algebras starting at sl(2) and with the Wronskians as the brackets\; all t
 he structure constants\, unless zero due to repetitions\, equal ±1 in a s
 uitable basis.\nLet the base dimension d ⩾ 1 be arbitrary: ℝd ∋ (x1\
 ,…\,xd). We proved in arXiv:math.RA/04110185 that the complete generalis
 ed 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 wit
 h dimension d and order k but the steps\, as k ↦ k + 1\, grow as well: o
 ver d > 1 the gaps get larger and larger. In a recent work arXiv:2511.0384
 8 [math.RA] we prove that by allowing the multivariate Wronskians be incom
 plete in their top differential order k > 1\, we do preserve all the SH-Li
 e Jacobi identities.\nFor complete Wronskians of orders k ⩾ 1 over (mult
 i)dimension d ⩾ 1 as the brackets\, in a work in progress (joint with M.
  G. Ķēniņš) we exhaustively describe all the finite-dimensional polyno
 mial N-ary SH-Lie algebra generalisations of sl(2)\; we express their stru
 cture constants in terms of the multivariate Vandermonde determinants. Rel
 axing the finite-dimensionality assumption and taking the (Laurent-)monomi
 als in d variables for the generators\, we obtain multivariate analogues o
 f the Witt algebra from CFT.\n========\nPour être informé des prochains 
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LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/16555/
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