Analogs of Bol operators for pgl(a + 1 | b) ⊂ oect(a | b)

Sofiane Bouarroudj, Dimitry Leites, Irina Shchepochkina

Research output: Contribution to journalArticlepeer-review

Abstract

Bol operators (Bols for short) are differential operators invariant under the projective action of (2) ≃ (2) between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: (a + 1|b)-invariant differential operators between spaces of tensor fields on (a|b)-dimensional supermanifolds with irreducible, as (a|b)-modules, fibers of arbitrary, even infinite, dimension for certain "key"values of a and b - the ones for which the solution is describable. We discovered many new operators for (a|b) = (2|0), (0|3) and for the case of 1|1-dimensional general superstring, which looks like a most natural superization of Bol's result, additional to the cases of super analogs of Bols between spaces of weighted densities on the 1|n-dimensional superstrings with a contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension >1, there are (a + b - 1)-parameter families of Bols, whereas there are no non-scalar nonzero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here.

Original languageEnglish (US)
Pages (from-to)1345-1368
JournalInternational Journal of Algebra and Computation
Volume32
Issue number7
DOIs
StatePublished - Nov 1 2022

Keywords

  • Bol operator
  • Lie superalgebra
  • Veblen's problem

ASJC Scopus subject areas

  • Mathematics(all)

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