TY - JOUR

T1 - Analogs of Bol operators for (a + 1 | b) ⊂ oect (a | b)

AU - Bouarroudj, Sofiane

AU - Leites, Dimitry

AU - Shchepochkina, Irina

N1 - Publisher Copyright:
© 2022 World Scientific Publishing Company.

PY - 2022/11/1

Y1 - 2022/11/1

N2 - Bol operators (Bols for short) are differential operators invariant under the projective action of (2) ∼ el(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.

AB - Bol operators (Bols for short) are differential operators invariant under the projective action of (2) ∼ el(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.

KW - Bol operator

KW - Lie superalgebra

KW - Veblen's problem

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U2 - 10.1142/S021819672250059X

DO - 10.1142/S021819672250059X

M3 - Article

AN - SCOPUS:85137550353

VL - 32

SP - 1345

EP - 1368

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

IS - 7

ER -