Analogs of Bol operators for $\mathfrak{pgl}(a+1\vert b)\subset \mathfrak{vect}(a\vert b)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866917774998110208 |
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| author | Bouarroudj, Sofiane Leites, Dimitry Shchepochkina, Irina |
| author_facet | Bouarroudj, Sofiane Leites, Dimitry Shchepochkina, Irina |
| contents | Bol operators (Bols for short) are differential operators invariant under the projective action of $\mathfrak{pgl}(2)\simeq\mathfrak{sl}(2)$ between spaces of weighted densities on the 1-dimensional manifold.
Here, we described analogs of Bols: $\mathfrak{pgl}(a+1\vert b)$-invariant differential operators between spaces of tensor fields on $(a\vert b)$-dimensional supermanifolds with irreducible, as $\mathfrak{gl}(a\vert b)$-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 $1\vert 1$-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 $1\vert n$-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 non-zero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_01080 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Analogs of Bol operators for $\mathfrak{pgl}(a+1\vert b)\subset \mathfrak{vect}(a\vert b)$ Bouarroudj, Sofiane Leites, Dimitry Shchepochkina, Irina Representation Theory Differential Geometry {Primary 17B10 Secondary 53B99, 32Wxx Bol operators (Bols for short) are differential operators invariant under the projective action of $\mathfrak{pgl}(2)\simeq\mathfrak{sl}(2)$ between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: $\mathfrak{pgl}(a+1\vert b)$-invariant differential operators between spaces of tensor fields on $(a\vert b)$-dimensional supermanifolds with irreducible, as $\mathfrak{gl}(a\vert b)$-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 $1\vert 1$-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 $1\vert n$-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 non-zero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here. |
| title | Analogs of Bol operators for $\mathfrak{pgl}(a+1\vert b)\subset \mathfrak{vect}(a\vert b)$ |
| topic | Representation Theory Differential Geometry {Primary 17B10 Secondary 53B99, 32Wxx |
| url | https://arxiv.org/abs/2112.01080 |