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The space of m-ary differential operators as a module over the Lie algebra of vector fields

Sofiane Bouarroudj
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Abstract

The space of m-ary differential operators acting on weighted densities is a (m+1)-parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between this space and the corresponding space of symbols as sl(2)-modules. This yields to the notion of the sl(2)-equivariant symbol calculus for m-ary differential operators. We show, however, that these two modules cannot be isomorphic as sl(2)-modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules of second-order m-ary differential operators are isomorphic to each other, except for few modules called singular.

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Sofiane Bouarroudj. 2007-01-03. The space of m-ary differential operators as a module over the Lie algebra of vector fields. https://doi.org/10.1016/j.geomphys.2006.12.002

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