TY - JOUR

T1 - The space of m-ary differential operators as a module over the Lie algebra of vector fields

AU - Bouarroudj, Sofiane

N1 - Funding Information:
Many thanks to V. Ovsienko for his encouragement and many thanks to D. Leites, F. Wagemann and the reviewer for pointing out pertinent remarks. Author’s work was supported by Grant 03130211/05, UAE University.

PY - 2007/5

Y1 - 2007/5

N2 - The space Dunder(λ, -) ; μ, where under(λ, -) = (λ1, ..., λm), 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 the space Dunder(λ, -) ; μ and the corresponding space of symbols as s l (2)-modules. This yields to the notion of the s l (2)-equivariant symbol calculus for m-ary differential operators. We show, however, that these two modules cannot be isomorphic as s l (2)-modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules Dunder(λ, -) ; μ2 (i.e., the space of second-order operators) are isomorphic to each other, except for a few modules called singular.

AB - The space Dunder(λ, -) ; μ, where under(λ, -) = (λ1, ..., λm), 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 the space Dunder(λ, -) ; μ and the corresponding space of symbols as s l (2)-modules. This yields to the notion of the s l (2)-equivariant symbol calculus for m-ary differential operators. We show, however, that these two modules cannot be isomorphic as s l (2)-modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules Dunder(λ, -) ; μ2 (i.e., the space of second-order operators) are isomorphic to each other, except for a few modules called singular.

KW - Equivariant quantization

KW - Invariant operators

KW - Module of differential operators

UR - http://www.scopus.com/inward/record.url?scp=33846665882&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846665882&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2006.12.002

DO - 10.1016/j.geomphys.2006.12.002

M3 - Article

AN - SCOPUS:33846665882

VL - 57

SP - 1441

EP - 1456

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 6

ER -