19 research outputs found
Equivariant symbol calculus for differential operators acting on forms
We prove the existence and uniqueness of a projectively equivariant symbol
map (in the sense of Lecomte and Ovsienko) for the spaces of differential
operators transforming p-forms into functions. These results hold over a smooth
manifold endowed with a flat projective structure.
As an application, we classify the Vect(M)-equivariant maps from to
over any manifold M, recovering and improving earlier results by N.
Poncin. This provides the complete answer to a question raised by P. Lecomte
about the extension of a certain intrinsic homotopy operator.Comment: 14 page
Projectively equivariant quantizations over the superspace
We investigate the concept of projectively equivariant quantization in the
framework of super projective geometry. When the projective superalgebra
pgl(p+1|q) is simple, our result is similar to the classical one in the purely
even case: we prove the existence and uniqueness of the quantization except in
some critical situations. When the projective superalgebra is not simple (i.e.
in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a
one-parameter family of equivariant quantizations. We also provide explicit
formulas in terms of a generalized divergence operator acting on supersymmetric
tensor fields.Comment: 19 page
Natural and projectively equivariant quantizations by means of Cartan Connections
The existence of a natural and projectively equivariant quantization in the
sense of Lecomte [20] was proved recently by M. Bordemann [4], using the
framework of Thomas-Whitehead connections. We give a new proof of existence
using the notion of Cartan projective connections and we obtain an explicit
formula in terms of these connections. Our method yields the existence of a
projectively equivariant quantization if and only if an \sl(m+1,\R)-equivariant
quantization exists in the flat situation in the sense of [18], thus solving
one of the problems left open by M. Bordemann.Comment: 13 page
The ternary invariant differential operators acting on the spaces of weighted densities
Over n-dimensional manifolds, I classify ternary differential operators
acting on the spaces of weighted densities and invariant with respect to the
Lie algebra of vector fields. For n=1, some of these operators can be expressed
in terms of the de Rham exterior differential, the Poisson bracket, the Grozman
operator and the Feigin-Fuchs anti-symmetric operators; four of the operators
are new, up to dualizations and permutations. For n>1, I list multidimensional
conformal tranvectors, i.e.,operators acting on the spaces of weighted
densities and invariant with respect to o(p+1,q+1), where p+q=n. Except for the
scalar operator, these conformally invariant operators are not invariant with
respect to the whole Lie algebra of vector fields.Comment: 13 pages, no figures, to appear in Theor. Math. Phy