184 research outputs found
Conformal symbols and the action of contact vector fields over the superline
Let K be the Lie superalgebra of contact vector fields on the supersymmetric
line. We compute the action of K on the modules of differential and
pseudodifferential operators between spaces of tensor densities, in terms of
their conformal symbols. As applications we deduce the geometric subsymbols,
1-cohomology, and various uniserial subquotients of these modules. We also
outline the computation of the K-equivalences and symmetries of their
subquotients.Comment: 48 page
Linear differential operators on contact manifolds
We consider differential operators between sections of arbitrary powers of
the determinant line bundle over a contact manifold. We extend the standard
notions of the Heisenberg calculus: noncommutative symbolic calculus, the
principal symbol, and the contact order to such differential operators. Our
first main result is an intrinsically defined "subsymbol" of a differential
operator, which is a differential invariant of degree one lower than that of
the principal symbol. In particular, this subsymbol associates a contact vector
field to an arbitrary second order linear differential operator. Our second
main result is the construction of a filtration that strengthens the well-known
contact order filtration of the Heisenberg calculus
Centers and characters of Jacobi group-invariant differential operator algebras
We study the algebras of differential operators invariant with respect to the
scalar slash actions of real Jacobi groups of arbitrary rank. These algebras
are non-commutative and are generated by their elements of orders 2 and 3. We
prove that their centers are polynomial in one variable and are generated by
the Casimir operator. For slash actions with invertible indices we also compute
the characters of the IDO algebras: in rank exceeding 1 there are two, and in
rank 1 there are in general five. In rank 1 we compute in addition all
irreducible admissible representations of the IDO algebras.Comment: 16 page
Annihilators of tensor density modules
AbstractWe describe the two-sided ideals in the universal enveloping algebras of the Lie algebras of vector fields on the line and the circle which annihilate the tensor density modules. Both of these Lie algebras contain the projective subalgebra, a copy of sl2. The restrictions of the tensor density modules to this subalgebra are duals of Verma modules (of sl2) for Vec(R) and principal series modules (of sl2) for Vec(S1). Thus our results are related to the well-known theorem of Duflo describing the annihilating ideals of Verma modules of reductive Lie algebras. We find that, in general, the annihilator of a tensor density module of Vec(R) or Vec(S1) is generated by the Duflo generator of its annihilator over sl2 (the Casimir operator minus a scalar) together with one other generator, a cubic element of U(Vec(R)) not contained in U(sl2)
Relative extremal projectors
AbstractThis paper proves the existence of relative extremal projectors. An infinite factorization is given as well as a summation formula
- …