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Differential operators on the extended manifolds and operator pencils on the algebra of densities.QMAP Seminar
|Speaker: ||Hovhannes KHUDAVERDIAN, Manchester|
|Location: ||2112 MSB|
|Start time: ||Thu, May 22 2014, 4:10PM|
We consider operator pencils which act on densities of arbitrary
weights and which pass through operators acting on densities of a given
weight. The algebra of densities of arbitrary weight can be identified
with a special class of functions on the extended manifold
(weight of density plays the role of the additional coordinate),
and canoncial scalar product is well-defined on this space of functions.
Operator pencils can be identified with operators acting on the space of
functions on extended manifold. These data allow to perform geometrical
constructions in the spirit of Kaluza-Klein formalism. In particular we
show that self-adjointness uniquely defines $diff(M)$-equivariant pencil lifting for second order operators. To define pencil liftings for higher order operators we are forced to restrict equivariance of lifting to the
algebra of divergenceless vector fields.
Then we study liftings which are equivariant
with respect to finite-dimensional Lie algebra of projective
transformations. We consider all such the liftings on the basis of liftings which can be factored through full projective equivariant symbol map
constructed in works of Duval, Lecompte and Ovsienko.
It is interesting to note that in the case if we consider
the algebra of all smooth functions on the extended manifold
we will come to the classical Thomas construction of
lifitng of projective connection.
The talk is based on my works with Ted Voronov and our student