Mathematics Colloquia and Seminars

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Description

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 Adam Biggs.