Valentin Ovsienko

Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

We consider the supercircle S1|1 equipped with the standard contact structure. The Lie superalgebra

S^1

{\fancyscript{K}(1)}

Multi-parameter deformations of the module of symbols ofdifferential operators

of contact vector fields contains the Möbius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the canonical isomorphism between this space and the corresponding space of symbols and find all cases where such an isomorphism does not exist.

Liouville–Arnold integrability of the pentagram map on closed polygons

Abstract:We study deformations of the standard embedding of the Lie algebra Vect(S1) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle T*S1 (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of Vect(S1) into the Lie algebra of functions on ˙T*S1≔S1 which are Laurent polynomials on fibers, and (b) polynomial deformations of the Vect(S1) subalgebra inside the Lie algebra of formal Laurent series on ˙T*S1.

Deformations of Modules of Differential Forms

The space of symbols of differential operators on a smooth manifold (i.e., the space of symmetric contravariant tensor fields) is naturally a module over the Lie algebra of vector fields. We study, in the case of

Conformally equivariant quantum Hamiltonians

\bf R^n

Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor-densities on S1

with

Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2 + 1

n\geq2