On bilinear invariant differential operators acting on tensor fields on the symplectic manifold

Abstract

Let $M$ be an $n$-dimensional manifold, $V$ the space of a representation $\rho :GL(n)⟶GL(V)$. Locally, let $T(V)$ be the space of sections of the tensor bundle with fiber $V$ over a sufficiently small open set $U\subset M$, in other words, $T(V)$ is the space of tensor fields of type $V$ on $M$ on which the group $\Diff (M)$ of diffeomorphisms of $M$ naturally acts. Elsewhere, the author classified the $\Diff (M)$-invariant differential operators $D:T(V_1)\otimes T(V_2)⟶T(V_3)$ for irreducible fibers with lowest weight. Here the result is generalized to bilinear operators invariant with respect to the group $\Diff_ω(M)$ of symplectomorphisms of the symplectic manifold $(M,\omega )$. We classify all first order invariant operators; the list of other operators is conjectural. Among the new operators we mention a 2nd order one which determins an ``algebra'' structure on the space of metrics (symmetric forms) on $M$.

References in corpus (1)

• On bilinear invariant differential operators acting on tensor fields on the symplectic manifold

Cited by in corpus (3)

• Invariant operators on supermanifolds and Standard Models
• Invariant bilinear differential operators
• On bilinear invariant differential operators acting on tensor fields on the symplectic manifold