Volume 8, Issue 1, February 2001, Pages 31 - 37
On Bilinear Invariant Differential Operators Acting on Tensor Fields on the Symplectic Manifold
Received 6 May 2000, Revised 30 August 2000, Accepted 3 October 2000, Available Online 1 February 2001.
- https://doi.org/10.2991/jnmp.2001.8.1.3How to use a DOI?
- Let M be an n-dimensional manifold, V the space of a representation : 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 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(V1) T(V2) - T(V3) for irreducible fibers with lowest weight. Here the result is generalized to bilinear operators invariant with respect to the group Diff(M) of symplectomorphisms of the symplectic manifold (M, ). 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.
- Open Access
- This is an open access article distributed under the CC BY-NC license.
Cite this article
TY - JOUR AU - Pavel Grozman PY - 2001 DA - 2001/02 TI - On Bilinear Invariant Differential Operators Acting on Tensor Fields on the Symplectic Manifold JO - Journal of Nonlinear Mathematical Physics SP - 31 EP - 37 VL - 8 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2001.8.1.3 DO - https://doi.org/10.2991/jnmp.2001.8.1.3 ID - Grozman2001 ER -