Lie algebras of differential operators
by
Norbert Poncin
Centre Universitaire de Luxembourg
Coauthors: Janusz Grabowski

Lie algebras of differential operators

Norbert Poncin

The classical result of Pursell and Shanks [PS], which states that the Lie algebra of smooth vector fields of a smooth manifold characterizes the smooth structure of the variety, is the starting point of a multitude of works.

There are similar results in particular geometric situations-for instance for hamiltonian, contact or group invariant vector fields-for which specific tools have each time been constructed, [O, A, AG, HM], in the case of Lie algebras of vector fields that are modules over the corresponding rings of functions, [Am, G1, S], as well as for the Lie algebra of (not leaf but) foliation preserving vector fields, [G2].

First objective of the lecture is to prove that the Lie algebra D(M) of all linear differential operators D:C^\infty(M) --> C^\infty(M) of a smooth manifold M determines the smooth structure of M. Beyond this conclusion, it will be presented a description of all automorphisms of the Lie algebra D(M) (and even of the Lie subalgebra D¹(M) of all linear first-order differential operators of M) and of the Poisson algebra S(M)=Pol(T^*M) of polynomial functions on the cotangent bundle T^*M (the symbols of the operators in D(M)), the automorphisms of the two last algebras being of course canonically related with those of D(M). In each situation one obtains an explicit formula. For instance-in the case of D(M)-in terms of the automorphism of D(M) implemented by a diffeomorphism of M, the conjugation-automorphism of D(M), and the automorphism of D(M) generated by the derivation of D(M) associated to a closed 1-form of M.

The presented results have been obtained in joint work with Janusz Grabowski.

References

[A] Abe K, Pursell-Shanks type theorem for orbit spaces and G-manifolds, Publ. Res. Inst. Math. Sci. , 18 (1982), pp. 265-282
[Am] Amemiya I, Lie algebra of vector fields and complex structure, J. Math. Soc. Japan, 27 (1975), pp. 545-549
[AG] Atkin C J, Grabowski J, Homomorphisms of the Lie algebras associated with a symplectic manifold, Compos. Math., 76 (1990), pp. 315-348
[G1] Grabowski J, Isomorphisms and ideals of the Lie algebras of vector fields, Invent. math., 50 (1978), pp. 13-33
[G2] Grabowski J, Lie algebras of vector fields and generalized foliations, Publ. Matem., 37 (1993), pp 359-367
[HM] Hauser H, Müller G, Affine varieties and Lie algebras of vector fields, Manusc. Math., 80 (1993), pp. 309-337
[O] Omori H, Infinite dimensional Lie transformation groups, Lect. Notes in Math., 427 (1976), Springer Verlag
[PS] Shanks M E, Pursell L E, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc., 5 (1954), pp. 468-472
[S] Skryabin S M, The regular Lie rings of derivations of commutative rings, preprint WINITI 4403-W87 (1987)

Date received: April 14, 2003