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New invariant differential operators on supermanifolds and pseudo- (co)homology. (English) Zbl 0773.58004

General topology and applications, Proc. 5th Northeast Conf., New York/NY (USA) 1989, Lect. Notes Pure Appl. Math. 134, 217-238 (1991).
[For the entire collection see Zbl 0744.00027.]
This is an extremely terse summary of recent work done by the authors on discovering new differential operators on supermanifolds invariant under superdiffeomorphisms. Most of those results concentrate around Veblen’s general problem of classifying the \(n\)-ary \(\text{Diff}(B)\)-invariant smooth operators between the so-called spaces of geometric objects on a smooth manifold \(B\) [A. A. Kirillov, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 16, 3-29 (1980; Zbl 0453.53008)] – or, rather, an appropriate “super” analogue of that problem.
An approach to the theory of integration on supermanifolds due to Bernstein and the first author plays an important rôle. Certain Lie superalgebras of vector fields on supermanifolds and their representations are discussed. New possible cohomology theory (theories) for supermanifolds are sketched. Open problems are scattered in abundance throughout the text. The presentation is compressed to such a degree that often even new definitions are not given in rigorous form because of lack of space.
The present summary would be interesting to all those concerned with future directions in geometry and analysis incorporating odd variables, if only as an expression of the viewpoint shared by one of the mathematicians who participated in shaping the concept of supermanifold (the first author).

MSC:

58C50 Analysis on supermanifolds or graded manifolds
17B70 Graded Lie (super)algebras
58A50 Supermanifolds and graded manifolds
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