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Journal of Lie Theory 33 (2023), No. 4, 965--1004
Copyright Heldermann Verlag 2023

On Odd Parameters in Geometry

Dimitry Leites
Department of Mathematics, University of Stockholm, Sweden

(1) In 1976, looking at the list of simple finite-dimensional complex Lie superalgebras, J. Bernstein and I, and independently M. Duflo, observed that some divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that no other simple Lie superalgebras have such deformations. Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and relations with simple Lie (super)algebras over fields of positive characteristic.
(2) Any ringed space of the form (a manifold M, the sheaf of sections of the exterior algebra of a vector bundle over M) is called split supermanifold. K. Gawedzki (1977) and M. Batchelor (1979) proved that every smooth supermanifold is split. In 1982, P. Green and V. P. Palamodov showed that a complex-analytic supermanifold can be non-split. So far, researchers considered only even obstructions to splitness. This lead them to the conclusion that any supermanifold of superdimension m|1 is split. I'll show there are non-split supermanifolds of superdimension m|1; e.g., certain superstrings, the obstructions to their splitness depend on odd parameters.

Keywords: Simple Lie superalgebra, deformation, non-split supermanifold.

MSC: 58A50, 17B60.

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