Cohomology in algebra, geometry, physicsand statistics

G(3) supergeometry and a supersymmetric extension of the Hilbert-Cartan equation

I will report on the realization of the simple Lie superalgebra G(3) as supersymmetry of various geometric
structures – most importantly super-versions of the Hilbert–Cartan equation and Cartan’s involutive PDE system
that exhibit G(2) symmetry – and compute, via Spencer cohomology groups, the Tanaka-Weisfeiler prolongation
of the negatively graded Lie superalgebras associated with two particular choices of parabolics. I will then discuss
non-holonomic superdistributions with growth vector (2|4, 1|2, 2|0) obtained as super-deformations of rank 2 distributions
in a 5-dimensional space, and show that the second Spencer cohomology group gives a binary quadric,
thereby providing a “square-root” of Cartan’s classical binary quartic invariant for (2, 3, 5)-distributions.
If time allows, I will outline an extension of Tanaka’s geometric prolongation scheme to the case of supermanifolds.
This is a joint work with B. Kruglikov and D. The.

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