Jet functors in noncommutative geometry
Henrik Winther
Abstract: We construct an infinite family of endofunctors $J_d^n$ on the category of left $A$-modules, where $A$ is a unital associative algebra over a commutative ring $k$, equipped with an exterior algebra $\Omega^\bullet_d$. We prove that these functors generalize the corresponding classical notion of jet functors. The functor $J_d^n$ comes equipped with a natural transformation from the identity functor to itself, which plays the rĂ´le of the classical prolongation map. This allows us to define the notion of linear differential operator with respect to $\Omega^{\bullet}_d$. These retain most classical properties of differential operators, and operators such as partial derivatives and connections belong to this class. Moreover, we construct a functor of quantum symmetric forms $S^n_d$ associated to $\Omega^\bullet_d$, and proceed to introduce the corresponding noncommutative analogue of the Spencer $\delta$-complex. We give necessary and sufficient conditions under which the jet functor $J_d^n$ satisfies the jet exact sequence, $0\rightarrow S^n_d \rightarrow J_d^n \rightarrow J_d^{n-1} \rightarrow 0$. This involves imposing mild homological conditions on the exterior algebra, in particular on the Spencer cohomology $H^{\bullet,2}$.
This is a joint work with K. Flood and M. Mantegazza.
mathematical physicsanalysis of PDEsdifferential geometry
Audience: researchers in the topic
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Geometry of differential equations seminar
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