On construction of differential Z-graded varieties (Joint work with A. Hancharuk)

Abstract: Given a commutative unital algebra O, a proper ideal I⊂O, and a positively graded differential variety over O/I, we construct a Z-graded extension whose negative part is an arborescent Koszul–Tate resolution of O/I. This extension is obtained by means of an explicit algorithm that exploits the homotopy retract data of the arborescent Koszul–Tate resolution, thereby significantly reducing the number of homological computations required in the construction.

When the positively graded differential variety is defined over O and preserves the ideal I, the extension admits a canonical and explicit description in terms of decorated trees together with the associated computed data.

As a by-product, to every Lie–Rinehart algebra over the coordinate ring of an affine variety W, we associate an explicit differential Z-graded variety. Its negative component is the arborescent Koszul–Tate resolution of the coordinate ring​ of W, while its positive component is the universal dg-variety of the given Lie–Rinehart algebra.

These constructions also yield applications to singular foliation theory, extending results of C. Laurent-Gengoux, S. Lavau, and T. Strobl. Explicit examples are provided.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( paper | slides | video )

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Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.

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