2019/03/11 by Helge Øystein Maakestad, Maakestad, Helge Øystein · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1903.04285
The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a D-Lie algebra. A D-Lie algebra L is a Lie-Rinehart algebra over A/k equipped with an A⊗k A-module structure that is compatible with the Lie-structure. It may be viewed as a simultaneous generalization of a Lie-Rinehart algebra and an Atiyah algebra with additional structure. Given a D-Lie algebra L and an arbitrary connection (ρ, E) we construct the universal ring U⊗(L,ρ) of the connection (ρ, E). The associative unital ring U⊗(L,ρ) is in the case when A is Noetherian and L and E finitely generated A-modules, an almost commutative Noetherian sub ring of Diff(E) - the ring of differential operators on E. It is constructed using non-abelian extensions of D-Lie algebras. The non-flat connection (ρ, E) is a finitely generated U⊗(L,ρ)-module, hence we may speak of the characteristic variety Char(ρ,E) of (ρ, E) in the sense of D-modules. We may define the notion of holonomicity for non-flat connections using the universal ring U⊗(L,ρ). This was previously done for flat connections. We also define cohomology and homology of arbitrary non-flat connections. The cohomology and homology of a non-flat connection (ρ,E) is defined using Ext and Tor-groups of a non-Noetherian ring U. In the case when the A-module E is finitely generated we may always calculate cohomology and homology using a Noetherian quotient of U. This was previously done for flat connections.