2019/05/23 by Maakestad, Helge Øystein
#16S30 #17B35 #17B45 #17B56 #20G05 #20G10 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1905.09630
In a previous paper we introduced the notion of a D-Lie algebra L. A D-Lie algebra L is an A/k-Lie-Rinehart algebra with a right A-module structure and a canonical central element D satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of L-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient L of a D-Lie algebra L and use this to classify D-Lie algebras where L is projective as A-module. We define for any 2-cocycle f∈ Z2(Derk(A),A) a functor Ff(-) from the category of A/k-Lie-Rinehart algebras to the category of D-Lie algebras and classify D-Lie algebras with projective canoncial quotient using the functor Ff(-). We prove a similar classification for non-abelian extensions of D-Lie algebras. We classify L-connections in the case when the canonical quotient L of L is projective as A-module. Any L-connection is determined by a 2-cocycle f∈ Z2(Derk(A),A) and an L-connection (E,∇). We introduce the correspondence and Chow-operator of an L-connection. The aim of this construction is to relate connections on D-Lie algebras to algebraic cycles an the category of correspondences. The Chow-operator cannot be defined for an ordinary connection on an A/k-Lie-Rinehart algebra. It depends in a non-trivial way on the right A-module structure on L and the canonical quotient A/k-Lie-Rinehart algebra L has no such structure.