Multi derivation Maurer-Cartan algebras and sh-Lie-Rinehart algebras

Abstract

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we first develop a characterization of sh Lie-Rinehart algebras in terms of differential graded cocommutative coalgebras and Lie algebra twisting cochains that extends the nowadays standard characterization of an ordinary sh Lie algebra (equivalently: Linfty algebra) in terms of its associated generalized Cartan-Chevalley-Eilenberg coalgebra. Our approach avoids any higher brackets but reproduces these brackets in a conceptual manner. The new technical tool we develop is a notion of filtered multi derivation chain algebra, somewhat more general than the standard notion of a multicomplex endowed with a compatible algebra structure. The crucial observation, just as for ordinary Lie-Rinehart algebras, is this: For a general sh Lie-Rinehart algebra,the generalized Cartan-Chevalley-Eilenberg operator on the corresponding graded algebra involves two operators, one coming from the sh Lie algebra structure and the other from the generalized action on the corresponding algebra; the sum of the operators is defined on the algebra while the operators are individually defined only on a larger ambient algebra. We illustrate the structure with quasi Lie-Rinehart algebras.