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: L∞ 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 concept 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 one from the generalized action on the corresponding algebra; the sum of the two 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.