Triple Cohomology of Lie-Rinehart Algebras and the Canonical Class of Associative Algebras

We introduce a bicomplex which computes the triple cohomology of Lie–Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology \cite{Ri} provided the Lie–Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra.