Universal enveloping algebras of weighted differential Poisson algebras

The $\lambda$-differential operators and modified $\lambda$-differential operators are generalizations of classical differential operators. This paper introduces the notions of $\lambda$-differential Poisson ($\lambda$-DP for short) algebras and modified $\lambda$-differential Poisson ($\lambda$-mDP for short) algebras as generalizations of differential Poisson algebras. The $\lambda$-DP algebra is proved to be closed under tensor product, and a $\lambda$-DP algebra structure is provided on the cohomology algebra of the $\lambda$-DP algebra. These conclusions are also applied to $\lambda$-mDP algebras and their modules. Finally, the universal enveloping algebras of $\lambda$-DP algebras are generalized by constructing a $\mathcal{P}$-triple. Three isomorphisms among opposite algebras, tensor algebras and the universal enveloping algebras of $\lambda$-DP algebras are obtained.