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Abstract

This paper presents the concept of representations for matched pairs of Poisson algebras and constructs corresponding cohomology theories. The application of these cohomology groups within the framework of matched pairs is also examined. We begin by employing cohomology to study infinitesimal deformations and abelian extensions of such matched pairs. Subsequently, given an abelian extension along with a representation, we analyze the inducibility conditions for pairs of automorphisms, establishing both necessary and sufficient criteria. Finally, the Wells exact sequences for matched pairs of Poisson algebras are constructed, which provide a deeper characterization of the inducibility problem.