Classification of $U_q(\mathfrak{sl}_2)$-module algebra structures on the quantum plane

A complete list of $U_q(\mathfrak{sl}_2)$-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed. The classical limits of the $U_q(\mathfrak{sl}_2)$-module algebra structures are discussed.