The geometric reductivity of the quantum group $SL_q(2)$
We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if $G$ is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra $A$, then the algebra of invariants $A^G$ is finitely generated. We also prove that in characteristic $0$ a quantum group $G$ is geometrically reductive if and only if every rational $G$-module is semisimple, and that in positive characteristic every finite-dimensional quantum group is geometrically reductive. Both the concept of geometrically reductive quantum group and the above mentioned theorems are formulated in the language of Hopf algebras and generalize the results of Borsai and Ferrer Santos. The main theorem of the paper says that in positive characteristic the quantum group $SL_q(2)$ is geometrically reductive for any parameter $q$.