Dirac operator and its cohomology for the quantum group $U_q(\mathfrak{sl}_2)$

We introduce a Dirac operator $D$ for the quantum group $U_q(\mathfrak{sl}_2)$, as an element of the tensor product of $U_q(\mathfrak{sl}_2)$ with the Clifford algebra on two generators. We study the properties of $D$, including an analogue of Vogan's conjecture. We compute the cohomology of $D$ acting on various $U_q(\mathfrak{sl}_2)$-modules.