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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 studythe properties of $D$, including an analogue of Vogan's conjecture. We computethe cohomology of $D$ acting on various $U_q(\mathfrak{sl}_2)$-modules.

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