Cubic Dirac operator for \(U_q({\mathfrak {sl}}_2)\)

We construct the q-deformed Clifford algebra of \(\mathfrak {sl}_2\) and study its properties. This allows us to define the q-deformed noncommutative Weil algebra \(\mathcal {W}_q(\mathfrak {sl}_2)\) for \(U_q(\mathfrak {sl}_2)\) and the corresponding cubic Dirac operator \(D_q\). In the classical case this was done by Alekseev and Meinrenken in 2000. We show that the cubic Dirac operator \(D_q\) is invariant with respect to the \(U_q({\mathfrak {sl}}_2)\)-action and \(*\)-structures on \(\mathcal {W}_q(\mathfrak {sl}_2)\), moreover, the square of \(D_q\) is central in \(\mathcal {W}_q(\mathfrak {sl}_2)\). We compute the spectrum of the cubic element on finite-dimensional and Verma modules of \(U_q(\mathfrak {sl}_2)\) and the corresponding Dirac cohomology.