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

Abstract

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 it was done by Alekseev and Meinrenken. 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.