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.

20 pages

References in corpus (12)

• Quantum symmetric Kac-Moody pairs
• The non-commutative Weil algebra
• Supersymmetric quantum theory and non-commutative geometry
• The Weyl Character Formula, the half-spin representations, and equal rank subgroups
• Dirac Operators on Quantum Projective Spaces
• Quasitriangularity of quantum groups at roots of 1
• Unbraiding the braided tensor product
• Gauge theory on noncommutative Riemannian principal bundles
• Dirac operators on quantum two spheres
• Dirac operator and its cohomology for the quantum group $U_q({\mathfrak{sl}}_2)$
• On PBW-deformations of braided exterior algebras
• Quantised ${\mathfrak{sl}}_2$-differential algebras

Cited by in corpus (1)

• Group valued moment maps for even and odd simple $G$-modules